Optimal. Leaf size=83 \[ \frac{1}{6} c x^5 \sqrt{1-a^2 x^2}-\frac{c x^3 \sqrt{1-a^2 x^2}}{24 a^2}-\frac{c x \sqrt{1-a^2 x^2}}{16 a^4}+\frac{c \sin ^{-1}(a x)}{16 a^5} \]
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Rubi [A] time = 0.0708394, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6128, 279, 321, 216} \[ \frac{1}{6} c x^5 \sqrt{1-a^2 x^2}-\frac{c x^3 \sqrt{1-a^2 x^2}}{24 a^2}-\frac{c x \sqrt{1-a^2 x^2}}{16 a^4}+\frac{c \sin ^{-1}(a x)}{16 a^5} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 279
Rule 321
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^4 (c-a c x) \, dx &=c \int x^4 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{6} c x^5 \sqrt{1-a^2 x^2}+\frac{1}{6} c \int \frac{x^4}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c x^3 \sqrt{1-a^2 x^2}}{24 a^2}+\frac{1}{6} c x^5 \sqrt{1-a^2 x^2}+\frac{c \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{8 a^2}\\ &=-\frac{c x \sqrt{1-a^2 x^2}}{16 a^4}-\frac{c x^3 \sqrt{1-a^2 x^2}}{24 a^2}+\frac{1}{6} c x^5 \sqrt{1-a^2 x^2}+\frac{c \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{16 a^4}\\ &=-\frac{c x \sqrt{1-a^2 x^2}}{16 a^4}-\frac{c x^3 \sqrt{1-a^2 x^2}}{24 a^2}+\frac{1}{6} c x^5 \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{16 a^5}\\ \end{align*}
Mathematica [A] time = 0.0570437, size = 50, normalized size = 0.6 \[ \frac{c \left (a x \sqrt{1-a^2 x^2} \left (8 a^4 x^4-2 a^2 x^2-3\right )+3 \sin ^{-1}(a x)\right )}{48 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 91, normalized size = 1.1 \begin{align*}{\frac{c{x}^{5}}{6}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{c{x}^{3}}{24\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{cx}{16\,{a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{c}{16\,{a}^{4}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43528, size = 109, normalized size = 1.31 \begin{align*} \frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1} c x^{5} - \frac{\sqrt{-a^{2} x^{2} + 1} c x^{3}}{24 \, a^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} c x}{16 \, a^{4}} + \frac{c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72732, size = 155, normalized size = 1.87 \begin{align*} -\frac{6 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (8 \, a^{5} c x^{5} - 2 \, a^{3} c x^{3} - 3 \, a c x\right )} \sqrt{-a^{2} x^{2} + 1}}{48 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 11.942, size = 357, normalized size = 4.3 \begin{align*} - a^{2} c \left (\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left (a x \right )}}{16 a^{7}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left (a x \right )}}{16 a^{7}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left (a x \right )}}{8 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8 a^{5}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19992, size = 77, normalized size = 0.93 \begin{align*} \frac{1}{48} \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \,{\left (4 \, c x^{2} - \frac{c}{a^{2}}\right )} x^{2} - \frac{3 \, c}{a^{4}}\right )} x + \frac{c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \, a^{4}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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