Optimal. Leaf size=319 \[ \frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}-\frac{a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{4 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.250436, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {6171, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}-\frac{a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{4 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 80
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^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \sqrt [4]{1-\frac{x}{a}}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{8} a \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 )\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{2} a^2 \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 )\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{4} a^2 \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 )-\frac{1}{4} a^2 \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 )\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{8} a^2 \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}{8} a^2 \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^2 \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 )}{8 \sqrt{2}}+\frac{a^2 \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 )}{8 \sqrt{2}}\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{a^2 \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 )}{8 \sqrt{2}}-\frac{a^2 \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 )}{8 \sqrt{2}}-\frac{a^2 \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 )}{4 \sqrt{2}}+\frac{a^2 \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 )}{4 \sqrt{2}}\\ &=\frac{1}{4} a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{4 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{4 \sqrt{2}}+\frac{a^2 \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 )}{8 \sqrt{2}}-\frac{a^2 \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 )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0693259, size = 56, normalized size = 0.18 \[ -\frac{8}{3} a^2 e^{\frac{3}{2} \coth ^{-1}(a x)} \left (\text{Hypergeometric2F1}\left (\frac{3}{4},2,\frac{7}{4},-e^{2 \coth ^{-1}(a x)}\right )-2 \text{Hypergeometric2F1}\left (\frac{3}{4},3,\frac{7}{4},-e^{2 \coth ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.137, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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 = 1.54085, size = 306, normalized size = 0.96 \begin{align*} -\frac{1}{16} \,{\left (2 \, \sqrt{2} a \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} a \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} a \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} a \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 (5 \, a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{2 \,{\left (a x - 1\right )}}{a x + 1} + \frac{{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72257, size = 1035, normalized size = 3.24 \begin{align*} \frac{4 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \arctan \left (-\frac{a^{8} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right ) + 4 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \arctan \left (\frac{a^{8} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right ) - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \log \left (a^{4} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}\right ) + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \log \left (a^{4} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}\right ) + 4 \,{\left (3 \, a^{2} x^{2} + a x - 2\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{16 \, x^{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 \frac{\sqrt [4]{\frac{a x - 1}{a x + 1}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1465, size = 301, normalized size = 0.94 \begin{align*} -\frac{1}{16} \,{\left (2 \, \sqrt{2} a \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} a \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} a \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} a \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{5 \,{\left (a x - 1\right )} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} + a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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