Integrand size = 12, antiderivative size = 139 \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=-\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4} \]
[Out]
Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5884, 5953, 3393, 3388, 2211, 2235, 2236} \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4}-\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5884
Rule 5953
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {1}{8} a \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}} \, dx \\ & = \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{8 a^4} \\ & = \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{8 a^4} \\ & = -\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{64 a^4}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{16 a^4} \\ & = -\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{128 a^4}-\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{128 a^4}-\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{32 a^4}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{32 a^4} \\ & = -\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{64 a^4}-\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{64 a^4}-\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{16 a^4}-\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{16 a^4} \\ & = -\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\frac {\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-4 \text {arccosh}(a x)\right )+4 \sqrt {2} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-2 \text {arccosh}(a x)\right )+\sqrt {-\text {arccosh}(a x)} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},2 \text {arccosh}(a x)\right )+\Gamma \left (\frac {3}{2},4 \text {arccosh}(a x)\right )\right )}{128 a^4 \sqrt {-\text {arccosh}(a x)}} \]
[In]
[Out]
Time = 1.14 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {\sqrt {2}\, \left (8 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}-4 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }-\pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{64 \sqrt {\pi }\, a^{4}}-\frac {-64 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{4} x^{4}+64 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+\pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+\pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-8 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }}{256 \sqrt {\pi }\, a^{4}}\) | \(152\) |
[In]
[Out]
Exception generated. \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int x^{3} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int { x^{3} \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]
[In]
[Out]
Exception generated. \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int x^3\,\sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \]
[In]
[Out]