Optimal. Leaf size=96 \[ -\frac {3 \sinh ^{-1}(a x)^2}{32 a^4}-\frac {3 x^2}{32 a^2}-\frac {x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{8 a}+\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{16 a^3}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^2+\frac {x^4}{32} \]
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Rubi [A] time = 0.17, antiderivative size = 96, 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 = {5661, 5758, 5675, 30} \[ -\frac {3 x^2}{32 a^2}-\frac {x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{8 a}+\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{16 a^3}-\frac {3 \sinh ^{-1}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^2+\frac {x^4}{32} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5661
Rule 5675
Rule 5758
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a x)^2 \, dx &=\frac {1}{4} x^4 \sinh ^{-1}(a x)^2-\frac {1}{2} a \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)}{8 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^2+\frac {\int x^3 \, dx}{8}+\frac {3 \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a}\\ &=\frac {x^4}{32}+\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{16 a^3}-\frac {x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{8 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^2-\frac {3 \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 a^3}-\frac {3 \int x \, dx}{16 a^2}\\ &=-\frac {3 x^2}{32 a^2}+\frac {x^4}{32}+\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{16 a^3}-\frac {x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{8 a}-\frac {3 \sinh ^{-1}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.75 \[ \frac {\left (8 a^4 x^4-3\right ) \sinh ^{-1}(a x)^2+a^2 x^2 \left (a^2 x^2-3\right )-2 a x \sqrt {a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)}{32 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 92, normalized size = 0.96 \[ \frac {a^{4} x^{4} - 3 \, a^{2} x^{2} + {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{32 \, a^{4}} \]
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 = 0.38, size = 87, normalized size = 0.91 \[ \frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{8}+\frac {3 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{16}-\frac {3 \arcsinh \left (a x \right )^{2}}{32}+\frac {a^{4} x^{4}}{32}-\frac {3 a^{2} x^{2}}{32}-\frac {3}{32}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 109, normalized size = 1.14 \[ \frac {1}{4} \, x^{4} \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{32} \, {\left (\frac {x^{4}}{a^{2}} - \frac {3 \, x^{2}}{a^{4}} + \frac {3 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{6}}\right )} a^{2} - \frac {1}{16} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} x^{3}}{a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} a \operatorname {arsinh}\left (a x\right ) \]
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 \[ \int x^3\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.86, size = 90, normalized size = 0.94 \[ \begin {cases} \frac {x^{4} \operatorname {asinh}^{2}{\left (a x \right )}}{4} + \frac {x^{4}}{32} - \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{8 a} - \frac {3 x^{2}}{32 a^{2}} + \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{16 a^{3}} - \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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