Optimal. Leaf size=22 \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6128, 261} \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x (c-a c x) \, dx &=c \int x \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 1.32 \[ \frac {{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 18, normalized size = 0.82 \[ -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 1.50 \[ -\frac {\left (a x -1\right )^{2} \left (a x +1\right )^{2} c}{3 a^{2} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 37, normalized size = 1.68 \[ \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} c x^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 18, normalized size = 0.82 \[ -\frac {c\,{\left (1-a^2\,x^2\right )}^{3/2}}{3\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 42, normalized size = 1.91 \[ \begin {cases} \frac {c x^{2} \sqrt {- a^{2} x^{2} + 1}}{3} - \frac {c \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} & \text {for}\: a \neq 0 \\\frac {c x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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