Optimal. Leaf size=168 \[ -\frac{29 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{192 x}+\frac{11}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{11}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x^3}-\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4} \]
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Rubi [A] time = 0.0785711, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6126, 99, 151, 12, 93, 298, 203, 206} \[ -\frac{29 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{192 x}+\frac{11}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{11}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x^3}-\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 99
Rule 151
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{1}{2} \tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac{\sqrt [4]{1-a x}}{x^5 \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{1}{4} \int \frac{-\frac{7 a}{2}+3 a^2 x}{x^4 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{1}{12} \int \frac{-\frac{29 a^2}{4}+7 a^3 x}{x^3 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac{1}{24} \int \frac{-\frac{83 a^3}{8}+\frac{29 a^4 x}{4}}{x^2 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}-\frac{1}{24} \int -\frac{33 a^4}{16 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}+\frac{1}{128} \left (11 a^4\right ) \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}+\frac{1}{32} \left (11 a^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}-\frac{1}{64} \left (11 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac{1}{64} \left (11 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac{29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac{83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}+\frac{11}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{11}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0281805, size = 86, normalized size = 0.51 \[ \frac{\sqrt [4]{1-a x} \left (-66 a^4 x^4 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )+83 a^4 x^4+25 a^3 x^3-2 a^2 x^2+8 a x-48\right )}{192 x^4 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}}{\frac{1}{\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74222, size = 382, normalized size = 2.27 \begin{align*} \frac{66 \, a^{4} x^{4} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 33 \, a^{4} x^{4} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 33 \, a^{4} x^{4} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \,{\left (83 \, a^{3} x^{3} - 58 \, a^{2} x^{2} + 56 \, a x - 48\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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