Optimal. Leaf size=356 \[ \frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/4}+\frac{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/4}}{3 x}-\frac{17 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{17 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{17 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}+\frac{17 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.281635, 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, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/4}+\frac{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/4}}{3 x}-\frac{17 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{17 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{17 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}+\frac{17 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 90
Rule 80
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{e^{\frac{3}{2} \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (1+\frac{x}{a}\right )^{3/4}}{\left (1-\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}+\frac{1}{3} a^2 \operatorname{Subst}\left (\int \frac{\left (-1-\frac{3 x}{2 a}\right ) \left (1+\frac{x}{a}\right )^{3/4}}{\left (1-\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}-\frac{1}{24} \left (17 a^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/4}}{\left (1-\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}-\frac{1}{16} \left (17 a^2\right ) \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{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}+\frac{1}{4} \left (17 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )\\ &=\frac{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}+\frac{1}{4} \left (17 a^3\right ) \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{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}+\frac{1}{8} \left (17 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 (17 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{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}+\frac{1}{16} \left (17 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 (17 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 (17 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 (17 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{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}-\frac{17 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{17 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 (17 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 (17 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{17}{24} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{4} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}+\frac{a^2 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/4}}{3 x}-\frac{17 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{17 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{17 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{17 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 = 0.117398, size = 93, normalized size = 0.26 \[ \frac{1}{96} a^3 \left (51 \text{RootSum}\left [\text{$\#$1}^4+1\& ,\frac{\coth ^{-1}(a x)-2 \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right )}{\text{$\#$1}}\& \right ]+\frac{8 e^{\frac{3}{2} \coth ^{-1}(a x)} \left (30 e^{2 \coth ^{-1}(a x)}+45 e^{4 \coth ^{-1}(a x)}+17\right )}{\left (e^{2 \coth ^{-1}(a x)}+1\right )^3}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49628, size = 374, normalized size = 1.05 \begin{align*} \frac{1}{96} \,{\left (102 \, \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 ) + 102 \, \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 ) + 51 \, \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 ) - 51 \, \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 (17 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 30 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + 45 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{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 = 1.78408, size = 1133, normalized size = 3.18 \begin{align*} -\frac{204 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \arctan \left (-\frac{a^{12} + \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} \sqrt{a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{12}}}}{a^{12}}\right ) + 204 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \arctan \left (\frac{a^{12} - \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} \sqrt{a^{6} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{12}}}}{a^{12}}\right ) - 51 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \log \left (289 \, a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + 289 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 289 \, \sqrt{a^{12}}\right ) + 51 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \log \left (289 \, a^{6} \sqrt{\frac{a x - 1}{a x + 1}} - 289 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 289 \, \sqrt{a^{12}}\right ) - 4 \,{\left (23 \, a^{3} x^{3} + 37 \, a^{2} x^{2} + 22 \, a x + 8\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{96 \, x^{3}} \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 = 1.13451, size = 366, normalized size = 1.03 \begin{align*} \frac{1}{96} \,{\left (102 \, \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 ) + 102 \, \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 ) + 51 \, \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 ) - 51 \, \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{30 \,{\left (a x - 1\right )} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} + \frac{17 \,{\left (a x - 1\right )}^{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{2}} + 45 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{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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