Optimal. Leaf size=385 \[ -\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{9/4}-\frac{2 a^3 \left (\frac{1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}-\frac{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{55 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.31253, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {6171, 89, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{9/4}-\frac{2 a^3 \left (\frac{1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (\frac{1}{a x}+1\right )^{5/4}-\frac{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{55 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 89
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{5}{2} \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \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^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{5}{2 a}+\frac{x}{2 a^2}\right ) \left (1+\frac{x}{a}\right )^{5/4}}{\sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}+\frac{1}{2} \left (11 a^2\right ) \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{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}+\frac{1}{8} \left (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}+\frac{1}{16} \left (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}-\frac{1}{4} \left (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}-\frac{1}{4} \left (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}+\frac{1}{8} \left (55 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 (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}-\frac{1}{16} \left (55 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 (55 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 (55 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 (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}-\frac{55 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{55 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 (55 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 (55 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{55}{8} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{5/4}-\frac{2 a^3 \left (1+\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{3/4} \left (1+\frac{1}{a x}\right )^{9/4}+\frac{55 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{55 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{55 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{55 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 = 0.143025, size = 104, normalized size = 0.27 \[ a^3 \left (-\frac{55}{32} \text{RootSum}\left [\text{$\#$1}^4+1\& ,\frac{\coth ^{-1}(a x)-2 \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right )}{\text{$\#$1}^3}\& \right ]-\frac{e^{\frac{1}{2} \coth ^{-1}(a x)} \left (462 e^{2 \coth ^{-1}(a x)}+425 e^{4 \coth ^{-1}(a x)}+96 e^{6 \coth ^{-1}(a x)}+165\right )}{12 \left (e^{2 \coth ^{-1}(a x)}+1\right )^3}\right ) \]
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}^{4}} \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 = 1.67789, size = 389, normalized size = 1.01 \begin{align*} -\frac{1}{96} \,{\left (165 \,{\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 (\frac{425 \,{\left (a x - 1\right )} a^{2}}{a x + 1} + \frac{462 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{165 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + 96 \, a^{2}\right )}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} + 3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 3 \, \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 = 1.71029, size = 1319, normalized size = 3.43 \begin{align*} \frac{660 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}}{\left (a x^{4} - x^{3}\right )} \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 ) + 660 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}}{\left (a x^{4} - x^{3}\right )} \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 ) + 165 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}}{\left (a x^{4} - x^{3}\right )} \log \left (27680640625 \, a^{18} \sqrt{\frac{a x - 1}{a x + 1}} + 27680640625 \, \sqrt{a^{12}} a^{12} + 27680640625 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 165 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}}{\left (a x^{4} - x^{3}\right )} \log \left (27680640625 \, a^{18} \sqrt{\frac{a x - 1}{a x + 1}} + 27680640625 \, \sqrt{a^{12}} a^{12} - 27680640625 \, \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 (287 \, a^{4} x^{4} + 226 \, a^{3} x^{3} - 87 \, a^{2} x^{2} - 34 \, a x - 8\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{96 \,{\left (a x^{4} - x^{3}\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 = 1.21429, size = 393, normalized size = 1.02 \begin{align*} -\frac{1}{96} \,{\left (330 \, \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 ) + 330 \, \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 ) - 165 \, \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 ) + 165 \, \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{768 \, a^{2}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} + \frac{8 \,{\left (\frac{174 \,{\left (a x - 1\right )} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} + \frac{69 \,{\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}} + 137 \, 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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