Integrand size = 14, antiderivative size = 107 \[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=-\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]
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Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5887, 5556, 12, 3389, 2211, 2236, 2235} \[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}-\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]
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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 \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^2}+\frac {\text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^2} \\ & = -\frac {\text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b c^2}+\frac {\text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b c^2} \\ & = -\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \left (\text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )\right )}{4 \sqrt {b} c^2} \]
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\[\int \frac {x}{\sqrt {a +b \,\operatorname {arccosh}\left (c x \right )}}d x\]
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Exception generated. \[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int \frac {x}{\sqrt {a + b \operatorname {acosh}{\left (c x \right )}}}\, dx \]
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\[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int { \frac {x}{\sqrt {b \operatorname {arcosh}\left (c x\right ) + a}} \,d x } \]
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\[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int { \frac {x}{\sqrt {b \operatorname {arcosh}\left (c x\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int \frac {x}{\sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )}} \,d x \]
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