Optimal. Leaf size=111 \[ \frac {3 \sin ^{-1}(a x)}{8 a^5}-\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {(45 a x+64) \sqrt {1-a^2 x^2}}{120 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2}}{15 a^3} \]
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Rubi [A] time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6124, 833, 780, 216} \[ -\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {4 x^2 \sqrt {1-a^2 x^2}}{15 a^3}-\frac {(45 a x+64) \sqrt {1-a^2 x^2}}{120 a^5}+\frac {3 \sin ^{-1}(a x)}{8 a^5} \]
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^4 \, dx &=\int \frac {x^4 (1+a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}-\frac {\int \frac {x^3 \left (-4 a-5 a^2 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{5 a^2}\\ &=-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}+\frac {\int \frac {x^2 \left (15 a^2+16 a^3 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{20 a^4}\\ &=-\frac {4 x^2 \sqrt {1-a^2 x^2}}{15 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}-\frac {\int \frac {x \left (-32 a^3-45 a^4 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{60 a^6}\\ &=-\frac {4 x^2 \sqrt {1-a^2 x^2}}{15 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}-\frac {(64+45 a x) \sqrt {1-a^2 x^2}}{120 a^5}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}\\ &=-\frac {4 x^2 \sqrt {1-a^2 x^2}}{15 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {x^4 \sqrt {1-a^2 x^2}}{5 a}-\frac {(64+45 a x) \sqrt {1-a^2 x^2}}{120 a^5}+\frac {3 \sin ^{-1}(a x)}{8 a^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.54 \[ \frac {45 \sin ^{-1}(a x)-\sqrt {1-a^2 x^2} \left (24 a^4 x^4+30 a^3 x^3+32 a^2 x^2+45 a x+64\right )}{120 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 73, normalized size = 0.66 \[ -\frac {{\left (24 \, a^{4} x^{4} + 30 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 45 \, a x + 64\right )} \sqrt {-a^{2} x^{2} + 1} + 90 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 127, normalized size = 1.14 \[ -\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a}-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{3}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{5}}-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{4}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{4} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 105, normalized size = 0.95 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{5 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{4 \, a^{2}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{15 \, a^{3}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{4}} + \frac {3 \, \arcsin \left (a x\right )}{8 \, a^{5}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 112, normalized size = 1.01 \[ \frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^4\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8}{15\,a^3\,\sqrt {-a^2}}+\frac {a\,x^4}{5\,\sqrt {-a^2}}-\frac {3\,x\,\sqrt {-a^2}}{8\,a^4}+\frac {4\,x^2}{15\,a\,\sqrt {-a^2}}+\frac {x^3\,{\left (-a^2\right )}^{3/2}}{4\,a^4}\right )}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.75, size = 221, normalized size = 1.99 \[ a \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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