Optimal. Leaf size=162 \[ -\frac {4 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{9 a}-\frac {8 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{27 a}-\frac {160 x}{27 a^2}-\frac {8 x \sinh ^{-1}(a x)^2}{3 a^2}+\frac {8 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{9 a^3}+\frac {160 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{27 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4+\frac {4}{9} x^3 \sinh ^{-1}(a x)^2+\frac {8 x^3}{81} \]
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Rubi [A] time = 0.36, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5661, 5758, 5717, 5653, 8, 30} \[ -\frac {4 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{9 a}+\frac {8 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{9 a^3}-\frac {8 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{27 a}+\frac {160 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{27 a^3}-\frac {160 x}{27 a^2}-\frac {8 x \sinh ^{-1}(a x)^2}{3 a^2}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4+\frac {4}{9} x^3 \sinh ^{-1}(a x)^2+\frac {8 x^3}{81} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5653
Rule 5661
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}(a x)^4 \, dx &=\frac {1}{3} x^3 \sinh ^{-1}(a x)^4-\frac {1}{3} (4 a) \int \frac {x^3 \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {4 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4+\frac {4}{3} \int x^2 \sinh ^{-1}(a x)^2 \, dx+\frac {8 \int \frac {x \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{9 a}\\ &=\frac {4}{9} x^3 \sinh ^{-1}(a x)^2+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4-\frac {8 \int \sinh ^{-1}(a x)^2 \, dx}{3 a^2}-\frac {1}{9} (8 a) \int \frac {x^3 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {8 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a}-\frac {8 x \sinh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \sinh ^{-1}(a x)^2+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4+\frac {8 \int x^2 \, dx}{27}+\frac {16 \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{27 a}+\frac {16 \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a}\\ &=\frac {8 x^3}{81}+\frac {160 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a^3}-\frac {8 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a}-\frac {8 x \sinh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \sinh ^{-1}(a x)^2+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4-\frac {16 \int 1 \, dx}{27 a^2}-\frac {16 \int 1 \, dx}{3 a^2}\\ &=-\frac {160 x}{27 a^2}+\frac {8 x^3}{81}+\frac {160 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a^3}-\frac {8 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a}-\frac {8 x \sinh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \sinh ^{-1}(a x)^2+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A] time = 0.07, size = 112, normalized size = 0.69 \[ \frac {27 a^3 x^3 \sinh ^{-1}(a x)^4+8 a x \left (a^2 x^2-60\right )-36 \left (a^2 x^2-2\right ) \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3+36 a x \left (a^2 x^2-6\right ) \sinh ^{-1}(a x)^2-24 \left (a^2 x^2-20\right ) \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{81 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 154, normalized size = 0.95 \[ \frac {27 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 8 \, a^{3} x^{3} - 36 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 36 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 24 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 480 \, a x}{81 \, a^{3}} \]
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 = 140, normalized size = 0.86 \[ \frac {\frac {a^{3} x^{3} \arcsinh \left (a x \right )^{4}}{3}+\frac {8 \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{9}-\frac {4 a^{2} x^{2} \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{9}-\frac {8 a x \arcsinh \left (a x \right )^{2}}{3}+\frac {160 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )}{27}-\frac {160 a x}{27}+\frac {4 a^{3} x^{3} \arcsinh \left (a x \right )^{2}}{9}-\frac {8 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{27}+\frac {8 a^{3} x^{3}}{81}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 143, normalized size = 0.88 \[ \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right )^{4} - \frac {4}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{3} - \frac {4}{81} \, {\left (2 \, a {\left (\frac {3 \, {\left (\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )} \operatorname {arsinh}\left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} - 60 \, x}{a^{4}}\right )} - \frac {9 \, {\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )^{2}}{a^{3}}\right )} a \]
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 \[ \int x^2\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.08, size = 158, normalized size = 0.98 \[ \begin {cases} \frac {x^{3} \operatorname {asinh}^{4}{\left (a x \right )}}{3} + \frac {4 x^{3} \operatorname {asinh}^{2}{\left (a x \right )}}{9} + \frac {8 x^{3}}{81} - \frac {4 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{9 a} - \frac {8 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{27 a} - \frac {8 x \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{2}} - \frac {160 x}{27 a^{2}} + \frac {8 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{9 a^{3}} + \frac {160 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{27 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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