Integrand size = 8, antiderivative size = 68 \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{a} \]
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Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5880, 5953, 3388, 2211, 2235, 2236} \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5880
Rule 5953
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+(2 a) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}} \, dx \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a}+\frac {2 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{a} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )}{a \sqrt {\text {arccosh}(a x)}} \]
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Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {-2 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}+\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )+\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{\sqrt {\pi }\, a \,\operatorname {arccosh}\left (a x \right )}\) | \(66\) |
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Exception generated. \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]
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