Optimal. Leaf size=130 \[ \frac{x \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{10 \sqrt [4]{1-\frac{1}{a x}}}{a \sqrt [4]{\frac{1}{a x}+1}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]
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Rubi [A] time = 0.0464704, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6170, 94, 93, 298, 203, 206} \[ \frac{x \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{10 \sqrt [4]{1-\frac{1}{a x}}}{a \sqrt [4]{\frac{1}{a x}+1}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6170
Rule 94
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^{-\frac{5}{2} \coth ^{-1}(a x)} \, dx &=-\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 )\\ &=\frac{\left (1-\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{5 \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 )}{2 a}\\ &=\frac{10 \sqrt [4]{1-\frac{1}{a x}}}{a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\left (1-\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{5 \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 )}{2 a}\\ &=\frac{10 \sqrt [4]{1-\frac{1}{a x}}}{a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\left (1-\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{10 \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 )}{a}\\ &=\frac{10 \sqrt [4]{1-\frac{1}{a x}}}{a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\left (1-\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{5 \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 )}{a}+\frac{5 \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 )}{a}\\ &=\frac{10 \sqrt [4]{1-\frac{1}{a x}}}{a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\left (1-\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}\\ \end{align*}
Mathematica [C] time = 0.0659613, size = 31, normalized size = 0.24 \[ \frac{8 e^{-\frac{1}{2} \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (-\frac{1}{4},2,\frac{3}{4},e^{2 \coth ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.325, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47956, size = 178, normalized size = 1.37 \begin{align*} -\frac{1}{2} \, a{\left (\frac{4 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac{10 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{2}} + \frac{5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} - \frac{5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{2}} - \frac{16 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79123, size = 232, normalized size = 1.78 \begin{align*} \frac{2 \,{\left (a x + 9\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 10 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23107, size = 174, normalized size = 1.34 \begin{align*} -\frac{1}{2} \, a{\left (\frac{10 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{2}} + \frac{5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} - \frac{5 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{2}} - \frac{16 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{2}} + \frac{4 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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