Optimal. Leaf size=55 \[ \frac{a x+1}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x}{3 a c^2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0637577, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6148, 778, 191} \[ \frac{a x+1}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x}{3 a c^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 778
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{x (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac{1+a x}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a c^2}\\ &=\frac{1+a x}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x}{3 a c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0193405, size = 44, normalized size = 0.8 \[ \frac{-a^2 x^2+a x-1}{3 a^2 c^2 (a x-1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 41, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}{x}^{2}-ax+1}{ \left ( 3\,ax-3 \right ){c}^{2}{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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*} a \int \frac{x^{2}}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}\,{d x} + \frac{1}{3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52715, size = 170, normalized size = 3.09 \begin{align*} \frac{a^{3} x^{3} - a^{2} x^{2} - a x +{\left (a^{2} x^{2} - a x + 1\right )} \sqrt{-a^{2} x^{2} + 1} + 1}{3 \,{\left (a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} - a^{3} c^{2} x + a^{2} 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{x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- 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{{\left (a x + 1\right )} x}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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