Integrand size = 12, antiderivative size = 167 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {\sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {2 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {\sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4} \]
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Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5779, 5818, 5780, 5556, 3389, 2211, 2235, 2236, 12} \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {\sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {2 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {\sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5779
Rule 5780
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx}{a}+\frac {1}{3} (8 a) \int \frac {x^4}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {64}{3} \int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {8 \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx}{a^2} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^4}+\frac {64 \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4}+\frac {4 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^4}-\frac {16 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {4 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4}+\frac {4 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4}-\frac {2 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^4}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4}-\frac {8 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {8 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {8 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {4 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^4}+\frac {4 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^4}+\frac {16 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {16 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {\sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {2 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {\sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {-4 e^{-4 \text {arcsinh}(a x)} \text {arcsinh}(a x)+4 e^{-2 \text {arcsinh}(a x)} \text {arcsinh}(a x)+4 e^{2 \text {arcsinh}(a x)} \text {arcsinh}(a x)-4 e^{4 \text {arcsinh}(a x)} \text {arcsinh}(a x)-8 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+4 \sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )-4 \sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )+8 \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )+2 \sinh (2 \text {arcsinh}(a x))-\sinh (4 \text {arcsinh}(a x))}{12 a^4 \text {arcsinh}(a x)^{3/2}} \]
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\[\int \frac {x^{3}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^{3}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]
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