\(\int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx\) [112]

Optimal result

Integrand size = 14, antiderivative size = 287 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1003 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \] Output:

26111/1920*(1-1/a/x)^(1/4)/a^5/(1+1/a/x)^(1/4)+5533/1920*(1-1/a/x)^(1/4)*x 
/a^4/(1+1/a/x)^(1/4)-1189/960*(1-1/a/x)^(1/4)*x^2/a^3/(1+1/a/x)^(1/4)+181/ 
240*(1-1/a/x)^(1/4)*x^3/a^2/(1+1/a/x)^(1/4)-21/40*(1-1/a/x)^(1/4)*x^4/a/(1 
+1/a/x)^(1/4)+1/5*(1-1/a/x)^(1/4)*x^5/(1+1/a/x)^(1/4)+1003/128*arctan((1+1 
/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5-1003/128*arctanh((1+1/a/x)^(1/4)/(1-1/a/x 
)^(1/4))/a^5
 

Mathematica [A] (verified)

Time = 5.43 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.69 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {8 e^{-\frac {1}{2} \coth ^{-1}(a x)}-\frac {32 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{-2 \coth ^{-1}(a x)}\right )^5}-\frac {122 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{-2 \coth ^{-1}(a x)}\right )^4}-\frac {233 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{6 \left (-1+e^{-2 \coth ^{-1}(a x)}\right )^3}-\frac {1661 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{48 \left (-1+e^{-2 \coth ^{-1}(a x)}\right )^2}-\frac {4117 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{192 \left (-1+e^{-2 \coth ^{-1}(a x)}\right )}-\frac {1003}{128} \arctan \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )+\frac {1003}{256} \log \left (1-e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )-\frac {1003}{256} \log \left (1+e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )}{a^5} \] Input:

Integrate[x^4/E^((5*ArcCoth[a*x])/2),x]
 

Output:

(8/E^(ArcCoth[a*x]/2) - 32/(5*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]) 
)^5) - 122/(5*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^4) - 233/(6*E^ 
(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^3) - 1661/(48*E^(ArcCoth[a*x]/ 
2)*(-1 + E^(-2*ArcCoth[a*x]))^2) - 4117/(192*E^(ArcCoth[a*x]/2)*(-1 + E^(- 
2*ArcCoth[a*x]))) - (1003*ArcTan[E^(-1/2*ArcCoth[a*x])])/128 + (1003*Log[1 
 - E^(-1/2*ArcCoth[a*x])])/256 - (1003*Log[1 + E^(-1/2*ArcCoth[a*x])])/256 
)/a^5
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.08, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {6721, 109, 27, 168, 27, 168, 27, 168, 27, 168, 27, 172, 27, 104, 25, 827, 216, 219}

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.

\(\displaystyle \int x^4 e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^{5/4} x^6}{\left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{5} \int \frac {\left (21 a-\frac {20}{x}\right ) x^5}{2 a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (21 a-\frac {20}{x}\right ) x^5}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {1}{4} \int \frac {\left (181 a-\frac {168}{x}\right ) x^4}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\left (181 a-\frac {168}{x}\right ) x^4}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {\left (1189 a-\frac {1086}{x}\right ) x^3}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\left (1189 a-\frac {1086}{x}\right ) x^3}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {-\frac {1}{2} \int \frac {\left (5533 a-\frac {4756}{x}\right ) x^2}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {\left (5533 a-\frac {4756}{x}\right ) x^2}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\int \frac {\left (15045 a-\frac {11066}{x}\right ) x}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {\int \frac {\left (15045 a-\frac {11066}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {2 a \int \frac {15045 x}{2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {15045 a \int \frac {x}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {60180 a \int -\frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-60180 a \int \frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {60180 a \left (\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {60180 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {60180 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {52222 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {5533 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {181 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a}-\frac {21 a x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}}{10 a^2}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}\)

Input:

Int[x^4/E^((5*ArcCoth[a*x])/2),x]
 

Output:

((1 - 1/(a*x))^(1/4)*x^5)/(5*(1 + 1/(a*x))^(1/4)) + ((-21*a*(1 - 1/(a*x))^ 
(1/4)*x^4)/(4*(1 + 1/(a*x))^(1/4)) - ((-181*a*(1 - 1/(a*x))^(1/4)*x^3)/(3* 
(1 + 1/(a*x))^(1/4)) - ((-1189*a*(1 - 1/(a*x))^(1/4)*x^2)/(2*(1 + 1/(a*x)) 
^(1/4)) - ((-5533*a*(1 - 1/(a*x))^(1/4)*x)/(1 + 1/(a*x))^(1/4) - ((52222*a 
*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4) + 60180*a*(ArcTan[(1 + 1/(a*x))^ 
(1/4)/(1 - 1/(a*x))^(1/4)]/2 - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^( 
1/4)]/2))/(2*a))/(4*a))/(6*a))/(8*a))/(10*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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + Simp[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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[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 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 827

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

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]

\[\int x^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}d x\]

Input:

int(x^4*((a*x-1)/(a*x+1))^(5/4),x)
 

Output:

int(x^4*((a*x-1)/(a*x+1))^(5/4),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.41 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {2 \, {\left (384 \, a^{5} x^{5} - 1008 \, a^{4} x^{4} + 1448 \, a^{3} x^{3} - 2378 \, a^{2} x^{2} + 5533 \, a x + 26111\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{3840 \, a^{5}} \] Input:

integrate(x^4*((a*x-1)/(a*x+1))^(5/4),x, algorithm="fricas")
 

