Integrand size = 8, antiderivative size = 112 \[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}-\frac {8 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5773, 5818, 5819, 3389, 2211, 2235, 2236} \[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {8 \sqrt {a^2 x^2+1}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5773
Rule 5818
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {1}{5} (2 a) \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}+\frac {4}{15} \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}-\frac {8 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {1}{15} (8 a) \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}-\frac {8 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}-\frac {8 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a}+\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}-\frac {8 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {8 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a}+\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x}{15 \text {arcsinh}(a x)^{3/2}}-\frac {8 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {-2 e^{\text {arcsinh}(a x)} \left (3+2 \text {arcsinh}(a x)+4 \text {arcsinh}(a x)^2\right )+8 (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+e^{-\text {arcsinh}(a x)} \left (-6+4 \text {arcsinh}(a x)-8 \text {arcsinh}(a x)^2+8 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{30 a \text {arcsinh}(a x)^{5/2}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {2 \left (2 \operatorname {arcsinh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-2 \operatorname {arcsinh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+4 \sqrt {a^{2} x^{2}+1}\, \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {\pi }+2 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x +3 \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\right )}{15 \sqrt {\pi }\, a \operatorname {arcsinh}\left (a x \right )^{3}}\) | \(105\) |
[In]
[Out]
Exception generated. \[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {1}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]
[In]
[Out]