Integrand size = 10, antiderivative size = 63 \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2} \]
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Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5887, 5556, 12, 3389, 2211, 2235, 2236} \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a^2} \\ & = -\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{4 a^2}+\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{4 a^2} \\ & = -\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2 a^2}+\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2 a^2} \\ & = -\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (-\text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{4 a^2} \]
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Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{8 a^{2}}\) | \(37\) |
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Exception generated. \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {x}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \]
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\[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {x}{\sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \]
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\[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {x}{\sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {x}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \]
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