Optimal. Leaf size=268 \[ -a \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}-\frac{a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{\sqrt{2}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.218081, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {6171, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -a \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}-\frac{a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{\sqrt{2}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{e^{-\frac{1}{2} \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+(2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,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}+(2 a) \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 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+a \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )+a \operatorname{Subst}\left (\int \frac{1+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}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{2 \sqrt{2}}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{2 \sqrt{2}}\\ &=-a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{a \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{2 \sqrt{2}}+\frac{a \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{2 \sqrt{2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{\sqrt{2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{\sqrt{2}}\\ &=-a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{\sqrt{2}}+\frac{a \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{\sqrt{2}}-\frac{a \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{2 \sqrt{2}}+\frac{a \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.050332, size = 33, normalized size = 0.12 \[ -\frac{8}{3} a e^{\frac{3}{2} \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{3}{4},2,\frac{7}{4},-e^{2 \coth ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\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 = 1.50447, size = 251, normalized size = 0.94 \begin{align*} \frac{1}{4} \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + \sqrt{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \sqrt{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \frac{8 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\frac{a x - 1}{a x + 1} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7223, size = 990, normalized size = 3.69 \begin{align*} -\frac{4 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \arctan \left (-\frac{a^{4} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right ) + 4 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \arctan \left (\frac{a^{4} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right ) - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \log \left (a^{2} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}\right ) + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \log \left (a^{2} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}\right ) + 4 \,{\left (a x + 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{4 \, x} \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{\sqrt [4]{\frac{a x - 1}{a x + 1}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17205, size = 251, normalized size = 0.94 \begin{align*} \frac{1}{4} \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + \sqrt{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \sqrt{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \frac{8 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\frac{a x - 1}{a x + 1} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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