Optimal. Leaf size=97 \[ -\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3 \sinh ^{-1}(a x)}{8 a^2}+\frac {3}{4} x^2 \sinh ^{-1}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a}+\frac {\sinh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5776, 5812,
5783, 327, 221} \begin {gather*} -\frac {3 x \sqrt {a^2 x^2+1}}{8 a}-\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 a}+\frac {\sinh ^{-1}(a x)^3}{4 a^2}+\frac {3 \sinh ^{-1}(a x)}{8 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^3+\frac {3}{4} x^2 \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int x \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{2} x^2 \sinh ^{-1}(a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^3+\frac {3}{2} \int x \sinh ^{-1}(a x) \, dx+\frac {3 \int \frac {\sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{4 a}\\ &=\frac {3}{4} x^2 \sinh ^{-1}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a}+\frac {\sinh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3}{4} x^2 \sinh ^{-1}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a}+\frac {\sinh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^3+\frac {3 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a}\\ &=-\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3 \sinh ^{-1}(a x)}{8 a^2}+\frac {3}{4} x^2 \sinh ^{-1}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a}+\frac {\sinh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 80, normalized size = 0.82 \begin {gather*} \frac {-3 a x \sqrt {1+a^2 x^2}+\left (3+6 a^2 x^2\right ) \sinh ^{-1}(a x)-6 a x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2+\left (2+4 a^2 x^2\right ) \sinh ^{-1}(a x)^3}{8 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.10, size = 88, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {\left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{3}}{2}-\frac {3 a x \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{4}-\frac {\arcsinh \left (a x \right )^{3}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )}{4}-\frac {3 a x \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \arcsinh \left (a x \right )}{8}}{a^{2}}\) | \(88\) |
default | \(\frac {\frac {\left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{3}}{2}-\frac {3 a x \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{4}-\frac {\arcsinh \left (a x \right )^{3}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )}{4}-\frac {3 a x \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \arcsinh \left (a x \right )}{8}}{a^{2}}\) | \(88\) |
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 = 0.33, size = 112, normalized size = 1.15 \begin {gather*} -\frac {6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} + 1} a x - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{8 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 92, normalized size = 0.95 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {asinh}^{3}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}{\left (a x \right )}}{4} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {a^{2} x^{2} + 1}}{8 a} + \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asinh}{\left (a x \right )}}{8 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 )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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