Optimal. Leaf size=55 \[ \frac{(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{3 \coth ^{-1}(a x)}}{15 a c^2} \]
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Rubi [A] time = 0.0650561, 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} \[ \frac{(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{3 \coth ^{-1}(a x)}}{15 a c^2} \]
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 = 0.149389, 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 = 0.131, 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 ){c}^{2}a} \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 = 1.31571, size = 74, normalized size = 1.35 \begin{align*} \frac{\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49653, size = 161, normalized size = 2.93 \begin{align*} -\frac{{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, 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 = 1.14126, size = 93, normalized size = 1.69 \begin{align*} \frac{{\left (a x + 1\right )}^{2}{\left (\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3\right )}}{60 \,{\left (a x - 1\right )}^{2} a c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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