Optimal. Leaf size=80 \[ -\frac {2 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{9 a}-\frac {4 x}{9 a^2}+\frac {4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^2+\frac {2 x^3}{27} \]
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Rubi [A] time = 0.12, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5661, 5758, 5717, 8, 30} \[ -\frac {2 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{9 a}+\frac {4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^3}-\frac {4 x}{9 a^2}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^2+\frac {2 x^3}{27} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5661
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}(a x)^2 \, dx &=\frac {1}{3} x^3 \sinh ^{-1}(a x)^2-\frac {1}{3} (2 a) \int \frac {x^3 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^2+\frac {2 \int x^2 \, dx}{9}+\frac {4 \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{9 a}\\ &=\frac {2 x^3}{27}+\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^3}-\frac {2 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^2-\frac {4 \int 1 \, dx}{9 a^2}\\ &=-\frac {4 x}{9 a^2}+\frac {2 x^3}{27}+\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^3}-\frac {2 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 0.74 \[ \frac {1}{27} \left (2 x \left (x^2-\frac {6}{a^2}\right )-\frac {6 \left (a^2 x^2-2\right ) \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{a^3}+9 x^3 \sinh ^{-1}(a x)^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 82, normalized size = 1.02 \[ \frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a^{3} x^{3} - 6 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 12 \, a x}{27 \, a^{3}} \]
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 = 72, normalized size = 0.90 \[ \frac {\frac {a^{3} x^{3} \arcsinh \left (a x \right )^{2}}{3}+\frac {4 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )}{9}-\frac {2 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{9}-\frac {4 a x}{9}+\frac {2 a^{3} x^{3}}{27}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 70, normalized size = 0.88 \[ \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right )^{2} - \frac {2}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right ) + \frac {2 \, {\left (a^{2} x^{3} - 6 \, x\right )}}{27 \, a^{2}} \]
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 \[ \int x^2\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 76, normalized size = 0.95 \[ \begin {cases} \frac {x^{3} \operatorname {asinh}^{2}{\left (a x \right )}}{3} + \frac {2 x^{3}}{27} - \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a} - \frac {4 x}{9 a^{2}} + \frac {4 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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