Optimal. Leaf size=120 \[ -\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.24, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5663, 5779, 3312, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 3312
Rule 5663
Rule 5779
Rubi steps
\begin {align*} \int x^2 \sqrt {\sinh ^{-1}(a x)} \, dx &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}-\frac {i \operatorname {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {x}}-\frac {i \sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{24 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^{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 )}{16 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{24 a^3}-\frac {\operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{24 a^3}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\sinh ^{-1}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{48 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 101, normalized size = 0.84 \[ \frac {\sqrt {3} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-3 \sinh ^{-1}(a x)\right )-9 \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-\sinh ^{-1}(a x)\right )+\sqrt {-\sinh ^{-1}(a x)} \left (9 \Gamma \left (\frac {3}{2},\sinh ^{-1}(a x)\right )-\sqrt {3} \Gamma \left (\frac {3}{2},3 \sinh ^{-1}(a x)\right )\right )}{72 a^3 \sqrt {-\sinh ^{-1}(a x)}} \]
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 \[ \int x^{2} \sqrt {\operatorname {arsinh}\left (a x\right )}\,{d x} \]
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 \[ \int x^{2} \sqrt {\arcsinh \left (a x \right )}\, dx \]
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 \[ \int x^{2} \sqrt {\operatorname {arsinh}\left (a x\right )}\,{d x} \]
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 \[ \int x^2\,\sqrt {\mathrm {asinh}\left (a\,x\right )} \,d x \]
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 \[ \int x^{2} \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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