Optimal. Leaf size=87 \[ \frac {3 \sin ^{-1}(a x)}{8 a^4}-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{24 a^4} \]
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Rubi [A] time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6124, 833, 780, 216} \[ -\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2}-\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{24 a^4}+\frac {3 \sin ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 833
Rule 6124
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1+a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {\int \frac {x^2 \left (-3 a-4 a^2 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {\int \frac {x \left (8 a^2+9 a^3 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{12 a^4}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {(16+9 a x) \sqrt {1-a^2 x^2}}{24 a^4}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {(16+9 a x) \sqrt {1-a^2 x^2}}{24 a^4}+\frac {3 \sin ^{-1}(a x)}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.60 \[ \frac {9 \sin ^{-1}(a x)-\sqrt {1-a^2 x^2} \left (6 a^3 x^3+8 a^2 x^2+9 a x+16\right )}{24 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 65, normalized size = 0.75 \[ -\frac {{\left (6 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 \, a x + 16\right )} \sqrt {-a^{2} x^{2} + 1} + 18 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{24 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 59, normalized size = 0.68 \[ -\frac {1}{24} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{a} + \frac {4}{a^{2}}\right )} + \frac {9}{a^{3}}\right )} x + \frac {16}{a^{4}}\right )} + \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, a^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 107, normalized size = 1.23 \[ -\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a}-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{3}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{3} \sqrt {a^{2}}}-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 85, normalized size = 0.98 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{4 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac {3 \, \arcsin \left (a x\right )}{8 \, a^{4}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 97, normalized size = 1.11 \[ \frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^3\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,{\left (-a^2\right )}^{3/2}}+\frac {3\,x\,\sqrt {-a^2}}{8\,a^3}+\frac {a^2\,x^2}{3\,{\left (-a^2\right )}^{3/2}}-\frac {x^3\,{\left (-a^2\right )}^{3/2}}{4\,a^3}\right )}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.46, size = 199, normalized size = 2.29 \[ a \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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