Optimal. Leaf size=278 \[ -\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac{3 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac{5 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \tanh ^{-1}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.225189, antiderivative size = 278, 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} \[ -\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac{3 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac{5 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \tanh ^{-1}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7} \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=\frac{\left (a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{(-1+a x)^5 (1+a x)^2} \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=\frac{\left (a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (\frac{1}{4 (-1+a x)^5}-\frac{1}{4 (-1+a x)^4}+\frac{3}{16 (-1+a x)^3}-\frac{1}{8 (-1+a x)^2}-\frac{1}{32 (1+a x)^2}+\frac{5}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{\left (5 a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{32 \left (c-a^2 c x^2\right )^{7/2}}\\ &=-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac{5 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0875844, size = 99, normalized size = 0.36 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (-15 a^4 x^4+45 a^3 x^3-35 a^2 x^2-15 a x+15 (a x-1)^4 (a x+1) \tanh ^{-1}(a x)+32\right )}{96 c^3 (a x-1)^4 (a x+1) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 241, normalized size = 0.9 \begin{align*} -{\frac{15\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-15\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-45\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}+45\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-30\,{x}^{4}{a}^{4}+30\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -30\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+90\,{x}^{3}{a}^{3}+30\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-30\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-70\,{a}^{2}{x}^{2}-45\,ax\ln \left ( ax+1 \right ) +45\,\ln \left ( ax-1 \right ) xa-30\,ax+15\,\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) +64}{192\, \left ( ax-1 \right ) ^{2} \left ( ax+1 \right ) ^{2} \left ({a}^{2}{x}^{2}-1 \right ){c}^{4}a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{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{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96986, size = 406, normalized size = 1.46 \begin{align*} -\frac{15 \,{\left (a^{6} x^{5} - 3 \, a^{5} x^{4} + 2 \, a^{4} x^{3} + 2 \, a^{3} x^{2} - 3 \, 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 (15 \, a^{4} x^{4} - 45 \, a^{3} x^{3} + 35 \, a^{2} x^{2} + 15 \, a x - 32\right )} \sqrt{-a^{2} c}}{192 \,{\left (a^{7} c^{4} x^{5} - 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x + a^{2} c^{4}\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{7}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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