Optimal. Leaf size=117 \[ \frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{25 a}-\frac {16 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{75 a^5}+\frac {8 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{75 a^3}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^2+\frac {2 x^5}{125} \]
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Rubi [A] time = 0.19, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {8 x^3}{225 a^2}-\frac {2 x^4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{25 a}+\frac {8 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{75 a^3}-\frac {16 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{75 a^5}+\frac {16 x}{75 a^4}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^2+\frac {2 x^5}{125} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5661
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int x^4 \sinh ^{-1}(a x)^2 \, dx &=\frac {1}{5} x^5 \sinh ^{-1}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 x^4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^2+\frac {2 \int x^4 \, dx}{25}+\frac {8 \int \frac {x^3 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{25 a}\\ &=\frac {2 x^5}{125}+\frac {8 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^2-\frac {16 \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{75 a^3}-\frac {8 \int x^2 \, dx}{75 a^2}\\ &=-\frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{75 a^5}+\frac {8 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^2+\frac {16 \int 1 \, dx}{75 a^4}\\ &=\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{75 a^5}+\frac {8 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.07, size = 75, normalized size = 0.64 \[ \frac {\frac {240 x}{a^4}-\frac {40 x^3}{a^2}-\frac {30 \sqrt {a^2 x^2+1} \left (3 a^4 x^4-4 a^2 x^2+8\right ) \sinh ^{-1}(a x)}{a^5}+225 x^5 \sinh ^{-1}(a x)^2+18 x^5}{1125} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 99, normalized size = 0.85 \[ \frac {225 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + 240 \, a x}{1125 \, a^{5}} \]
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 = 103, normalized size = 0.88 \[ \frac {\frac {a^{5} x^{5} \arcsinh \left (a x \right )^{2}}{5}-\frac {16 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )}{75}-\frac {2 a^{4} x^{4} \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{25}+\frac {8 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{75}+\frac {16 a x}{75}+\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 99, normalized size = 0.85 \[ \frac {1}{5} \, x^{5} \operatorname {arsinh}\left (a x\right )^{2} - \frac {2}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} + 1} x^{4}}{a^{2}} - \frac {4 \, \sqrt {a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} + 1}}{a^{6}}\right )} a \operatorname {arsinh}\left (a x\right ) + \frac {2 \, {\left (9 \, a^{4} x^{5} - 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \]
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 \[ \int x^4\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.03, size = 114, normalized size = 0.97 \[ \begin {cases} \frac {x^{5} \operatorname {asinh}^{2}{\left (a x \right )}}{5} + \frac {2 x^{5}}{125} - \frac {2 x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{25 a} - \frac {8 x^{3}}{225 a^{2}} + \frac {8 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{75 a^{3}} + \frac {16 x}{75 a^{4}} - \frac {16 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{75 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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