Optimal. Leaf size=110 \[ -\frac{9}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{9}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(1-a x)^{7/4} \sqrt [4]{a x+1}}{2 x^2}+\frac{3 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x} \]
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Rubi [A] time = 0.0421458, 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, 212, 206, 203} \[ -\frac{9}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{9}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(1-a x)^{7/4} \sqrt [4]{a x+1}}{2 x^2}+\frac{3 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
Rule 94
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{-\frac{3}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1-a x)^{3/4}}{x^3 (1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}-\frac{1}{4} (3 a) \int \frac{(1-a x)^{3/4}}{x^2 (1+a x)^{3/4}} \, dx\\ &=\frac{3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}+\frac{1}{8} \left (9 a^2\right ) \int \frac{1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac{3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}+\frac{1}{2} \left (9 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}-\frac{1}{4} \left (9 a^2\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}{4} \left (9 a^2\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{3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}-\frac{9}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{9}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0205215, size = 70, normalized size = 0.64 \[ \frac{(1-a x)^{3/4} \left (-6 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )+5 a^2 x^2+3 a x-2\right )}{4 x^2 (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{3}{2}}}}\, 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} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67503, size = 331, normalized size = 3.01 \begin{align*} -\frac{18 \, a^{2} x^{2} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 9 \, a^{2} x^{2} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 9 \, a^{2} x^{2} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \,{\left (5 \, a^{2} x^{2} - 7 \, 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(-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^{3} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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