Optimal. Leaf size=182 \[ \frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{6 (a x+1)^3 \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.198197, antiderivative size = 182, 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} \[ \frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{6 (a x+1)^3 \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-3 \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^{-3 \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) (1+a x)^4} \, 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}{2 (1+a x)^4}-\frac{1}{4 (1+a x)^3}-\frac{1}{8 (1+a x)^2}+\frac{1}{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}{6 (1+a x)^3 \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}{8 (1+a x) \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^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}{6 (1+a x)^3 \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}{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 \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0630013, size = 71, normalized size = 0.39 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (-3 a^2 x^2-9 a x+3 (a x+1)^3 \tanh ^{-1}(a x)-10\right )}{24 c^2 (a x+1)^3 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.144, 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}+9\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-9\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-6\,{a}^{2}{x}^{2}+9\,ax\ln \left ( ax+1 \right ) -9\,\ln \left ( ax-1 \right ) xa-18\,ax+3\,\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ) -20}{ \left ( 48\,ax+48 \right ) \left ( ax-1 \right ) \left ({a}^{2}{x}^{2}-1 \right ){c}^{3}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\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 = 1.60945, size = 290, normalized size = 1.59 \begin{align*} -\frac{3 \,{\left (a^{4} x^{3} + 3 \, 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 (3 \, a^{2} x^{2} + 9 \, a x + 10\right )} \sqrt{-a^{2} c}}{48 \,{\left (a^{5} c^{3} x^{3} + 3 \, a^{4} c^{3} x^{2} + 3 \, 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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\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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