Optimal. Leaf size=110 \[ \frac{1}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{1}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(1-a x)^{5/4} (a x+1)^{3/4}}{2 x^2}+\frac{a \sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x} \]
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Rubi [A] time = 0.0455312, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6126, 96, 94, 93, 298, 203, 206} \[ \frac{1}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{1}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(1-a x)^{5/4} (a x+1)^{3/4}}{2 x^2}+\frac{a \sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
Rule 94
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{1}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{\sqrt [4]{1-a x}}{x^3 \sqrt [4]{1+a x}} \, dx\\ &=-\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}-\frac{1}{4} a \int \frac{\sqrt [4]{1-a x}}{x^2 \sqrt [4]{1+a x}} \, dx\\ &=\frac{a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac{1}{8} a^2 \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac{a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac{1}{2} a^2 \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{a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}-\frac{1}{4} a^2 \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}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac{1}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{1}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0205784, size = 69, normalized size = 0.63 \[ \frac{\sqrt [4]{1-a x} \left (-2 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )+3 a^2 x^2+a x-2\right )}{4 x^2 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.107, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}{\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^{3} \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 = 1.74879, size = 332, normalized size = 3.02 \begin{align*} \frac{2 \, a^{2} x^{2} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - a^{2} x^{2} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + a^{2} x^{2} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (3 \, a x - 2\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx \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^{3} \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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