Optimal. Leaf size=142 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^2}+\frac{1}{2} x^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (\frac{1}{a x}+1\right )^{3/4}-\frac{x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{4 a} \]
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Rubi [A] time = 0.0602726, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^2}+\frac{1}{2} x^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (\frac{1}{a x}+1\right )^{3/4}-\frac{x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{4 a} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 96
Rule 94
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^{-\frac{1}{2} \coth ^{-1}(a x)} x \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{x^3 \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4} x^2+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{x^2 \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 a}\\ &=-\frac{\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{4 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4} x^2-\frac{\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{\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{4 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4} x^2-\frac{\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{\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{4 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4} x^2+\frac{\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{\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{\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{4 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4} x^2-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.154832, size = 66, normalized size = 0.46 \[ \frac{-\frac{2 e^{\frac{3}{2} \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)}-5\right )}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}-\tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+\tanh ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int x\sqrt [4]{{\frac{ax-1}{ax+1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53275, size = 204, normalized size = 1.44 \begin{align*} -\frac{1}{8} \, a{\left (\frac{4 \,{\left (5 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} - \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{2 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{3}} - \frac{\log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{3}} + \frac{\log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63593, size = 244, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (2 \, a^{2} x^{2} - a x - 3\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 2 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt [4]{\frac{a x - 1}{a x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19221, size = 189, normalized size = 1.33 \begin{align*} \frac{1}{8} \, a{\left (\frac{2 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{3}} + \frac{\log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{3}} - \frac{\log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{3}} + \frac{4 \,{\left (\frac{5 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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