Output:

1/3840*(2*(384*a^5*x^5 - 1008*a^4*x^4 + 1448*a^3*x^3 - 2378*a^2*x^2 + 5533 
*a*x + 26111)*((a*x - 1)/(a*x + 1))^(1/4) - 30090*arctan(((a*x - 1)/(a*x + 
 1))^(1/4)) - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 15045*log(((a*x 
 - 1)/(a*x + 1))^(1/4) - 1))/a^5
 

Sympy [F(-1)]

Timed out. \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\text {Timed out} \] Input:

integrate(x**4*((a*x-1)/(a*x+1))**(5/4),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.97 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {4 \, {\left (20585 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 49120 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 61130 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 33816 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 7365 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{6}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} + \frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}} - \frac {30720 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{6}}\right )} \] Input:

integrate(x^4*((a*x-1)/(a*x+1))^(5/4),x, algorithm="maxima")
 

Output:

-1/3840*a*(4*(20585*((a*x - 1)/(a*x + 1))^(17/4) - 49120*((a*x - 1)/(a*x + 
 1))^(13/4) + 61130*((a*x - 1)/(a*x + 1))^(9/4) - 33816*((a*x - 1)/(a*x + 
1))^(5/4) + 7365*((a*x - 1)/(a*x + 1))^(1/4))/(5*(a*x - 1)*a^6/(a*x + 1) - 
 10*(a*x - 1)^2*a^6/(a*x + 1)^2 + 10*(a*x - 1)^3*a^6/(a*x + 1)^3 - 5*(a*x 
- 1)^4*a^6/(a*x + 1)^4 + (a*x - 1)^5*a^6/(a*x + 1)^5 - a^6) + 30090*arctan 
(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 15045*log(((a*x - 1)/(a*x + 1))^(1/4) 
+ 1)/a^6 - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6 - 30720*((a*x - 
1)/(a*x + 1))^(1/4)/a^6)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.89 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {15045 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} - \frac {30720 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{6}} - \frac {4 \, {\left (\frac {33816 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {61130 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {49120 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {20585 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{4}} - 7365 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \] Input:

integrate(x^4*((a*x-1)/(a*x+1))^(5/4),x, algorithm="giac")
 

Output:

-1/3840*a*(30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 15045*log(((a*x 
 - 1)/(a*x + 1))^(1/4) + 1)/a^6 - 15045*log(abs(((a*x - 1)/(a*x + 1))^(1/4 
) - 1))/a^6 - 30720*((a*x - 1)/(a*x + 1))^(1/4)/a^6 - 4*(33816*(a*x - 1)*( 
(a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) - 61130*(a*x - 1)^2*((a*x - 1)/(a*x + 
 1))^(1/4)/(a*x + 1)^2 + 49120*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(1/4)/(a* 
x + 1)^3 - 20585*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^4 - 736 
5*((a*x - 1)/(a*x + 1))^(1/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))
 

Mupad [B] (verification not implemented)

Time = 24.06 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.88 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {8\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a^5}+\frac {\frac {491\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{64}-\frac {1409\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{40}+\frac {6113\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{96}-\frac {307\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}}{6}+\frac {4117\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}}{192}}{a^5+\frac {10\,a^5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a^5\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a^5\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a^5\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {5\,a^5\,\left (a\,x-1\right )}{a\,x+1}}-\frac {1003\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}+\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )\,1003{}\mathrm {i}}{128\,a^5} \] Input:

int(x^4*((a*x - 1)/(a*x + 1))^(5/4),x)
 

Output:

(atan(((a*x - 1)/(a*x + 1))^(1/4)*1i)*1003i)/(128*a^5) + (8*((a*x - 1)/(a* 
x + 1))^(1/4))/a^5 + ((491*((a*x - 1)/(a*x + 1))^(1/4))/64 - (1409*((a*x - 
 1)/(a*x + 1))^(5/4))/40 + (6113*((a*x - 1)/(a*x + 1))^(9/4))/96 - (307*(( 
a*x - 1)/(a*x + 1))^(13/4))/6 + (4117*((a*x - 1)/(a*x + 1))^(17/4))/192)/( 
a^5 + (10*a^5*(a*x - 1)^2)/(a*x + 1)^2 - (10*a^5*(a*x - 1)^3)/(a*x + 1)^3 
+ (5*a^5*(a*x - 1)^4)/(a*x + 1)^4 - (a^5*(a*x - 1)^5)/(a*x + 1)^5 - (5*a^5 
*(a*x - 1))/(a*x + 1)) - (1003*atan(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5 
)
 

Reduce [F]

\[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\left (\int \frac {\left (a x -1\right )^{\frac {1}{4}} x^{5}}{\left (a x +1\right )^{\frac {1}{4}} a x +\left (a x +1\right )^{\frac {1}{4}}}d x \right ) a -\left (\int \frac {\left (a x -1\right )^{\frac {1}{4}} x^{4}}{\left (a x +1\right )^{\frac {1}{4}} a x +\left (a x +1\right )^{\frac {1}{4}}}d x \right ) \] Input:

int(x^4*((a*x-1)/(a*x+1))^(5/4),x)
 

Output:

int(((a*x - 1)**(1/4)*x**5)/((a*x + 1)**(1/4)*a*x + (a*x + 1)**(1/4)),x)*a 
 - int(((a*x - 1)**(1/4)*x**4)/((a*x + 1)**(1/4)*a*x + (a*x + 1)**(1/4)),x 
)