Optimal. Leaf size=170 \[ -\frac{3 \left (1-a^2 x^2\right )^{3/2}}{a^4 x^3 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (a x+1)^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}-\frac{3 \left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.197348, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6160, 6150, 43} \[ -\frac{3 \left (1-a^2 x^2\right )^{3/2}}{a^4 x^3 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (a x+1)^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}-\frac{3 \left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{3/2} \int \frac{e^{-3 \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2} \int \frac{x^3}{(1+a x)^3} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2} \int \left (\frac{1}{a^3}-\frac{1}{a^3 (1+a x)^3}+\frac{3}{a^3 (1+a x)^2}-\frac{3}{a^3 (1+a x)}\right ) \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}+\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3 (1+a x)^2}-\frac{3 \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3 (1+a x)}-\frac{3 \left (1-a^2 x^2\right )^{3/2} \log (1+a x)}{a^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ \end{align*}
Mathematica [A] time = 0.0557708, size = 86, normalized size = 0.51 \[ -\frac{\sqrt{1-a^2 x^2} \left (a^2 x^2-1\right ) \left (\frac{x}{a^3}-\frac{3}{a^4 (a x+1)}+\frac{1}{2 a^4 (a x+1)^2}-\frac{3 \log (a x+1)}{a^4}\right )}{x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.177, size = 106, normalized size = 0.6 \begin{align*}{\frac{ \left ( -2\,{x}^{3}{a}^{3}+6\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-4\,{a}^{2}{x}^{2}+12\,ax\ln \left ( ax+1 \right ) +4\,ax+6\,\ln \left ( ax+1 \right ) +5 \right ) \left ( ax-1 \right ) }{ \left ( 2\,ax+2 \right ){a}^{4}{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a^{2} x^{2}}\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 = 2.1977, size = 994, normalized size = 5.85 \begin{align*} \left [-\frac{3 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt{-c} \log \left (\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x +{\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) +{\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right )}}, \frac{6 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt{c} \arctan \left (\frac{{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) -{\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right )}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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