Integrand size = 12, antiderivative size = 229 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4} \]
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Time = 0.30 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5779, 5818, 5778, 3388, 2211, 2235, 2236} \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {128 x^3 \sqrt {a^2 x^2+1}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {16 x \sqrt {a^2 x^2+1}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5778
Rule 5779
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6 \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}+\frac {64}{15} \int \frac {x^3}{\text {arcsinh}(a x)^{3/2}} \, dx+\frac {8 \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx}{5 a^2} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^4}+\frac {128 \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^4}+\frac {8 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^4}-\frac {64 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {32 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4}-\frac {32 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4}-\frac {32 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^4}+\frac {16 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^4}+\frac {16 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^4}+\frac {64 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {64 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {64 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^4} \\ & = -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {4 \text {arcsinh}(a x) \left (e^{-2 \text {arcsinh}(a x)} (1-4 \text {arcsinh}(a x))+e^{2 \text {arcsinh}(a x)} (1+4 \text {arcsinh}(a x))+4 \sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+4 \sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )-4 \text {arcsinh}(a x) \left (e^{-4 \text {arcsinh}(a x)} (1-8 \text {arcsinh}(a x))+e^{4 \text {arcsinh}(a x)} (1+8 \text {arcsinh}(a x))+16 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+16 \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )\right )+6 \sinh (2 \text {arcsinh}(a x))-3 \sinh (4 \text {arcsinh}(a x))}{60 a^4 \text {arcsinh}(a x)^{5/2}} \]
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\[\int \frac {x^{3}}{\operatorname {arcsinh}\left (a x \right )^{\frac {7}{2}}}d x\]
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Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^{3}}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]
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