Optimal. Leaf size=43 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5657, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5657
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sinh ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a}\\ &=\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a}+\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a}\\ &=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 1.09 \[ \frac {\frac {\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}-\Gamma \left (\frac {1}{2},\sinh ^{-1}(a x)\right )}{2 a} \]
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 \[ \int \frac {1}{\sqrt {\operatorname {arsinh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 24, normalized size = 0.56 \[ \frac {\sqrt {\pi }\, \left (\erf \left (\sqrt {\arcsinh \left (a x \right )}\right )+\erfi \left (\sqrt {\arcsinh \left (a x \right )}\right )\right )}{2 a} \]
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 \[ \int \frac {1}{\sqrt {\operatorname {arsinh}\left (a x\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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