Optimal. Leaf size=86 \[ \frac {3 \sinh ^{-1}(a x)^2}{16 a^5}+\frac {3 x^2}{16 a^3}+\frac {x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{8 a^4}-\frac {x^4}{16 a} \]
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Rubi [A] time = 0.15, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5758, 5675, 30} \[ \frac {3 x^2}{16 a^3}+\frac {x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{8 a^4}+\frac {3 \sinh ^{-1}(a x)^2}{16 a^5}-\frac {x^4}{16 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5675
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^4 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx &=\frac {x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{4 a^2}-\frac {3 \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}-\frac {\int x^3 \, dx}{4 a}\\ &=-\frac {x^4}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{4 a^2}+\frac {3 \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^4}+\frac {3 \int x \, dx}{8 a^3}\\ &=\frac {3 x^2}{16 a^3}-\frac {x^4}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{4 a^2}+\frac {3 \sinh ^{-1}(a x)^2}{16 a^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.73 \[ \frac {-a^4 x^4+3 a^2 x^2+2 a x \sqrt {a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)+3 \sinh ^{-1}(a x)^2}{16 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 83, normalized size = 0.97 \[ -\frac {a^{4} x^{4} - 3 \, a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{16 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 74, normalized size = 0.86 \[ \frac {4 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-x^{4} a^{4}-6 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +3 a^{2} x^{2}+3 \arcsinh \left (a x \right )^{2}+3}{16 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 83, normalized size = 0.97 \[ -\frac {1}{16} \, {\left (\frac {x^{4}}{a^{2}} - \frac {3 \, x^{2}}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )^{2}}{a^{6}}\right )} a + \frac {1}{8} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} x^{3}}{a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} \operatorname {arsinh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 82, normalized size = 0.95 \[ \begin {cases} - \frac {x^{4}}{16 a} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{2}}{16 a^{3}} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{8 a^{4}} + \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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