Optimal. Leaf size=359 \[ \frac{\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1)^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{\left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (a x+1)^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{\left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1)^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{201 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{64 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.251201, antiderivative size = 359, 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} \[ \frac{\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1)^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{\left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (a x+1)^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{\left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1)^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{201 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{64 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{7/2} \int \frac{e^{-3 \tanh ^{-1}(a x)} x^7}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (1-a^2 x^2\right )^{7/2} \int \frac{x^7}{(1-a x)^2 (1+a x)^5} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (1-a^2 x^2\right )^{7/2} \int \left (\frac{1}{a^7}+\frac{1}{32 a^7 (-1+a x)^2}+\frac{9}{64 a^7 (-1+a x)}-\frac{1}{4 a^7 (1+a x)^5}+\frac{3}{2 a^7 (1+a x)^4}-\frac{59}{16 a^7 (1+a x)^3}+\frac{75}{16 a^7 (1+a x)^2}-\frac{201}{64 a^7 (1+a x)}\right ) \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (1-a^2 x^2\right )^{7/2}}{a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}+\frac{\left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^4}-\frac{\left (1-a^2 x^2\right )^{7/2}}{2 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^3}+\frac{59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^2}-\frac{75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)}+\frac{9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}-\frac{201 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{64 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ \end{align*}
Mathematica [A] time = 0.121583, size = 146, normalized size = 0.41 \[ \frac{\sqrt{1-a^2 x^2} \left (-2 \left (32 a^6 x^6+96 a^5 x^5-87 a^4 x^4-309 a^3 x^3-59 a^2 x^2+207 a x+104\right )-9 (a x-1) (a x+1)^4 \log (1-a x)+201 (a x-1) (a x+1)^4 \log (a x+1)\right )}{64 a^2 c^3 x (a x-1) (a x+1)^4 \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.173, size = 248, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2} \left ( -64\,{x}^{6}{a}^{6}+201\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-9\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-192\,{x}^{5}{a}^{5}+603\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-27\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}+174\,{x}^{4}{a}^{4}+402\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -18\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+618\,{x}^{3}{a}^{3}-402\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+18\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+118\,{a}^{2}{x}^{2}-603\,ax\ln \left ( ax+1 \right ) +27\,\ln \left ( ax-1 \right ) xa-414\,ax-201\,\ln \left ( ax+1 \right ) +9\,\ln \left ( ax-1 \right ) -208 \right ) }{ \left ( 64\,ax+64 \right ){a}^{8}{x}^{7}}\sqrt{-{a}^{2}{x}^{2}+1} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{7}{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{7}{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^{8} x^{8} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} + 3 \, a^{8} c^{4} x^{8} - 8 \, a^{6} c^{4} x^{6} - 6 \, a^{5} c^{4} x^{5} + 6 \, a^{4} c^{4} x^{4} + 8 \, a^{3} c^{4} x^{3} - 3 \, a c^{4} x - c^{4}}, 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{{\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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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