\(\int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx\) [134]

Optimal result

Integrand size = 14, antiderivative size = 354 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=\frac {37 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{96 a^2}+\frac {3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x^2}{8 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x^3-\frac {11 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 \sqrt {2} a^3}+\frac {11 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 \sqrt {2} a^3}+\frac {11 \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}+\frac {11 \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}+\frac {11 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\left (1+\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right ) \sqrt [8]{1-\frac {1}{a x}}}\right )}{64 \sqrt {2} a^3} \] Output:

37/96*(1-1/a/x)^(7/8)*(1+1/a/x)^(1/8)*x/a^2+3/8*(1-1/a/x)^(7/8)*(1+1/a/x)^ 
(1/8)*x^2/a+1/3*(1-1/a/x)^(7/8)*(1+1/a/x)^(1/8)*x^3-11/128*arctan(1-2^(1/2 
)*(1+1/a/x)^(1/8)/(1-1/a/x)^(1/8))*2^(1/2)/a^3+11/128*arctan(1+2^(1/2)*(1+ 
1/a/x)^(1/8)/(1-1/a/x)^(1/8))*2^(1/2)/a^3+11/64*arctan((1+1/a/x)^(1/8)/(1- 
1/a/x)^(1/8))/a^3+11/64*arctanh((1+1/a/x)^(1/8)/(1-1/a/x)^(1/8))/a^3+11/12 
8*arctanh(2^(1/2)*(1+1/a/x)^(1/8)/(1+(1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/(1-1 
/a/x)^(1/8))*2^(1/2)/a^3
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 5.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.47 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=\frac {-4 \left (-\frac {1024 e^{\frac {1}{4} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}-\frac {1600 e^{\frac {1}{4} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}-\frac {840 e^{\frac {1}{4} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}-66 \arctan \left (e^{\frac {1}{4} \coth ^{-1}(a x)}\right )+33 \log \left (1-e^{\frac {1}{4} \coth ^{-1}(a x)}\right )-33 \log \left (1+e^{\frac {1}{4} \coth ^{-1}(a x)}\right )\right )-33 \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\coth ^{-1}(a x)-4 \log \left (e^{\frac {1}{4} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]}{1536 a^3} \] Input:

Integrate[E^(ArcCoth[a*x]/4)*x^2,x]
 

Output:

(-4*((-1024*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x]))^3 - (1600*E^(Arc 
Coth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x]))^2 - (840*E^(ArcCoth[a*x]/4))/(-1 + 
 E^(2*ArcCoth[a*x])) - 66*ArcTan[E^(ArcCoth[a*x]/4)] + 33*Log[1 - E^(ArcCo 
th[a*x]/4)] - 33*Log[1 + E^(ArcCoth[a*x]/4)]) - 33*RootSum[1 + #1^4 & , (A 
rcCoth[a*x] - 4*Log[E^(ArcCoth[a*x]/4) - #1])/#1^3 & ])/(1536*a^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.92 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.10, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {6721, 110, 27, 168, 27, 168, 27, 104, 758, 755, 756, 216, 219, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 e^{\frac {1}{4} \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \frac {\sqrt [8]{1+\frac {1}{a x}} x^4}{\sqrt [8]{1-\frac {1}{a x}}}d\frac {1}{x}\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {1}{3} \int \frac {\left (9 a+\frac {8}{x}\right ) x^3}{4 a^2 \sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\int \frac {\left (9 a+\frac {8}{x}\right ) x^3}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}}{12 a^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {-\frac {1}{2} \int -\frac {\left (37 a+\frac {36}{x}\right ) x^2}{4 a \sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {\int \frac {\left (37 a+\frac {36}{x}\right ) x^2}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {-\int -\frac {33 x}{4 \sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {\frac {33}{4} \int \frac {x}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \int \frac {1}{\frac {1}{x^8}-1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 758

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {\frac {66 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )-37 a x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a}-\frac {9}{2} a x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{12 a^2}\)

Input:

Int[E^(ArcCoth[a*x]/4)*x^2,x]
 

Output:

((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^3)/3 - ((-9*a*(1 - 1/(a*x))^(7/ 
8)*(1 + 1/(a*x))^(1/8)*x^2)/2 + (-37*a*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^( 
1/8)*x + 66*((-1/2*ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)] - ArcTa 
nh[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/2)/2 + ((ArcTan[1 - (Sqrt[2]*( 
1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/Sqrt[2] - ArcTan[1 + (Sqrt[2]*(1 
+ 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*(1 + 
 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + x^(-2)]/(2*Sqrt[2]) - Log[1 + (Sqrt 
[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + x^(-2)]/(2*Sqrt[2]))/2)/2)) 
/(8*a))/(12*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 758

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b 
, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^(n/2)), 
 x], x] + Simp[r/(2*a)   Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] 
 && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 6721

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x 
/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && 
!IntegerQ[n] && IntegerQ[m]
 

Maple [F]

\[\int \frac {x^{2}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}}}d x\]

Input:

int(1/((a*x-1)/(a*x+1))^(1/8)*x^2,x)
 

Output:

int(1/((a*x-1)/(a*x+1))^(1/8)*x^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.68 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=\frac {8 \, {\left (32 \, a^{3} x^{3} + 68 \, a^{2} x^{2} + 73 \, a x + 37\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}} - 66 \, \sqrt {2} \arctan \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right ) - 66 \, \sqrt {2} \arctan \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right ) + 33 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 33 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 132 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + 66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right ) - 66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right )}{768 \, a^{3}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/8)*x^2,x, algorithm="fricas")
 

