Integrand size = 12, antiderivative size = 102 \[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=x \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c} \]
[Out]
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 5953, 3388, 2211, 2236, 2235} \[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}+x \sqrt {a+b \text {arccosh}(c x)} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5879
Rule 5953
Rubi steps \begin{align*} \text {integral}& = x \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} (b c) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx \\ & = x \sqrt {a+b \text {arccosh}(c x)}-\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 c} \\ & = x \sqrt {a+b \text {arccosh}(c x)}-\frac {\text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 c}-\frac {\text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 c} \\ & = x \sqrt {a+b \text {arccosh}(c x)}-\frac {\text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 c}-\frac {\text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 c} \\ & = x \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.98 \[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}}\right )}{2 c} \]
[In]
[Out]
\[\int \sqrt {a +b \,\operatorname {arccosh}\left (c x \right )}d x\]
[In]
[Out]
Exception generated. \[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=\int \sqrt {a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
[In]
[Out]
\[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=\int { \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=\int { \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {a+b \text {arccosh}(c x)} \, dx=\int \sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
[In]
[Out]