Optimal. Leaf size=265 \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{\left (1-a^2 x^2\right )^{5/2}}{a^6 x^5 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (a x+1)^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{7 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{23 \left (1-a^2 x^2\right )^{5/2} \log (a x+1)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.211101, antiderivative size = 265, 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, 88} \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{\left (1-a^2 x^2\right )^{5/2}}{a^6 x^5 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (a x+1)^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{7 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{23 \left (1-a^2 x^2\right )^{5/2} \log (a x+1)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \frac{e^{-\tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \frac{x^5}{(1-a x)^2 (1+a x)^3} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \left (\frac{1}{a^5}+\frac{1}{8 a^5 (-1+a x)^2}+\frac{7}{16 a^5 (-1+a x)}-\frac{1}{4 a^5 (1+a x)^3}+\frac{1}{a^5 (1+a x)^2}-\frac{23}{16 a^5 (1+a x)}\right ) \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}+\frac{\left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 (1+a x)^2}-\frac{\left (1-a^2 x^2\right )^{5/2}}{a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 (1+a x)}+\frac{7 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{23 \left (1-a^2 x^2\right )^{5/2} \log (1+a x)}{16 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ \end{align*}
Mathematica [A] time = 0.124133, size = 88, normalized size = 0.33 \[ \frac{\left (1-a^2 x^2\right )^{5/2} \left (2 \left (8 a x+\frac{1}{1-a x}-\frac{8}{a x+1}+\frac{1}{(a x+1)^2}\right )+7 \log (1-a x)-23 \log (a x+1)\right )}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 167, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( -16\,{x}^{4}{a}^{4}+23\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -7\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-16\,{x}^{3}{a}^{3}+23\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-7\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+34\,{a}^{2}{x}^{2}-23\,ax\ln \left ( ax+1 \right ) +7\,\ln \left ( ax-1 \right ) xa+18\,ax-23\,\ln \left ( ax+1 \right ) +7\,\ln \left ( ax-1 \right ) -12 \right ) }{16\,{a}^{6}{x}^{5}}\sqrt{-{a}^{2}{x}^{2}+1} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{5}{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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1} a^{6} x^{6} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{7} c^{3} x^{7} + a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} + 3 \, a^{2} c^{3} x^{2} - a c^{3} x - c^{3}}, 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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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