Integrand size = 10, antiderivative size = 93 \[ \int x \sqrt {\text {arccosh}(a x)} \, dx=-\frac {\sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a^2} \]
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Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5884, 5953, 3393, 3388, 2211, 2235, 2236} \[ \int x \sqrt {\text {arccosh}(a x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a^2}-\frac {\sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5884
Rule 5953
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}} \, dx \\ & = \frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{4 a^2} \\ & = \frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a^2} \\ & = -\frac {\sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{8 a^2} \\ & = -\frac {\sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{16 a^2}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{16 a^2} \\ & = -\frac {\sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{8 a^2}-\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{8 a^2} \\ & = -\frac {\sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.70 \[ \int x \sqrt {\text {arccosh}(a x)} \, dx=\frac {8 \sqrt {\text {arccosh}(a x)} \cosh (2 \text {arccosh}(a x))-\sqrt {2 \pi } \left (\text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{32 a^2} \]
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Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\sqrt {2}\, \left (8 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}-4 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }-\pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{32 \sqrt {\pi }\, a^{2}}\) | \(75\) |
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Exception generated. \[ \int x \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \sqrt {\text {arccosh}(a x)} \, dx=\int x \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]
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\[ \int x \sqrt {\text {arccosh}(a x)} \, dx=\int { x \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int x \sqrt {\text {arccosh}(a x)} \, dx=\int { x \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int x \sqrt {\text {arccosh}(a x)} \, dx=\int x\,\sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \]
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