Optimal. Leaf size=176 \[ -\frac {25 \sqrt [4]{1-\frac {1}{a x}}}{2 a^2 \sqrt [4]{\frac {1}{a x}+1}}-\frac {25 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {x^2 \left (1-\frac {1}{a x}\right )^{9/4}}{2 \sqrt [4]{\frac {1}{a x}+1}}-\frac {5 x \left (1-\frac {1}{a x}\right )^{5/4}}{4 a \sqrt [4]{\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6171, 96, 94, 93, 298, 203, 206} \[ -\frac {25 \sqrt [4]{1-\frac {1}{a x}}}{2 a^2 \sqrt [4]{\frac {1}{a x}+1}}-\frac {25 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {x^2 \left (1-\frac {1}{a x}\right )^{9/4}}{2 \sqrt [4]{\frac {1}{a x}+1}}-\frac {5 x \left (1-\frac {1}{a x}\right )^{5/4}}{4 a \sqrt [4]{\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 298
Rule 6171
Rubi steps
\begin {align*} \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/4}}{x^3 \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\left (1-\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/4}}{x^2 \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{4 a}\\ &=-\frac {5 \left (1-\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\left (1-\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {25 \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{x \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=-\frac {25 \sqrt [4]{1-\frac {1}{a x}}}{2 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {5 \left (1-\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\left (1-\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {25 \operatorname {Subst}\left (\int \frac {1}{x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=-\frac {25 \sqrt [4]{1-\frac {1}{a x}}}{2 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {5 \left (1-\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\left (1-\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {25 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a^2}\\ &=-\frac {25 \sqrt [4]{1-\frac {1}{a x}}}{2 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {5 \left (1-\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\left (1-\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1+\frac {1}{a x}}}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}-\frac {25 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}\\ &=-\frac {25 \sqrt [4]{1-\frac {1}{a x}}}{2 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {5 \left (1-\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\left (1-\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {25 \tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 121, normalized size = 0.69 \[ -\frac {e^{-\frac {1}{2} \coth ^{-1}(a x)} \left (-90 e^{2 \coth ^{-1}(a x)}+50 e^{4 \coth ^{-1}(a x)}+25 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)}-1\right )^2 \tan ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-25 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)}-1\right )^2 \tanh ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+32\right )}{4 a^2 \left (e^{2 \coth ^{-1}(a x)}-1\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.90, size = 95, normalized size = 0.54 \[ \frac {2 \, {\left (2 \, a^{2} x^{2} - 9 \, a x - 43\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 50 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 161, normalized size = 0.91 \[ \frac {1}{8} \, a {\left (\frac {50 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} + \frac {25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} - \frac {25 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{3}} - \frac {64 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{3}} + \frac {4 \, {\left (\frac {13 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - 9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int x \left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 172, normalized size = 0.98 \[ -\frac {1}{8} \, a {\left (\frac {4 \, {\left (13 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - 9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac {50 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} - \frac {25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{3}} + \frac {64 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 145, normalized size = 0.82 \[ \frac {25\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2}-\frac {8\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a^2}-\frac {\frac {9\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{2}-\frac {13\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{4\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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