Integrand size = 8, antiderivative size = 43 \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5774, 3388, 2211, 2235, 2236} \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5774
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a} \\ & = \frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a} \\ & = \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-\Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )}{2 a} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )\right )}{2 a}\) | \(24\) |
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Exception generated. \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \]
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