Optimal. Leaf size=132 \[ -\frac {x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a}-\frac {4 x \sinh ^{-1}(a x)}{3 a^2}-\frac {2 \left (a^2 x^2+1\right )^{3/2}}{27 a^3}+\frac {14 \sqrt {a^2 x^2+1}}{9 a^3}+\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3+\frac {2}{9} x^3 \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.22, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5661, 5758, 5717, 5653, 261, 266, 43} \[ -\frac {2 \left (a^2 x^2+1\right )^{3/2}}{27 a^3}+\frac {14 \sqrt {a^2 x^2+1}}{9 a^3}-\frac {x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a}+\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^3}-\frac {4 x \sinh ^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3+\frac {2}{9} x^3 \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 5653
Rule 5661
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{3} x^3 \sinh ^{-1}(a x)^3-a \int \frac {x^3 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3+\frac {2}{3} \int x^2 \sinh ^{-1}(a x) \, dx+\frac {2 \int \frac {x \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{3 a}\\ &=\frac {2}{9} x^3 \sinh ^{-1}(a x)+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3-\frac {4 \int \sinh ^{-1}(a x) \, dx}{3 a^2}-\frac {1}{9} (2 a) \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {4 x \sinh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \sinh ^{-1}(a x)+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3+\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx}{3 a}-\frac {1}{9} a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac {4 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {4 x \sinh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \sinh ^{-1}(a x)+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3-\frac {1}{9} a \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {14 \sqrt {1+a^2 x^2}}{9 a^3}-\frac {2 \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \sinh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \sinh ^{-1}(a x)+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.06, size = 93, normalized size = 0.70 \[ \frac {9 a^3 x^3 \sinh ^{-1}(a x)^3-2 \left (a^2 x^2-20\right ) \sqrt {a^2 x^2+1}-9 \left (a^2 x^2-2\right ) \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2+6 a x \left (a^2 x^2-6\right ) \sinh ^{-1}(a x)}{27 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 124, normalized size = 0.94 \[ \frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )}}{27 \, a^{3}} \]
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 = 116, normalized size = 0.88 \[ \frac {\frac {a^{3} x^{3} \arcsinh \left (a x \right )^{3}}{3}+\frac {2 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3}-\frac {\arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{3}-\frac {4 a x \arcsinh \left (a x \right )}{3}+\frac {40 \sqrt {a^{2} x^{2}+1}}{27}+\frac {2 a^{3} x^{3} \arcsinh \left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{27}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 116, normalized size = 0.88 \[ \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{2} - \frac {2}{27} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )}{a^{3}}\right )} \]
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 \[ \int x^2\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.80, size = 128, normalized size = 0.97 \[ \begin {cases} \frac {x^{3} \operatorname {asinh}^{3}{\left (a x \right )}}{3} + \frac {2 x^{3} \operatorname {asinh}{\left (a x \right )}}{9} - \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1}}{27 a} - \frac {4 x \operatorname {asinh}{\left (a x \right )}}{3 a^{2}} + \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{3}} + \frac {40 \sqrt {a^{2} x^{2} + 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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