Optimal. Leaf size=110 \[ \frac {3 x^2}{4}-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac {3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4 \]
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Rubi [A]
time = 0.15, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 5812,
5783, 30} \begin {gather*} -\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{2 a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2+\frac {3 x^2}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 5776
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int x \sinh ^{-1}(a x)^4 \, dx &=\frac {1}{2} x^2 \sinh ^{-1}(a x)^4-(2 a) \int \frac {x^2 \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+3 \int x \sinh ^{-1}(a x)^2 \, dx+\frac {\int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{a}\\ &=\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4-(3 a) \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+\frac {3 \int x \, dx}{2}+\frac {3 \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 a}\\ &=\frac {3 x^2}{4}-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac {3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 94, normalized size = 0.85 \begin {gather*} \frac {3 a^2 x^2-6 a x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+\left (3+6 a^2 x^2\right ) \sinh ^{-1}(a x)^2-4 a x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3+\left (1+2 a^2 x^2\right ) \sinh ^{-1}(a x)^4}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.09, size = 105, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{4}}{2}-a x \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}-\frac {\arcsinh \left (a x \right )^{4}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{2}}{2}-\frac {3 a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{2}-\frac {3 \arcsinh \left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}+\frac {3}{4}}{a^{2}}\) | \(105\) |
default | \(\frac {\frac {\left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{4}}{2}-a x \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}-\frac {\arcsinh \left (a x \right )^{4}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{2}}{2}-\frac {3 a x \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{2}-\frac {3 \arcsinh \left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}+\frac {3}{4}}{a^{2}}\) | \(105\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 138, normalized size = 1.25 \begin {gather*} -\frac {4 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 3 \, a^{2} x^{2} + 6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 104, normalized size = 0.95 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {asinh}^{4}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{2} + \frac {3 x^{2}}{4} - \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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