Optimal. Leaf size=351 \[ -\frac{2 a^2 \left (\frac{1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}-\frac{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{25 a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}+\frac{25 a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}+\frac{25 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{4 \sqrt{2}}-\frac{25 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.284915, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {6171, 78, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{2 a^2 \left (\frac{1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}-\frac{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{25 a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}+\frac{25 a^2 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}+\frac{25 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{4 \sqrt{2}}-\frac{25 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 78
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{5}{2} \coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (1+\frac{x}{a}\right )^{5/4}}{\left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}+(5 a) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/4}}{\sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{4} (25 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{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{8} (25 a) \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{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{2} \left (25 a^2\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{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{2} \left (25 a^2\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{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{4} \left (25 a^2\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}{4} \left (25 a^2\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{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{8} \left (25 a^2\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}{8} \left (25 a^2\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 (25 a^2\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 )}{8 \sqrt{2}}-\frac{\left (25 a^2\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 )}{8 \sqrt{2}}\\ &=-\frac{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{25 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{25 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{\left (25 a^2\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 )}{4 \sqrt{2}}+\frac{\left (25 a^2\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 )}{4 \sqrt{2}}\\ &=-\frac{25}{4} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{5}{2} a^2 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^2 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{25 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{25 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{25 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{25 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 [A] time = 0.2621, size = 186, normalized size = 0.53 \[ \frac{1}{16} a^2 \left (-128 e^{\frac{1}{2} \coth ^{-1}(a x)}-\frac{104 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}+\frac{32 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}+1\right )^2}-25 \sqrt{2} \log \left (-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right )+25 \sqrt{2} \log \left (\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right )-50 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+50 \sqrt{2} \tan ^{-1}\left (\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.336, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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 = 1.52576, size = 329, normalized size = 0.94 \begin{align*} -\frac{1}{16} \,{\left (25 \,{\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 + \frac{8 \,{\left (\frac{45 \,{\left (a x - 1\right )} a}{a x + 1} + \frac{25 \,{\left (a x - 1\right )}^{2} a}{{\left (a x + 1\right )}^{2}} + 16 \, a\right )}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7602, size = 1247, normalized size = 3.55 \begin{align*} \frac{100 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}}{\left (a x^{3} - x^{2}\right )} \arctan \left (-\frac{a^{8} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2} \sqrt{a^{12} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{a^{8}} a^{8} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\left (a^{8}\right )}^{\frac{1}{4}}}{a^{8}}\right ) + 100 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}}{\left (a x^{3} - x^{2}\right )} \arctan \left (\frac{a^{8} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2} \sqrt{a^{12} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{a^{8}} a^{8} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\left (a^{8}\right )}^{\frac{1}{4}}}{a^{8}}\right ) + 25 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}}{\left (a x^{3} - x^{2}\right )} \log \left (244140625 \, a^{12} \sqrt{\frac{a x - 1}{a x + 1}} + 244140625 \, \sqrt{a^{8}} a^{8} + 244140625 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 25 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}}{\left (a x^{3} - x^{2}\right )} \log \left (244140625 \, a^{12} \sqrt{\frac{a x - 1}{a x + 1}} + 244140625 \, \sqrt{a^{8}} a^{8} - 244140625 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 4 \,{\left (43 \, a^{3} x^{3} + 34 \, a^{2} x^{2} - 11 \, a x - 2\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{16 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18918, size = 328, normalized size = 0.93 \begin{align*} -\frac{1}{16} \,{\left (50 \, \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 ) + 50 \, \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 ) - 25 \, \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 ) + 25 \, \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{128 \, a}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} + \frac{8 \,{\left (\frac{9 \,{\left (a x - 1\right )} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} + 13 \, a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{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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