Optimal. Leaf size=252 \[ \frac{a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^4 x^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{7 a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.271751, antiderivative size = 252, 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} \[ \frac{a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^4 x^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{7 a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
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^3 \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^6} \, 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^3 (-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^3}+\frac{a}{x^2}+\frac{2 a^2}{x}+\frac{a^3}{2 (-1+a x)^2}-\frac{7 a^3}{4 (-1+a x)}-\frac{a^3}{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^2}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{a^5 \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{2 a^5 \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{7 a^5 \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^5 \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 = 0.080199, size = 94, normalized size = 0.37 \[ \frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{2 a^2}{1-a x}+8 a^2 \log (x)-7 a^2 \log (1-a x)-a^2 \log (a x+1)-\frac{4 a}{x}-\frac{2}{x^2}\right )}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.143, size = 138, normalized size = 0.6 \begin{align*}{\frac{8\,{a}^{3}\ln \left ( x \right ){x}^{3}-{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -7\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-8\,{a}^{2}\ln \left ( x \right ){x}^{2}+\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+7\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-6\,{a}^{2}{x}^{2}+2\,ax+2}{ \left ( 4\,{a}^{2}{x}^{2}-4 \right ){c}^{2}{x}^{2}}\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^{3} \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 = 1.72366, size = 232, normalized size = 0.92 \begin{align*} -\frac{{\left (6 \, a^{2} x^{2} - 2 \, a x +{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 7 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 8 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 2\right )} \sqrt{-a^{2} c}}{4 \,{\left (a^{2} c^{2} x^{3} - a c^{2} x^{2}\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^{3} \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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