Output:

1/768*(8*(32*a^3*x^3 + 68*a^2*x^2 + 73*a*x + 37)*((a*x - 1)/(a*x + 1))^(7/ 
8) - 66*sqrt(2)*arctan(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + 1) - 66*sqrt( 
2)*arctan(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) - 1) + 33*sqrt(2)*log(sqrt(2 
)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) - 33*sqrt 
(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) 
 + 1) - 132*arctan(((a*x - 1)/(a*x + 1))^(1/8)) + 66*log(((a*x - 1)/(a*x + 
 1))^(1/8) + 1) - 66*log(((a*x - 1)/(a*x + 1))^(1/8) - 1))/a^3
 

Sympy [F]

\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=\int \frac {x^{2}}{\sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \] Input:

integrate(1/((a*x-1)/(a*x+1))**(1/8)*x**2,x)
 

Output:

Integral(x**2/((a*x - 1)/(a*x + 1))**(1/8), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.96 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{768} \, a {\left (\frac {16 \, {\left (33 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {23}{8}} - 10 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{8}} + 105 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {33 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )\right )}}{a^{4}} + \frac {132 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{4}} - \frac {66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{4}} + \frac {66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right )}{a^{4}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/8)*x^2,x, algorithm="maxima")
 

Output:

-1/768*a*(16*(33*((a*x - 1)/(a*x + 1))^(23/8) - 10*((a*x - 1)/(a*x + 1))^( 
15/8) + 105*((a*x - 1)/(a*x + 1))^(7/8))/(3*(a*x - 1)*a^4/(a*x + 1) - 3*(a 
*x - 1)^2*a^4/(a*x + 1)^2 + (a*x - 1)^3*a^4/(a*x + 1)^3 - a^4) + 33*(2*sqr 
t(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8))) + 2*sqr 
t(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/8))) - sqrt 
(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) 
+ 1) + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x 
+ 1))^(1/4) + 1))/a^4 + 132*arctan(((a*x - 1)/(a*x + 1))^(1/8))/a^4 - 66*l 
og(((a*x - 1)/(a*x + 1))^(1/8) + 1)/a^4 + 66*log(((a*x - 1)/(a*x + 1))^(1/ 
8) - 1)/a^4)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.87 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{768} \, a {\left (\frac {66 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{4}} + \frac {66 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{4}} - \frac {33 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {33 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {132 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{4}} - \frac {66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{4}} + \frac {66 \, \log \left (-\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{4}} + \frac {16 \, {\left (33 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {23}{8}} - 10 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{8}} + 105 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/8)*x^2,x, algorithm="giac")
 

Output:

-1/768*a*(66*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1)) 
^(1/8)))/a^4 + 66*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x 
 + 1))^(1/8)))/a^4 - 33*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + 
((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 + 33*sqrt(2)*log(-sqrt(2)*((a*x - 1)/ 
(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 + 132*arctan(((a*x 
 - 1)/(a*x + 1))^(1/8))/a^4 - 66*log(((a*x - 1)/(a*x + 1))^(1/8) + 1)/a^4 
+ 66*log(-((a*x - 1)/(a*x + 1))^(1/8) + 1)/a^4 + 16*(33*((a*x - 1)/(a*x + 
1))^(23/8) - 10*((a*x - 1)/(a*x + 1))^(15/8) + 105*((a*x - 1)/(a*x + 1))^( 
7/8))/(a^4*((a*x - 1)/(a*x + 1) - 1)^3))
 

Mupad [B] (verification not implemented)

Time = 24.13 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.64 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=\frac {\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{16}-\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{24}+\frac {11\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{23/8}}{16}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,11{}\mathrm {i}}{64\,a^3}-\frac {11\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{64\,a^3}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {11}{128}+\frac {11}{128}{}\mathrm {i}\right )}{a^3}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {11}{128}-\frac {11}{128}{}\mathrm {i}\right )}{a^3} \] Input:

int(x^2/((a*x - 1)/(a*x + 1))^(1/8),x)
 

Output:

((35*((a*x - 1)/(a*x + 1))^(7/8))/16 - (5*((a*x - 1)/(a*x + 1))^(15/8))/24 
 + (11*((a*x - 1)/(a*x + 1))^(23/8))/16)/(a^3 + (3*a^3*(a*x - 1)^2)/(a*x + 
 1)^2 - (a^3*(a*x - 1)^3)/(a*x + 1)^3 - (3*a^3*(a*x - 1))/(a*x + 1)) - (at 
an(((a*x - 1)/(a*x + 1))^(1/8)*1i)*11i)/(64*a^3) - (11*atan(((a*x - 1)/(a* 
x + 1))^(1/8)))/(64*a^3) - (2^(1/2)*atan(2^(1/2)*((a*x - 1)/(a*x + 1))^(1/ 
8)*(1/2 - 1i/2))*(11/128 - 11i/128))/a^3 - (2^(1/2)*atan(2^(1/2)*((a*x - 1 
)/(a*x + 1))^(1/8)*(1/2 + 1i/2))*(11/128 + 11i/128))/a^3
 

Reduce [F]

\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx=\int \frac {\left (a x +1\right )^{\frac {1}{8}} x^{2}}{\left (a x -1\right )^{\frac {1}{8}}}d x \] Input:

int(1/((a*x-1)/(a*x+1))^(1/8)*x^2,x)
 

Output:

int(((a*x + 1)**(1/8)*x**2)/(a*x - 1)**(1/8),x)