Optimal. Leaf size=55 \[ \frac{(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{-\coth ^{-1}(a x)}}{3 a c^2} \]
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Rubi [A] time = 0.0622484, 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} \[ \frac{(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{-\coth ^{-1}(a x)}}{3 a c^2} \]
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 = 0.140154, size = 48, normalized size = 0.87 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+2 a x-1\right )}{3 (a x-1) (a c x+c)^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 49, normalized size = 0.9 \begin{align*} -{\frac{2\,{a}^{2}{x}^{2}+2\,ax-1}{ \left ( 3\,{a}^{2}{x}^{2}-3 \right ) a{c}^{2}}\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 = 0.994994, size = 90, normalized size = 1.64 \begin{align*} \frac{1}{12} \, a{\left (\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 6 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac{3}{a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64564, size = 105, normalized size = 1.91 \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} - 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{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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