Optimal. Leaf size=53 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a}+x \sqrt {\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.11, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5653, 5779, 3308, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a}+x \sqrt {\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5653
Rule 5779
Rubi steps
\begin {align*} \int \sqrt {\sinh ^{-1}(a x)} \, dx &=x \sqrt {\sinh ^{-1}(a x)}-\frac {1}{2} a \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=x \sqrt {\sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a}\\ &=x \sqrt {\sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}\\ &=x \sqrt {\sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a}-\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a}\\ &=x \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 45, normalized size = 0.85 \[ -\frac {\frac {\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\Gamma \left (\frac {3}{2},\sinh ^{-1}(a x)\right )}{2 a} \]
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 \[ \int \sqrt {\operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 42, normalized size = 0.79 \[ \frac {4 \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, x a +\pi \erf \left (\sqrt {\arcsinh \left (a x \right )}\right )-\pi \erfi \left (\sqrt {\arcsinh \left (a x \right )}\right )}{4 \sqrt {\pi }\, a} \]
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 \[ \int \sqrt {\operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\mathrm {asinh}\left (a\,x\right )} \,d x \]
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 \[ \int \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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