Integrand size = 10, antiderivative size = 62 \[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=-\frac {1}{10} i \arcsin \left (a x^5\right )^2+\frac {1}{5} \arcsin \left (a x^5\right ) \log \left (1-e^{2 i \arcsin \left (a x^5\right )}\right )-\frac {1}{10} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (a x^5\right )}\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4914, 3798, 2221, 2317, 2438} \[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=-\frac {1}{10} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (a x^5\right )}\right )-\frac {1}{10} i \arcsin \left (a x^5\right )^2+\frac {1}{5} \arcsin \left (a x^5\right ) \log \left (1-e^{2 i \arcsin \left (a x^5\right )}\right ) \]
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4914
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int x \cot (x) \, dx,x,\arcsin \left (a x^5\right )\right ) \\ & = -\frac {1}{10} i \arcsin \left (a x^5\right )^2-\frac {2}{5} i \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin \left (a x^5\right )\right ) \\ & = -\frac {1}{10} i \arcsin \left (a x^5\right )^2+\frac {1}{5} \arcsin \left (a x^5\right ) \log \left (1-e^{2 i \arcsin \left (a x^5\right )}\right )-\frac {1}{5} \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin \left (a x^5\right )\right ) \\ & = -\frac {1}{10} i \arcsin \left (a x^5\right )^2+\frac {1}{5} \arcsin \left (a x^5\right ) \log \left (1-e^{2 i \arcsin \left (a x^5\right )}\right )+\frac {1}{10} i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin \left (a x^5\right )}\right ) \\ & = -\frac {1}{10} i \arcsin \left (a x^5\right )^2+\frac {1}{5} \arcsin \left (a x^5\right ) \log \left (1-e^{2 i \arcsin \left (a x^5\right )}\right )-\frac {1}{10} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (a x^5\right )}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94 \[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=\frac {1}{5} \left (\arcsin \left (a x^5\right ) \log \left (1-e^{2 i \arcsin \left (a x^5\right )}\right )-\frac {1}{2} i \left (\arcsin \left (a x^5\right )^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (a x^5\right )}\right )\right )\right ) \]
[In]
[Out]
\[\int \frac {\arcsin \left (a \,x^{5}\right )}{x}d x\]
[In]
[Out]
\[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=\int { \frac {\arcsin \left (a x^{5}\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=\int \frac {\operatorname {asin}{\left (a x^{5} \right )}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=\int { \frac {\arcsin \left (a x^{5}\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=\int { \frac {\arcsin \left (a x^{5}\right )}{x} \,d x } \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int \frac {\arcsin \left (a x^5\right )}{x} \, dx=-\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (a\,x^5\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{10}+\frac {\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (a\,x^5\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (a\,x^5\right )}{5}-\frac {{\mathrm {asin}\left (a\,x^5\right )}^2\,1{}\mathrm {i}}{10} \]
[In]
[Out]