Optimal. Leaf size=51 \[ \frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.0604559, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6185, 6183} \[ \frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{\coth ^{-1}(a x)} (1-2 a x)}{3 a c^2 \left (1-a^2 x^2\right )}+\frac{2 \int \frac{e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{3 c}\\ &=\frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{e^{\coth ^{-1}(a x)} (1-2 a x)}{3 a c^2 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.143481, size = 50, normalized size = 0.98 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-2 a x-1\right )}{3 c^2 (a x-1)^2 (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.045, size = 49, normalized size = 1. \begin{align*}{\frac{2\,{a}^{2}{x}^{2}-2\,ax-1}{ \left ( 3\,{a}^{2}{x}^{2}-3 \right ){c}^{2}a}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06119, size = 88, normalized size = 1.73 \begin{align*} \frac{1}{12} \, a{\left (\frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac{\frac{6 \,{\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57265, size = 123, normalized size = 2.41 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 2 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10668, size = 104, normalized size = 2.04 \begin{align*} \frac{1}{12} \, a{\left (\frac{{\left (a x + 1\right )}{\left (\frac{6 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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