Optimal. Leaf size=299 \[ -\frac{4 a \left (\frac{1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{5 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.256712, antiderivative size = 299, 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, 47, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{4 a \left (\frac{1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{5 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 47
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^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/4}}{\left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}+5 \operatorname{Subst}\left (\int \frac{\sqrt [4]{1+\frac{x}{a}}}{\sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{5}{2} \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 )\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-(10 a) \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-(10 a) \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 )\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}+(5 a) \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 )-(5 a) \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 )\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{2} (5 a) \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}{2} (5 a) \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{(5 a) \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 )}{2 \sqrt{2}}-\frac{(5 a) \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 )}{2 \sqrt{2}}\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{5 a \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 )}{2 \sqrt{2}}+\frac{5 a \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 )}{2 \sqrt{2}}-\frac{(5 a) \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 )}{\sqrt{2}}+\frac{(5 a) \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 )}{\sqrt{2}}\\ &=-5 a \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}}-\frac{4 a \left (1+\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{5 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{\sqrt{2}}-\frac{5 a \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 )}{2 \sqrt{2}}+\frac{5 a \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 )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.372751, size = 173, normalized size = 0.58 \[ a \left (-\frac{10 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}-\frac{8 e^{\frac{5}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}-\frac{5 \log \left (-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right )}{2 \sqrt{2}}+\frac{5 \log \left (\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right )}{2 \sqrt{2}}-\frac{5 \tan ^{-1}\left (1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{\sqrt{2}}+\frac{5 \tan ^{-1}\left (\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )}{\sqrt{2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.355, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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 = 1.5586, size = 275, normalized size = 0.92 \begin{align*} -\frac{1}{4} \,{\left (10 \, \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 ) + 10 \, \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 ) - 5 \, \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 ) + 5 \, \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 ) + \frac{8 \,{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\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 = 1.8141, size = 1169, normalized size = 3.91 \begin{align*} \frac{20 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}{\left (a x^{2} - x\right )} \arctan \left (-\frac{a^{4} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2} \sqrt{a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{a^{4}} a^{4} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\left (a^{4}\right )}^{\frac{1}{4}}}{a^{4}}\right ) + 20 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}{\left (a x^{2} - x\right )} \arctan \left (\frac{a^{4} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2} \sqrt{a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{a^{4}} a^{4} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\left (a^{4}\right )}^{\frac{1}{4}}}{a^{4}}\right ) + 5 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}{\left (a x^{2} - x\right )} \log \left (15625 \, a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + 15625 \, \sqrt{a^{4}} a^{4} + 15625 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 5 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}{\left (a x^{2} - x\right )} \log \left (15625 \, a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + 15625 \, \sqrt{a^{4}} a^{4} - 15625 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 4 \,{\left (9 \, a^{2} x^{2} + 8 \, a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{4 \,{\left (a x^{2} - x\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 = 1.20863, size = 293, normalized size = 0.98 \begin{align*} -\frac{1}{4} \,{\left (10 \, \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 ) + 10 \, \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 ) - 5 \, \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 ) + 5 \, \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 ) + \frac{8 \,{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\frac{{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} + \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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