Integrand size = 15, antiderivative size = 85 \[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2 i}{b n}\right ),\frac {1}{2} \left (3+\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2 i+b n) x^2} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4606, 4602, 371} \[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2 i}{b n}\right ),\frac {1}{2} \left (3+\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (b n+2 i)} \]
[In]
[Out]
Rule 371
Rule 4602
Rule 4606
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n x^2} \\ & = -\frac {\left (2 i e^{i a} \left (c x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b-\frac {2}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x^2} \\ & = -\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2 i}{b n}\right ),\frac {1}{2} \left (3+\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2 i+b n) x^2} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {i}{b n},\frac {3}{2}+\frac {i}{b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{(2 i+b n) x^2} \]
[In]
[Out]
\[\int \frac {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}d x\]
[In]
[Out]
\[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {1}{x^3\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
[In]
[Out]