Integrand size = 10, antiderivative size = 118 \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2} \]
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Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5779, 5818, 5780, 5556, 12, 3389, 2211, 2235, 2236, 5783} \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (4 a) \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {16}{3} \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {4 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^2}+\frac {4 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {8 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^2}+\frac {8 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \text {arcsinh}(a x) \left (e^{-2 \text {arcsinh}(a x)}+e^{2 \text {arcsinh}(a x)}-\sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )-\sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )+\sinh (2 \text {arcsinh}(a x))}{3 a^2 \text {arcsinh}(a x)^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (4 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x +2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+2 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{3 \sqrt {\pi }\, a^{2} \operatorname {arcsinh}\left (a x \right )^{2}}\) | \(119\) |
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Exception generated. \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]
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