Optimal. Leaf size=163 \[ -\frac {3 \sinh ^{-1}(a x)^3}{32 a^4}-\frac {45 \sinh ^{-1}(a x)}{256 a^4}-\frac {9 x^2 \sinh ^{-1}(a x)}{32 a^2}-\frac {3 x^3 \sqrt {a^2 x^2+1}}{128 a}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{16 a}+\frac {45 x \sqrt {a^2 x^2+1}}{256 a^3}+\frac {9 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{32 a^3}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3+\frac {3}{32} x^4 \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.30, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5661, 5758, 5675, 321, 215} \[ -\frac {3 x^3 \sqrt {a^2 x^2+1}}{128 a}+\frac {45 x \sqrt {a^2 x^2+1}}{256 a^3}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{16 a}-\frac {9 x^2 \sinh ^{-1}(a x)}{32 a^2}+\frac {9 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{32 a^3}-\frac {3 \sinh ^{-1}(a x)^3}{32 a^4}-\frac {45 \sinh ^{-1}(a x)}{256 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3+\frac {3}{32} x^4 \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5758
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{4} x^4 \sinh ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{16 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3+\frac {3}{8} \int x^3 \sinh ^{-1}(a x) \, dx+\frac {9 \int \frac {x^2 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a}\\ &=\frac {3}{32} x^4 \sinh ^{-1}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{16 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3-\frac {9 \int \frac {\sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{32 a^3}-\frac {9 \int x \sinh ^{-1}(a x) \, dx}{16 a^2}-\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {9 x^2 \sinh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \sinh ^{-1}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{16 a}-\frac {3 \sinh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3+\frac {9 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{128 a}+\frac {9 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{32 a}\\ &=\frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {9 x^2 \sinh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \sinh ^{-1}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{16 a}-\frac {3 \sinh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3-\frac {9 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{256 a^3}-\frac {9 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3}\\ &=\frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {45 \sinh ^{-1}(a x)}{256 a^4}-\frac {9 x^2 \sinh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \sinh ^{-1}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{16 a}-\frac {3 \sinh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.07, size = 110, normalized size = 0.67 \[ \frac {8 \left (8 a^4 x^4-3\right ) \sinh ^{-1}(a x)^3+3 a x \left (15-2 a^2 x^2\right ) \sqrt {a^2 x^2+1}-24 a x \sqrt {a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)^2+3 \left (8 a^4 x^4-24 a^2 x^2-15\right ) \sinh ^{-1}(a x)}{256 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 142, normalized size = 0.87 \[ \frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 24 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{256 \, a^{4}} \]
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 = 141, normalized size = 0.87 \[ \frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16}+\frac {9 a x \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{32}-\frac {3 \arcsinh \left (a x \right )^{3}}{32}+\frac {3 \arcsinh \left (a x \right ) a^{4} x^{4}}{32}-\frac {3 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{128}+\frac {45 a x \sqrt {a^{2} x^{2}+1}}{256}+\frac {27 \arcsinh \left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )}{32}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - \int \frac {3 \, {\left (a^{3} x^{6} + \sqrt {a^{2} x^{2} + 1} a^{2} x^{5} + a x^{4}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{4 \, {\left (a^{3} x^{3} + a x + {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}\right )}}\,{d x} \]
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 \[ \int x^3\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.15, size = 160, normalized size = 0.98 \[ \begin {cases} \frac {x^{4} \operatorname {asinh}^{3}{\left (a x \right )}}{4} + \frac {3 x^{4} \operatorname {asinh}{\left (a x \right )}}{32} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1}}{128 a} - \frac {9 x^{2} \operatorname {asinh}{\left (a x \right )}}{32 a^{2}} + \frac {9 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{32 a^{3}} + \frac {45 x \sqrt {a^{2} x^{2} + 1}}{256 a^{3}} - \frac {3 \operatorname {asinh}^{3}{\left (a x \right )}}{32 a^{4}} - \frac {45 \operatorname {asinh}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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