Integrand size = 15, antiderivative size = 49 \[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\frac {e^{-2 i a} \left (1-c^4 e^{2 i a} x^4\right )}{2 c^4 x^3 \sin ^{\frac {3}{2}}(a-2 i \log (c x))} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 49, 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 = {4571, 4569, 267} \[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\frac {e^{-2 i a} \left (1-e^{2 i a} c^4 x^4\right )}{2 c^4 x^3 \sin ^{\frac {3}{2}}(a-2 i \log (c x))} \]
[In]
[Out]
Rule 267
Rule 4569
Rule 4571
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (x))} \, dx,x,c x\right )}{c} \\ & = \frac {\left (1-c^4 e^{2 i a} x^4\right )^{3/2} \text {Subst}\left (\int \frac {x^3}{\left (1-e^{2 i a} x^4\right )^{3/2}} \, dx,x,c x\right )}{c^4 x^3 \sin ^{\frac {3}{2}}(a-2 i \log (c x))} \\ & = \frac {e^{-2 i a} \left (1-c^4 e^{2 i a} x^4\right )}{2 c^4 x^3 \sin ^{\frac {3}{2}}(a-2 i \log (c x))} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\frac {x (\cos (a)-i \sin (a)) \sqrt {\frac {-2 i \left (-1+c^4 x^4\right ) \cos (a)+2 \left (1+c^4 x^4\right ) \sin (a)}{c^2 x^2}}}{\left (-1+c^4 x^4\right ) \cos (a)+i \left (1+c^4 x^4\right ) \sin (a)} \]
[In]
[Out]
\[\int \frac {1}{\sin \left (a -2 i \ln \left (c x \right )\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-i \, c^{4} x^{4} + i \, e^{\left (-2 i \, a\right )}} e^{\left (-\frac {3}{2} i \, a\right )}}{c^{5} x^{4} - c e^{\left (-2 i \, a\right )}} \]
[In]
[Out]
\[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\int \frac {1}{\sin ^{\frac {3}{2}}{\left (a - 2 i \log {\left (c x \right )} \right )}}\, dx \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (36) = 72\).
Time = 0.43 (sec) , antiderivative size = 402, normalized size of antiderivative = 8.20 \[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\frac {{\left ({\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} c^{4} x^{4} + 2 \, c^{2} x^{2} \cos \left (a\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} c^{4} x^{4} - 2 \, c^{2} x^{2} \cos \left (a\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left ({\left (c^{4} x^{4} {\left (\left (i + 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} - \left (i + 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right ) + {\left (c^{4} x^{4} {\left (\left (i - 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) - \left (i + 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} - \left (i - 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) - \left (i + 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), c^{2} x^{2} \cos \left (a\right ) + 1\right )\right ) + {\left ({\left (c^{4} x^{4} {\left (-\left (i - 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) + \left (i + 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} + \left (i - 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) + \left (i + 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right ) + {\left (c^{4} x^{4} {\left (\left (i + 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} - \left (i + 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), c^{2} x^{2} \cos \left (a\right ) + 1\right )\right )\right )}}{{\left ({\left (\cos \left (a\right )^{4} + 2 \, \cos \left (a\right )^{2} \sin \left (a\right )^{2} + \sin \left (a\right )^{4}\right )} c^{8} x^{8} - 2 \, {\left (\cos \left (a\right )^{2} - \sin \left (a\right )^{2}\right )} c^{4} x^{4} + 1\right )} c} \]
[In]
[Out]
\[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\int { \frac {1}{\sin \left (a - 2 i \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Time = 27.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx=\frac {2\,x\,\sqrt {\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,c^2\,x^2}-\frac {c^2\,x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{c^4\,x^4\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-1} \]
[In]
[Out]