Optimal. Leaf size=183 \[ \frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{4 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{3 a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.197224, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6192, 6193, 44, 207} \[ \frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{4 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{3 a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{\left (a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{(-1+a x)^2 (1+a x)^3} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{\left (a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (\frac{1}{8 (-1+a x)^2}+\frac{1}{4 (1+a x)^3}+\frac{1}{4 (1+a x)^2}-\frac{3}{8 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x)^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{4 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{8 \left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x)^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{4 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac{3 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0658235, size = 81, normalized size = 0.44 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (-3 a^2 x^2-3 a x+3 (a x-1) (a x+1)^2 \tanh ^{-1}(a x)+2\right )}{8 (a x-1) (a c x+c)^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.146, size = 169, normalized size = 0.9 \begin{align*} -{\frac{3\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+3\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-3\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-6\,{a}^{2}{x}^{2}-3\,ax\ln \left ( ax+1 \right ) +3\,\ln \left ( ax-1 \right ) xa-6\,ax-3\,\ln \left ( ax+1 \right ) +3\,\ln \left ( ax-1 \right ) +4}{ \left ( 16\,ax+16 \right ) \left ({a}^{2}{x}^{2}-1 \right ){c}^{3}a \left ( ax-1 \right ) }\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93605, size = 278, normalized size = 1.52 \begin{align*} -\frac{3 \,{\left (a^{4} x^{3} + a^{3} x^{2} - a^{2} x - a\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c} \sqrt{-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \,{\left (3 \, a^{2} x^{2} + 3 \, a x - 2\right )} \sqrt{-a^{2} c}}{16 \,{\left (a^{5} c^{3} x^{3} + a^{4} c^{3} x^{2} - a^{3} c^{3} x - a^{2} c^{3}\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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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