Integrand size = 10, antiderivative size = 70 \[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {5 \text {Chi}(2 \text {arcsinh}(a x))}{16 a^6}-\frac {\text {Chi}(4 \text {arcsinh}(a x))}{2 a^6}+\frac {3 \text {Chi}(6 \text {arcsinh}(a x))}{16 a^6} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5778, 3382} \[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=\frac {5 \text {Chi}(2 \text {arcsinh}(a x))}{16 a^6}-\frac {\text {Chi}(4 \text {arcsinh}(a x))}{2 a^6}+\frac {3 \text {Chi}(6 \text {arcsinh}(a x))}{16 a^6}-\frac {x^5 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)} \]
[In]
[Out]
Rule 3382
Rule 5778
Rubi steps \begin{align*} \text {integral}& = -\frac {x^5 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {5 \cosh (2 x)}{16 x}-\frac {\cosh (4 x)}{2 x}+\frac {3 \cosh (6 x)}{16 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^6} \\ & = -\frac {x^5 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {3 \text {Subst}\left (\int \frac {\cosh (6 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^6}+\frac {5 \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^6}-\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^6} \\ & = -\frac {x^5 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {5 \text {Chi}(2 \text {arcsinh}(a x))}{16 a^6}-\frac {\text {Chi}(4 \text {arcsinh}(a x))}{2 a^6}+\frac {3 \text {Chi}(6 \text {arcsinh}(a x))}{16 a^6} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=-\frac {-10 \text {arcsinh}(a x) \text {Chi}(2 \text {arcsinh}(a x))+16 \text {arcsinh}(a x) \text {Chi}(4 \text {arcsinh}(a x))-6 \text {arcsinh}(a x) \text {Chi}(6 \text {arcsinh}(a x))+5 \sinh (2 \text {arcsinh}(a x))-4 \sinh (4 \text {arcsinh}(a x))+\sinh (6 \text {arcsinh}(a x))}{32 a^6 \text {arcsinh}(a x)} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {-\frac {5 \sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}+\frac {5 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}+\frac {\sinh \left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{2}-\frac {\sinh \left (6 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (6 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{6}}\) | \(78\) |
default | \(\frac {-\frac {5 \sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}+\frac {5 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}+\frac {\sinh \left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{2}-\frac {\sinh \left (6 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (6 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{6}}\) | \(78\) |
[In]
[Out]
\[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{5}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{5}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{5}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^5}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^5}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
[In]
[Out]