Optimal. Leaf size=356 \[ \frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}+\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}}{3 x}+\frac{3 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{3 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{3 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}+\frac{3 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.291752, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {6171, 90, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}+\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}}{3 x}+\frac{3 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{3 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{3 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}+\frac{3 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 90
Rule 80
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \sqrt [4]{1+\frac{x}{a}}}{\sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}+\frac{1}{3} a^2 \operatorname{Subst}\left (\int \frac{\left (-1-\frac{x}{2 a}\right ) \sqrt [4]{1+\frac{x}{a}}}{\sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}-\frac{1}{8} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt [4]{1+\frac{x}{a}}}{\sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}-\frac{1}{16} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}+\frac{1}{4} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )\\ &=\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}+\frac{1}{4} \left (3 a^3\right ) \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 )\\ &=\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}-\frac{1}{8} \left (3 a^3\right ) \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}{8} \left (3 a^3\right ) \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{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}+\frac{1}{16} \left (3 a^3\right ) \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}{16} \left (3 a^3\right ) \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{\left (3 a^3\right ) \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 )}{16 \sqrt{2}}+\frac{\left (3 a^3\right ) \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 )}{16 \sqrt{2}}\\ &=\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}+\frac{3 a^3 \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 )}{16 \sqrt{2}}-\frac{3 a^3 \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 )}{16 \sqrt{2}}+\frac{\left (3 a^3\right ) \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 )}{8 \sqrt{2}}-\frac{\left (3 a^3\right ) \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 )}{8 \sqrt{2}}\\ &=\frac{3}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}+\frac{1}{12} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}}{3 x}-\frac{3 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{8 \sqrt{2}}+\frac{3 a^3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{8 \sqrt{2}}+\frac{3 a^3 \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 )}{16 \sqrt{2}}-\frac{3 a^3 \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 )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.106194, size = 93, normalized size = 0.26 \[ \frac{1}{96} a^3 \left (9 \text{RootSum}\left [\text{$\#$1}^4+1\& ,\frac{\coth ^{-1}(a x)-2 \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right )}{\text{$\#$1}^3}\& \right ]+\frac{8 e^{\frac{1}{2} \coth ^{-1}(a x)} \left (6 e^{2 \coth ^{-1}(a x)}+29 e^{4 \coth ^{-1}(a x)}+9\right )}{\left (e^{2 \coth ^{-1}(a x)}+1\right )^3}\right ) \]
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}^{4}}{\frac{1}{\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 = 1.49274, size = 365, normalized size = 1.03 \begin{align*} \frac{1}{96} \,{\left (9 \,{\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 )\right )} a^{2} + \frac{8 \,{\left (9 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{4}} + 6 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{4}} + 29 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}\right )}}{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66981, size = 1160, normalized size = 3.26 \begin{align*} -\frac{36 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \arctan \left (-\frac{a^{12} + \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2} \sqrt{a^{18} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{a^{12}} a^{12} + \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\left (a^{12}\right )}^{\frac{1}{4}}}{a^{12}}\right ) + 36 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \arctan \left (\frac{a^{12} - \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2} \sqrt{a^{18} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{a^{12}} a^{12} - \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\left (a^{12}\right )}^{\frac{1}{4}}}{a^{12}}\right ) + 9 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \log \left (729 \, a^{18} \sqrt{\frac{a x - 1}{a x + 1}} + 729 \, \sqrt{a^{12}} a^{12} + 729 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 9 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \log \left (729 \, a^{18} \sqrt{\frac{a x - 1}{a x + 1}} + 729 \, \sqrt{a^{12}} a^{12} - 729 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 4 \,{\left (11 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 18 \, a x + 8\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{96 \, x^{3}} \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}{x^{4} \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 = 1.17358, size = 366, normalized size = 1.03 \begin{align*} \frac{1}{96} \,{\left (18 \, \sqrt{2} a^{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 ) + 18 \, \sqrt{2} a^{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 ) - 9 \, \sqrt{2} a^{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 ) + 9 \, \sqrt{2} a^{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{6 \,{\left (a x - 1\right )} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} + \frac{9 \,{\left (a x - 1\right )}^{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{2}} + 29 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}\right )}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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