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# integral e^arctanx/(1+x^2)^1/2

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integral e^arctanx/(1+x^2)^1/2

Guest Dec 28, 2015
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#1
+91451
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http://www.wolframalpha.com/input/?i=integral%28%28e%5Eatanx%29%2Fsqrt%281%2Bx%5E2%29%29

I have no idea how to do this myself.  Sorry

Melody  Dec 28, 2015
edited by Melody  Dec 28, 2015
#2
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I don't even know what its supposed to be.

A few extra brackets to remove the ambiguity would help.

Guest Dec 28, 2015
#3
+91451
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Hi Guest, it is awefully quiet around here tonight :/

I am not aware of any ambiguity.

It is just a very advanced question.     :(

Melody  Dec 28, 2015
#4
0

What are the upper and lower bounds of the integral? If the lower bound and upper bound are equal then integral equals 0.

Guest Dec 28, 2015
#5
+91451
0

True but irrelevant.

Integrals do not have to have bounds.

This is presented as an indefinite integral.

You did make me thingk about what the graph looked like though.

I mean the function that is being integrated.

Here it is:

https://www.desmos.com/calculator/nldfhv0hot

Looking at the graph, I do not understand why the integral has an imaginary element???

Melody  Dec 28, 2015
edited by Melody  Dec 28, 2015
#6
0

1/2 i (polygamma(0, 1/4-i/4)-polygamma(0, 3/4-i/4))+(-1/2+i/2) x ((1+i)+polygamma(0, 1/4-i/4)-polygamma(0, 5/4-i/4))+1/2 x^2 ((-3-4 i)-polygamma(0, 1/4-i/4)-(2+i) polygamma(0, 3/4-i/4)+polygamma(0, 5/4-i/4)+(2+i) polygamma(0, 7/4-i/4))+(-1/6-i/6) x^3 ((6+18 i)+(1+i) polygamma(0, 1/4-i/4)+(5+5 i) polygamma(0, 3/4-i/4)+(3+6 i) polygamma(0, 5/4-i/4)-(5+5 i) polygamma(0, 7/4-i/4)-(4+7 i) polygamma(0, 9/4-i/4))+1/12 x^4 ((102+24 i)+(4-i) polygamma(0, 1/4-i/4)+(24-3 i) polygamma(0, 3/4-i/4)+(50-2 i) polygamma(0, 5/4-i/4)+(16+8 i) polygamma(0, 7/4-i/4)-(54-3 i) polygamma(0, 9/4-i/4)-(40+5 i) polygamma(0, 11/4-i/4))+O(x^5)
(Taylor series)

AT x=0

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((371/812825-(433 i)/812825) Gamma(3/2-i/2) exp(-(1+i) pi floor(((3 pi)/2-arg(x+i))/(2 pi))-2 i tan^(-1)(x)) ((2010+155 i) Gamma(1/2-i/2)^2 Gamma(1/2+i/2) (e^(2 i tan^(-1)(x)))^(1/2+i/2) exp((1/4+i/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4+i/4) (x+i)+(1/16-i/16) (x+i)^2-(1/48+i/48) (x+i)^3-(1/128-i/128) (x+i)^4+(1/320+i/320) (x+i)^5+O((x+i)^6))+(2010+155 i) Gamma(-1/2-i/2) exp(((1/4-i/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4-i/4) (x+i)-(1/16+i/16) (x+i)^2-(1/48-i/48) (x+i)^3+(1/128+i/128) (x+i)^4+(1/320-i/320) (x+i)^5+O((x+i)^6))+2 i tan^(-1)(x))-(773+464 i) Gamma(-1/2-i/2) exp(((1/4-(3 i)/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4-(3 i)/4) (x+i)-(3/16+i/16) (x+i)^2-(1/48-i/16) (x+i)^3+(3/128+i/128) (x+i)^4+(1/320-(3 i)/320) (x+i)^5+O((x+i)^6))+2 i tan^(-1)(x))+(440+345 i) Gamma(-1/2-i/2) exp(((1/4-(5 i)/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4-(5 i)/4) (x+i)-(5/16+i/16) (x+i)^2-(1/48-(5 i)/48) (x+i)^3+(5/128+i/128) (x+i)^4+(1/320-i/64) (x+i)^5+O((x+i)^6))+2 i tan^(-1)(x))-(303+266 i) Gamma(-1/2-i/2) exp(((1/4-(7 i)/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4-(7 i)/4) (x+i)-(7/16+i/16) (x+i)^2-(1/48-(7 i)/48) (x+i)^3+(7/128+i/128) (x+i)^4+(1/320-(7 i)/320) (x+i)^5+O((x+i)^6))+2 i tan^(-1)(x))+(230+215 i) Gamma(-1/2-i/2) exp(((1/4-(9 i)/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4-(9 i)/4) (x+i)-(9/16+i/16) (x+i)^2-(1/48-(3 i)/16) (x+i)^3+(9/128+i/128) (x+i)^4+(1/320-(9 i)/320) (x+i)^5+O((x+i)^6))+2 i tan^(-1)(x))-(185+180 i) Gamma(-1/2-i/2) exp(((1/4-(11 i)/4) (2 i log(x+i)-2 i log(2)+pi)+(1/4-(11 i)/4) (x+i)-(11/16+i/16) (x+i)^2-(1/48-(11 i)/48) (x+i)^3+(11/128+i/128) (x+i)^4+(1/320-(11 i)/320) (x+i)^5+O((x+i)^6))+2 i tan^(-1)(x))))/Gamma(1/2-i/2)^2

