Optimal. Leaf size=124 \[ -\frac {c^2 \sin ^{-1}(a x)}{16 a^4}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3} \]
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Rubi [A] time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6128, 833, 780, 195, 216} \[ \frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \sin ^{-1}(a x)}{16 a^4} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 780
Rule 833
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^3 (c-a c x)^2 \, dx &=c \int x^3 (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c \int x^2 \left (3 a c-6 a^2 c x\right ) \sqrt {1-a^2 x^2} \, dx}{6 a^2}\\ &=-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}+\frac {c \int x \left (12 a^2 c-15 a^3 c x\right ) \sqrt {1-a^2 x^2} \, dx}{30 a^4}\\ &=-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \int \sqrt {1-a^2 x^2} \, dx}{8 a^3}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \sin ^{-1}(a x)}{16 a^4}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 89, normalized size = 0.72 \[ -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (40 a^5 x^5-48 a^4 x^4-10 a^3 x^3+16 a^2 x^2-15 a x+32\right )-60 \sin ^{-1}(a x)-150 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 104, normalized size = 0.84 \[ \frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (40 \, a^{5} c^{2} x^{5} - 48 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 16 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x + 32 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 92, normalized size = 0.74 \[ -\frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, a c^{2} x - 6 \, c^{2}\right )} x - \frac {5 \, c^{2}}{a}\right )} x + \frac {8 \, c^{2}}{a^{2}}\right )} x - \frac {15 \, c^{2}}{a^{3}}\right )} x + \frac {32 \, c^{2}}{a^{4}}\right )} - \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, a^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 163, normalized size = 1.31 \[ -\frac {c^{2} a \,x^{5} \sqrt {-a^{2} x^{2}+1}}{6}+\frac {c^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a}+\frac {c^{2} x \sqrt {-a^{2} x^{2}+1}}{16 a^{3}}-\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{3} \sqrt {a^{2}}}+\frac {c^{2} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {c^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {2 c^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 141, normalized size = 1.14 \[ -\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{5} + \frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{4} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x^{3}}{24 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x^{2}}{15 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x}{16 \, a^{3}} - \frac {c^{2} \arcsin \left (a x\right )}{16 \, a^{4}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{15 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 154, normalized size = 1.24 \[ \frac {c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {2\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a^4}+\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{16\,a^3}-\frac {a\,c^2\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^3\,\sqrt {-a^2}}+\frac {c^2\,x^3\,\sqrt {1-a^2\,x^2}}{24\,a}-\frac {c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.36, size = 486, normalized size = 3.92 \[ a^{3} c^{2} \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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