Integrand size = 10, antiderivative size = 147 \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5779, 5818, 5778, 3388, 2211, 2235, 2236, 5783} \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}-\frac {32 x \sqrt {a^2 x^2+1}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {2 x \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5778
Rule 5779
Rule 5783
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (4 a) \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}+\frac {16}{15} \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {32 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^2}+\frac {16 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {32 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {32 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 \text {arcsinh}(a x) \left (e^{-2 \text {arcsinh}(a x)} (1-4 \text {arcsinh}(a x))+e^{2 \text {arcsinh}(a x)} (1+4 \text {arcsinh}(a x))+4 \sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+4 \sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )+3 \sinh (2 \text {arcsinh}(a x))}{15 a^2 \text {arcsinh}(a x)^{5/2}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (16 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x +4 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -8 \operatorname {arcsinh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-8 \operatorname {arcsinh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+2 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{15 \sqrt {\pi }\, a^{2} \operatorname {arcsinh}\left (a x \right )^{3}}\) | \(147\) |
[In]
[Out]
Exception generated. \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]
[In]
[Out]