Integrand size = 12, antiderivative size = 139 \[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=-\frac {3 \sqrt {\text {arcsinh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4} \]
[Out]
Time = 0.19 (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 = {5777, 5819, 3393, 3388, 2211, 2235, 2236} \[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {3 \sqrt {\text {arcsinh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5777
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {1}{8} a \int \frac {x^4}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = \frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4} \\ & = \frac {1}{4} x^4 \sqrt {\text {arcsinh}(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 {arcsinh}(a x)\right )}{8 a^4} \\ & = -\frac {3 \sqrt {\text {arcsinh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^4}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^4} \\ & = -\frac {3 \sqrt {\text {arcsinh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{128 a^4}-\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{128 a^4}+\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^4}+\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^4} \\ & = -\frac {3 \sqrt {\text {arcsinh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{64 a^4}-\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{64 a^4}+\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a^4}+\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a^4} \\ & = -\frac {3 \sqrt {\text {arcsinh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arcsinh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71 \[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=\frac {\frac {\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {3}{2},-4 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {4 \sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {3}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-4 \sqrt {2} \Gamma \left (\frac {3}{2},2 \text {arcsinh}(a x)\right )+\Gamma \left (\frac {3}{2},4 \text {arcsinh}(a x)\right )}{128 a^4} \]
[In]
[Out]
\[\int x^{3} \sqrt {\operatorname {arcsinh}\left (a x \right )}d x\]
[In]
[Out]
Exception generated. \[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=\int x^{3} \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=\int { x^{3} \sqrt {\operatorname {arsinh}\left (a x\right )} \,d x } \]
[In]
[Out]
Exception generated. \[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx=\int x^3\,\sqrt {\mathrm {asinh}\left (a\,x\right )} \,d x \]
[In]
[Out]