Optimal. Leaf size=70 \[ -\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 \sin ^{-1}(a x)}{8 a^2} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6128, 780, 195, 216} \[ -\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 \sin ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 780
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x (c-a c x)^2 \, dx &=c \int x (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \int \sqrt {1-a^2 x^2} \, dx}{4 a}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \sin ^{-1}(a x)}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 67, normalized size = 0.96 \[ -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (6 a^3 x^3-8 a^2 x^2-3 a x+8\right )-6 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 82, normalized size = 1.17 \[ \frac {6 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 69, normalized size = 0.99 \[ -\frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, a {\left | a \right |}} - \frac {1}{24} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (3 \, a c^{2} x - 4 \, c^{2}\right )} x - \frac {3 \, c^{2}}{a}\right )} x + \frac {8 \, c^{2}}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 117, normalized size = 1.67 \[ -\frac {c^{2} a \,x^{3} \sqrt {-a^{2} x^{2}+1}}{4}+\frac {c^{2} x \sqrt {-a^{2} x^{2}+1}}{8 a}-\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a \sqrt {a^{2}}}+\frac {c^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 95, normalized size = 1.36 \[ -\frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{3} + \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{2} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x}{8 \, a} - \frac {c^{2} \arcsin \left (a x\right )}{8 \, a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 108, normalized size = 1.54 \[ \frac {c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2}+\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8\,a}-\frac {a\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.17, size = 330, normalized size = 4.71 \[ a^{3} c^{2} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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