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 = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 43
Rule 266
Rule 6128
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.51, size = 39, normalized size = 0.87 \[ \frac {{\left (3 \, a^{4} c x^{4} - a^{2} c x^{2} - 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{15 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 47, normalized size = 1.04 \[ \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c - 5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{15 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 0.96 \[ -\frac {\left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (3 a^{2} x^{2}+2\right ) c}{15 a^{4} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 58, normalized size = 1.29 \[ \frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} c x^{4} - \frac {\sqrt {-a^{2} x^{2} + 1} c x^{2}}{15 \, a^{2}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c}{15 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.80 \[ -\frac {5\,c\,{\left (1-a^2\,x^2\right )}^{3/2}-3\,c\,{\left (1-a^2\,x^2\right )}^{5/2}}{15\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 66, normalized size = 1.47 \[ \begin {cases} \frac {c x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {c x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 c \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {c x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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