Optimal. Leaf size=97 \[ x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]
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Rubi [A] time = 0.0355157, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]
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{1}{2} \coth ^{-1}(a x)} \, dx &=-\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 )\\ &=\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x+\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 )}{2 a}\\ &=\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x+\frac{2 \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}\\ &=\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x-\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 )}{a}+\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 )}{a}\\ &=\sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}-\frac{\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 = 0.0430654, size = 33, normalized size = 0.34 \[ -\frac{8 e^{\frac{3}{2} \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{3}{4},2,\frac{7}{4},e^{2 \coth ^{-1}(a x)}\right )}{3 a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.123, size = 0, normalized size = 0. \begin{align*} \int \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 = 1.48446, size = 150, normalized size = 1.55 \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{2 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{2}} + \frac{\log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} - \frac{\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 = 1.65827, size = 225, normalized size = 2.32 \begin{align*} \frac{2 \,{\left (a x + 1\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 )}{2 \, a} \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 \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 = 1.14995, size = 146, normalized size = 1.51 \begin{align*} -\frac{1}{2} \, a{\left (\frac{2 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{2}} + \frac{\log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} - \frac{\log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{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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