Optimal. Leaf size=113 \[ \frac {c^2 \sin ^{-1}(a x)}{8 a^3}-\frac {c^2 x \left (1-a^2 x^2\right )^{3/2}}{4 a^2}+\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a^2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3} \]
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Rubi [A] time = 0.12, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6128, 797, 641, 195, 216} \[ -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {c^2 x \left (1-a^2 x^2\right )^{3/2}}{4 a^2}+\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a^2}+\frac {c^2 \sin ^{-1}(a x)}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 797
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^2 (c-a c x)^2 \, dx &=c \int x^2 (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c \int (c-a c x) \sqrt {1-a^2 x^2} \, dx}{a^2}-\frac {c \int (c-a c x) \left (1-a^2 x^2\right )^{3/2} \, dx}{a^2}\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \int \sqrt {1-a^2 x^2} \, dx}{a^2}-\frac {c^2 \int \left (1-a^2 x^2\right )^{3/2} \, dx}{a^2}\\ &=\frac {c^2 x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {c^2 x \left (1-a^2 x^2\right )^{3/2}}{4 a^2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}-\frac {\left (3 c^2\right ) \int \sqrt {1-a^2 x^2} \, dx}{4 a^2}\\ &=\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {c^2 x \left (1-a^2 x^2\right )^{3/2}}{4 a^2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \sin ^{-1}(a x)}{2 a^3}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2}\\ &=\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {c^2 x \left (1-a^2 x^2\right )^{3/2}}{4 a^2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \sin ^{-1}(a x)}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 75, normalized size = 0.66 \[ -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (24 a^4 x^4-30 a^3 x^3-8 a^2 x^2+15 a x-16\right )+30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 92, normalized size = 0.81 \[ -\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c^{2} x^{4} - 30 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 16 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 81, normalized size = 0.72 \[ -\frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (4 \, a c^{2} x - 5 \, c^{2}\right )} x - \frac {4 \, c^{2}}{a}\right )} x + \frac {15 \, c^{2}}{a^{2}}\right )} x - \frac {16 \, c^{2}}{a^{3}}\right )} + \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, a^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 140, normalized size = 1.24 \[ -\frac {c^{2} a \,x^{4} \sqrt {-a^{2} x^{2}+1}}{5}+\frac {c^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a}+\frac {2 c^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{3}}+\frac {c^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{4}-\frac {c^{2} x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 118, normalized size = 1.04 \[ -\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{4} + \frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{3} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x^{2}}{15 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x}{8 \, a^{2}} + \frac {c^{2} \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 131, normalized size = 1.16 \[ \frac {2\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a^3}+\frac {c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8\,a^2}-\frac {a\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5}+\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^2\,\sqrt {-a^2}}+\frac {c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.52, size = 374, normalized size = 3.31 \[ a^{3} c^{2} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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