Optimal. Leaf size=121 \[ \frac {3 c^3 \sin ^{-1}(a x)}{16 a^3}+\frac {2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {3 c^3 x \sqrt {1-a^2 x^2}}{16 a^2}-\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3} \]
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Rubi [A] time = 0.20, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6128, 1809, 833, 780, 195, 216} \[ -\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac {3 c^3 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {3 c^3 \sin ^{-1}(a x)}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 780
Rule 833
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^2 (c-a c x)^3 \, dx &=c \int x^2 (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x^2 \left (-9 a^2 c^2+12 a^3 c^2 x\right ) \sqrt {1-a^2 x^2} \, dx}{6 a^2}\\ &=\frac {2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {c \int x \left (-24 a^3 c^2+45 a^4 c^2 x\right ) \sqrt {1-a^2 x^2} \, dx}{30 a^4}\\ &=\frac {2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac {\left (3 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx}{8 a^2}\\ &=\frac {3 c^3 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^2}\\ &=\frac {3 c^3 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac {3 c^3 \sin ^{-1}(a x)}{16 a^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.69 \[ \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (40 a^5 x^5-96 a^4 x^4+50 a^3 x^3+32 a^2 x^2-45 a x+64\right )-90 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 104, normalized size = 0.86 \[ -\frac {90 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (40 \, a^{5} c^{3} x^{5} - 96 \, a^{4} c^{3} x^{4} + 50 \, a^{3} c^{3} x^{3} + 32 \, a^{2} c^{3} x^{2} - 45 \, a c^{3} x + 64 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 92, normalized size = 0.76 \[ \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, a^{2} {\left | a \right |}} + \frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (\frac {16 \, c^{3}}{a} + {\left (25 \, c^{3} + 4 \, {\left (5 \, a^{2} c^{3} x - 12 \, a c^{3}\right )} x\right )} x\right )} x - \frac {45 \, c^{3}}{a^{2}}\right )} x + \frac {64 \, c^{3}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 163, normalized size = 1.35 \[ \frac {c^{3} a^{2} x^{5} \sqrt {-a^{2} x^{2}+1}}{6}+\frac {5 c^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{24}-\frac {3 c^{3} x \sqrt {-a^{2} x^{2}+1}}{16 a^{2}}+\frac {3 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{2} \sqrt {a^{2}}}-\frac {2 c^{3} a \,x^{4} \sqrt {-a^{2} x^{2}+1}}{5}+\frac {2 c^{3} x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a}+\frac {4 c^{3} \sqrt {-a^{2} x^{2}+1}}{15 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 141, normalized size = 1.17 \[ \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{5} - \frac {2}{5} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{4} + \frac {5}{24} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{3} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{2}}{15 \, a} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c^{3} x}{16 \, a^{2}} + \frac {3 \, c^{3} \arcsin \left (a x\right )}{16 \, a^{3}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 154, normalized size = 1.27 \[ \frac {4\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^3}+\frac {5\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{24}-\frac {3\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16\,a^2}-\frac {2\,a\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{5}+\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^2\,\sqrt {-a^2}}+\frac {2\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a}+\frac {a^2\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.48, size = 423, normalized size = 3.50 \[ - a^{4} c^{3} \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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