Optimal. Leaf size=94 \[ -\frac {1}{5} c^3 x^2 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (14-15 a x) \left (1-a^2 x^2\right )^{3/2}}{30 a^2}-\frac {c^3 x \sqrt {1-a^2 x^2}}{4 a}-\frac {c^3 \sin ^{-1}(a x)}{4 a^2} \]
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Rubi [A] time = 0.13, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6128, 1809, 780, 195, 216} \[ -\frac {1}{5} c^3 x^2 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (14-15 a x) \left (1-a^2 x^2\right )^{3/2}}{30 a^2}-\frac {c^3 x \sqrt {1-a^2 x^2}}{4 a}-\frac {c^3 \sin ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 780
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x (c-a c x)^3 \, dx &=c \int x (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {1}{5} c^3 x^2 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x \left (-7 a^2 c^2+10 a^3 c^2 x\right ) \sqrt {1-a^2 x^2} \, dx}{5 a^2}\\ &=-\frac {1}{5} c^3 x^2 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (14-15 a x) \left (1-a^2 x^2\right )^{3/2}}{30 a^2}-\frac {c^3 \int \sqrt {1-a^2 x^2} \, dx}{2 a}\\ &=-\frac {c^3 x \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{5} c^3 x^2 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (14-15 a x) \left (1-a^2 x^2\right )^{3/2}}{30 a^2}-\frac {c^3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {c^3 x \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{5} c^3 x^2 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (14-15 a x) \left (1-a^2 x^2\right )^{3/2}}{30 a^2}-\frac {c^3 \sin ^{-1}(a x)}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 75, normalized size = 0.80 \[ \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (12 a^4 x^4-30 a^3 x^3+16 a^2 x^2+15 a x-28\right )+30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{60 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 92, normalized size = 0.98 \[ \frac {30 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (12 \, a^{4} c^{3} x^{4} - 30 \, a^{3} c^{3} x^{3} + 16 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 28 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{60 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 81, normalized size = 0.86 \[ -\frac {c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{4 \, a {\left | a \right |}} + \frac {1}{60} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (\frac {15 \, c^{3}}{a} + 2 \, {\left (8 \, c^{3} + 3 \, {\left (2 \, a^{2} c^{3} x - 5 \, a c^{3}\right )} x\right )} x\right )} x - \frac {28 \, c^{3}}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 140, normalized size = 1.49 \[ \frac {c^{3} a^{2} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}+\frac {4 c^{3} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {7 c^{3} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {c^{3} a \,x^{3} \sqrt {-a^{2} x^{2}+1}}{2}+\frac {c^{3} x \sqrt {-a^{2} x^{2}+1}}{4 a}-\frac {c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{4 a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 118, normalized size = 1.26 \[ \frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{4} - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{3} + \frac {4}{15} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{2} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3} x}{4 \, a} - \frac {c^{3} \arcsin \left (a x\right )}{4 \, a^{2}} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{15 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 108, normalized size = 1.15 \[ \frac {c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{4\,a^3}-\frac {\frac {2\,c^3\,{\left (1-a^2\,x^2\right )}^{3/2}}{3}-\frac {c^3\,{\left (1-a^2\,x^2\right )}^{5/2}}{5}}{a^2}-\frac {\frac {c^3\,x\,\sqrt {1-a^2\,x^2}}{4}-\frac {c^3\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{2}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.71, size = 355, normalized size = 3.78 \[ - a^{4} c^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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