Optimal. Leaf size=179 \[ -\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {\sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.37, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5663, 5758, 5717, 5657, 3307, 2180, 2204, 2205, 5669, 5448} \[ -\frac {3 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac {3 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {\sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5448
Rule 5657
Rule 5663
Rule 5669
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac {1}{2} a \int \frac {x^3 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac {1}{12} \int \frac {x^2}{\sqrt {\sinh ^{-1}(a x)}} \, dx+\frac {\int \frac {x \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{3 a}\\ &=\frac {\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}-\frac {\int \frac {1}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{6 a^2}\\ &=\frac {\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac {\operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac {\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}\\ &=\frac {\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}-\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}\\ &=\frac {\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{12 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{12 a^3}+\frac {\operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}-\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac {\operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}\\ &=\frac {\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{96 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 102, normalized size = 0.57 \[ \frac {-\sqrt {3} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-3 \sinh ^{-1}(a x)\right )+27 \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-\sinh ^{-1}(a x)\right )+\sqrt {-\sinh ^{-1}(a x)} \left (27 \Gamma \left (\frac {5}{2},\sinh ^{-1}(a x)\right )-\sqrt {3} \Gamma \left (\frac {5}{2},3 \sinh ^{-1}(a x)\right )\right )}{216 a^3 \sqrt {-\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{2} \arcsinh \left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \int x^{2} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x^2\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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