Optimal. Leaf size=45 \[ \frac{c \left (1-a^2 x^2\right )^{5/2}}{5 a^4}-\frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a^4} \]
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Rubi [A] time = 0.0649259, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6128, 266, 43} \[ \frac{c \left (1-a^2 x^2\right )^{5/2}}{5 a^4}-\frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a^4} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^3 (c-a c x) \, dx &=c \int x^3 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{2} c \operatorname{Subst}\left (\int x \sqrt{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{2} c \operatorname{Subst}\left (\int \left (\frac{\sqrt{1-a^2 x}}{a^2}-\frac{\left (1-a^2 x\right )^{3/2}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a^4}+\frac{c \left (1-a^2 x^2\right )^{5/2}}{5 a^4}\\ \end{align*}
Mathematica [A] time = 0.0241801, size = 32, normalized size = 0.71 \[ -\frac{c \left (1-a^2 x^2\right )^{3/2} \left (3 a^2 x^2+2\right )}{15 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 43, normalized size = 1. \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2} \left ( ax+1 \right ) ^{2} \left ( 3\,{a}^{2}{x}^{2}+2 \right ) c}{15\,{a}^{4}}{\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 [A] time = 1.42526, size = 78, normalized size = 1.73 \begin{align*} \frac{1}{5} \, \sqrt{-a^{2} x^{2} + 1} c x^{4} - \frac{\sqrt{-a^{2} x^{2} + 1} c x^{2}}{15 \, a^{2}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c}{15 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5817, size = 82, normalized size = 1.82 \begin{align*} \frac{{\left (3 \, a^{4} c x^{4} - a^{2} c x^{2} - 2 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{15 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00177, size = 66, normalized size = 1.47 \begin{align*} \begin{cases} \frac{c x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{c x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 c \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{c x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20941, size = 63, normalized size = 1.4 \begin{align*} \frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} c - 5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{15 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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