Optimal. Leaf size=93 \[ -\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a^2}+\frac {\sqrt {\sinh ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.19, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5663, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a^2}+\frac {\sqrt {\sinh ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5663
Rule 5779
Rubi steps
\begin {align*} \int x \sqrt {\sinh ^{-1}(a x)} \, dx &=\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {\sqrt {\sinh ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^2}\\ &=\frac {\sqrt {\sinh ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^2}-\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^2}\\ &=\frac {\sqrt {\sinh ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a^2}-\frac {\operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a^2}\\ &=\frac {\sqrt {\sinh ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sinh ^{-1}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.56 \[ \frac {\frac {\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \sinh ^{-1}(a x)\right )}{\sqrt {-\sinh ^{-1}(a x)}}+\Gamma \left (\frac {3}{2},2 \sinh ^{-1}(a x)\right )}{8 \sqrt {2} a^2} \]
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 \[ \int x \sqrt {\operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 75, normalized size = 0.81 \[ \frac {\sqrt {2}\, \left (8 \sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, x^{2} a^{2}+4 \sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }-\pi \erf \left (\sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\right )-\pi \erfi \left (\sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\right )\right )}{32 \sqrt {\pi }\, a^{2}} \]
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 \[ \int x \sqrt {\operatorname {arsinh}\left (a x\right )}\,{d x} \]
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 \[ \int x\,\sqrt {\mathrm {asinh}\left (a\,x\right )} \,d x \]
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 \[ \int x \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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