Optimal. Leaf size=83 \[ \frac {c \sin ^{-1}(a x)}{16 a^5}+\frac {1}{6} c x^5 \sqrt {1-a^2 x^2}-\frac {c x^3 \sqrt {1-a^2 x^2}}{24 a^2}-\frac {c x \sqrt {1-a^2 x^2}}{16 a^4} \]
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Rubi [A] time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 216
Rule 279
Rule 321
Rule 6128
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*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.60 \[ \frac {c \left (a x \sqrt {1-a^2 x^2} \left (8 a^4 x^4-2 a^2 x^2-3\right )+3 \sin ^{-1}(a x)\right )}{48 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 69, normalized size = 0.83 \[ -\frac {6 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (8 \, a^{5} c x^{5} - 2 \, a^{3} c x^{3} - 3 \, a c x\right )} \sqrt {-a^{2} x^{2} + 1}}{48 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 57, normalized size = 0.69 \[ \frac {1}{48} \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, {\left (4 \, c x^{2} - \frac {c}{a^{2}}\right )} x^{2} - \frac {3 \, c}{a^{4}}\right )} x + \frac {c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, a^{4} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 91, normalized size = 1.10 \[ \frac {c \,x^{5} \sqrt {-a^{2} x^{2}+1}}{6}-\frac {c \,x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}-\frac {c x \sqrt {-a^{2} x^{2}+1}}{16 a^{4}}+\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{4} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 69, normalized size = 0.83 \[ \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} c x^{5} - \frac {\sqrt {-a^{2} x^{2} + 1} c x^{3}}{24 \, a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} c x}{16 \, a^{4}} + \frac {c \arcsin \left (a x\right )}{16 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 82, normalized size = 0.99 \[ \frac {c\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {c\,x^3\,\sqrt {1-a^2\,x^2}}{24\,a^2}-\frac {c\,x\,\sqrt {1-a^2\,x^2}}{16\,a^4}+\frac {c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^4\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.53, size = 192, normalized size = 2.31 \[ c \left (\begin {cases} \frac {i a^{2} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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