Optimal. Leaf size=52 \[ \frac{x}{3 c \sqrt{c-a^2 c x^2}}-\frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0642532, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6142, 653, 191} \[ \frac{x}{3 c \sqrt{c-a^2 c x^2}}-\frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6142
Rule 653
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac{(1-a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=-\frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac{1}{3} \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac{x}{3 c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0362213, size = 63, normalized size = 1.21 \[ -\frac{\sqrt{1-a x} (a x+2) \sqrt{1-a^2 x^2}}{3 a c (a x+1)^{3/2} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.029, size = 31, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2} \left ( ax+2 \right ) }{3\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.46851, size = 99, normalized size = 1.9 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x + 2\right )}}{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*} - \int \frac{a x}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} + c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int - \frac{1}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} + c \sqrt{- a^{2} c x^{2} + c}}\, dx \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{a^{2} x^{2} - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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