AT x=-i

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(1917/13818025+(1361 i)/13818025) e^((1+i) pi floor((pi-2 arg(x-i))/(4 pi))) ((1390-8195 i) exp((1/4+i/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+i/4) (x-i)-(1/16-i/16) (x-i)^2-(1/48+i/48) (x-i)^3+(1/128-i/128) (x-i)^4+(1/320+i/320) (x-i)^5+O((x-i)^6))+(1083+3556 i) exp((1/4+(3 i)/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+(3 i)/4) (x-i)-(3/16-i/16) (x-i)^2-(1/48+i/16) (x-i)^3+(3/128-i/128) (x-i)^4+(1/320+(3 i)/320) (x-i)^5+O((x-i)^6))-(940+2105 i) exp((1/4+(5 i)/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+(5 i)/4) (x-i)-(5/16-i/16) (x-i)^2-(1/48+(5 i)/48) (x-i)^3+(5/128-i/128) (x-i)^4+(1/320+i/64) (x-i)^5+O((x-i)^6))+(761+1478 i) exp((1/4+(7 i)/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+(7 i)/4) (x-i)-(7/16-i/16) (x-i)^2-(1/48+(7 i)/48) (x-i)^3+(7/128-i/128) (x-i)^4+(1/320+(7 i)/320) (x-i)^5+O((x-i)^6))-(630+1135 i) exp((1/4+(9 i)/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+(9 i)/4) (x-i)-(9/16-i/16) (x-i)^2-(1/48+(3 i)/16) (x-i)^3+(9/128-i/128) (x-i)^4+(1/320+(9 i)/320) (x-i)^5+O((x-i)^6))+(535+920 i) exp((1/4+(11 i)/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+(11 i)/4) (x-i)-(11/16-i/16) (x-i)^2-(1/48+(11 i)/48) (x-i)^3+(11/128-i/128) (x-i)^4+(1/320+(11 i)/320) (x-i)^5+O((x-i)^6))-(464+773 i) exp((1/4+(13 i)/4) (-2 i log(x-i)+2 i log(2)+pi)+(1/4+(13 i)/4) (x-i)-(13/16-i/16) (x-i)^2-(1/48+(13 i)/48) (x-i)^3+(13/128-i/128) (x-i)^4+(1/320+(13 i)/320) (x-i)^5+O((x-i)^6)))

AT x=i

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-1/2 e^(pi/2) (2 log(1/x)+i pi+2 gamma+log(4)+2 polygamma(0, 1/2-i/2))+(e^(pi/2) ((1+2 i)+(1+i) polygamma(0, 1/2-i/2)-(1+i) polygamma(0, 3/2-i/2)))/x-(i e^(pi/2) (polygamma(0, 1/2-i/2)-(3+i) polygamma(0, 3/2-i/2)+(2+i) polygamma(0, 5/2-i/2)))/x^2-((1/9-i/9) e^(pi/2) ((-2+4 i)+(3+3 i) polygamma(0, 1/2-i/2)-(18+18 i) polygamma(0, 3/2-i/2)+(27+36 i) polygamma(0, 5/2-i/2)-(12+21 i) polygamma(0, 7/2-i/2)))/x^3+(e^(pi/2) (1+(4+16 i) polygamma(0, 1/2-i/2)-(16+112 i) polygamma(0, 3/2-i/2)+(24+312 i) polygamma(0, 5/2-i/2)+(8-376 i) polygamma(0, 7/2-i/2)-(20-160 i) polygamma(0, 9/2-i/2)))/(24 x^4)+(e^(pi/2) ((3+12 i)+(15-5 i) polygamma(0, 1/2-i/2)-(155-45 i) polygamma(0, 3/2-i/2)+(590-130 i) polygamma(0, 5/2-i/2)-(1110-170 i) polygamma(0, 7/2-i/2)+(1025-75 i) polygamma(0, 9/2-i/2)-(365+5 i) polygamma(0, 11/2-i/2)))/(30 x^5)+O((1/x)^6)
(Puiseux series)

AT x=∞

Guest Dec 28, 2015
#7
+2493
0

Wow ! :)

Solveit  Dec 28, 2015

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