Optimal. Leaf size=122 \[ -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^2} \]
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Rubi [A]
time = 0.14, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5777, 5812,
5783, 5780, 5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^2}-\frac {3 x \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5777
Rule 5780
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int x \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}-\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}+\frac {3}{16} \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx+\frac {3 \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{8 a}\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^2}\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^2}+\frac {3 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^2}\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^2}+\frac {3 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^2}\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{8 a}+\frac {\sinh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 52, normalized size = 0.43 \begin {gather*} \frac {\frac {\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-2 \sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\Gamma \left (\frac {5}{2},2 \sinh ^{-1}(a x)\right )}{16 \sqrt {2} a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.47, size = 102, normalized size = 0.84
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-32 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {2}\, a^{2} x^{2}+24 \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, \sqrt {2}\, a x -16 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {2}+3 \pi \erf \left (\sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\right )-3 \pi \erfi \left (\sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\right )\right )}{128 \sqrt {\pi }\, a^{2}}\) | \(102\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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