Optimal. Leaf size=55 \[ \frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.0662282, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ \frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}-\frac{2 \int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{5 c}\\ &=\frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.141018, size = 43, normalized size = 0.78 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+6 a x+7\right )}{15 c^2 (a x+1)^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 49, normalized size = 0.9 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}+6\,ax+7}{ \left ( 15\,{a}^{2}{x}^{2}-15 \right ) a{c}^{2}} \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 [A] time = 1.04872, size = 81, normalized size = 1.47 \begin{align*} \frac{3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 15 \, \sqrt{\frac{a x - 1}{a x + 1}}}{60 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57377, size = 124, normalized size = 2.25 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} + 6 \, a x + 7\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\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 [A] time = 1.17797, size = 88, normalized size = 1.6 \begin{align*} -\frac{4 \,{\left (10 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{2} x^{2} + 5 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}}{15 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}^{5} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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