Optimal. Leaf size=130 \[ \frac{x \left (\frac{1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{10 \sqrt [4]{\frac{1}{a x}+1}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]
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Rubi [A] time = 0.0440145, 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, 212, 206, 203} \[ \frac{x \left (\frac{1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{10 \sqrt [4]{\frac{1}{a x}+1}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6170
Rule 94
Rule 93
Rule 212
Rule 206
Rule 203
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 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, 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{10 \operatorname{Subst}\left (\int \frac{1}{-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 [A] time = 0.133485, size = 67, normalized size = 0.52 \[ \frac{-\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)} \left (4 e^{2 \coth ^{-1}(a x)}-5\right )}{e^{2 \coth ^{-1}(a x)}-1}+5 \tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+5 \tanh ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.322, 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 = 1.48489, size = 177, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, a{\left (\frac{4 \,{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 4\right )}}{a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} - a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} + \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}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60109, size = 304, normalized size = 2.34 \begin{align*} -\frac{10 \,{\left (a x - 1\right )} \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 5 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) - 2 \,{\left (a^{2} x^{2} - 8 \, a x - 9\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{2 \,{\left (a^{2} x - a\right )}} \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 \frac{1}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20936, size = 190, normalized size = 1.46 \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{4 \,{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 4\right )}}{a^{2}{\left (\frac{{\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 )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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