Optimal. Leaf size=214 \[ \frac{a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{5 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.265469, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6192, 6193, 88} \[ \frac{a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{5 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^5} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{1}{x^2 (-1+a x)^2 (1+a x)} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (\frac{1}{x^2}+\frac{a}{x}+\frac{a^2}{2 (-1+a x)^2}-\frac{5 a^2}{4 (-1+a x)}+\frac{a^2}{4 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{5 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0651484, size = 79, normalized size = 0.37 \[ \frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{2 a}{1-a x}+4 a \log (x)-5 a \log (1-a x)+a \log (a x+1)-\frac{4}{x}\right )}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.136, size = 118, normalized size = 0.6 \begin{align*}{\frac{4\,{a}^{2}\ln \left ( x \right ){x}^{2}+\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-5\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-4\,a\ln \left ( x \right ) x-ax\ln \left ( ax+1 \right ) +5\,\ln \left ( ax-1 \right ) xa-6\,ax+4}{ \left ( 4\,{a}^{2}{x}^{2}-4 \right ){c}^{2}x}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{2} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72626, size = 197, normalized size = 0.92 \begin{align*} -\frac{\sqrt{-a^{2} c}{\left (6 \, a x -{\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 5 \,{\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \,{\left (a^{2} x^{2} - a x\right )} \log \left (x\right ) - 4\right )}}{4 \,{\left (a^{2} c^{2} x^{2} - a c^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{2} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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