Integrand size = 11, antiderivative size = 25 \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=\frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (i+\cot (x))} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3583, 2718} \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=\frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (\cot (x)+i)} \]
[In]
[Out]
Rule 2718
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {i \sin (x)}{3 (i+\cot (x))}-\frac {2}{3} i \int \sin (x) \, dx \\ & = \frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (i+\cot (x))} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=\frac {1}{6} (i \cos (x)+\sin (x)) (3+\cos (2 x)+2 i \sin (2 x)) \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {i {\mathrm e}^{-3 i x}}{12}+\frac {3 i \cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\) | \(19\) |
default | \(-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )+i\right )}-\frac {i}{\left (-i+\tan \left (\frac {x}{2}\right )\right )^{2}}+\frac {2}{3 \left (-i+\tan \left (\frac {x}{2}\right )\right )^{3}}+\frac {1}{-2 i+2 \tan \left (\frac {x}{2}\right )}\) | \(47\) |
norman | \(\frac {\frac {2 i \tan \left (x \right )^{2}}{3}-\frac {i \tan \left (\frac {x}{2}\right )^{2}}{3}-\frac {\tan \left (x \right ) \tan \left (\frac {x}{2}\right )^{2}}{3}+\frac {2 \tan \left (x \right )^{2} \tan \left (\frac {x}{2}\right )}{3}+\frac {4 i \tan \left (x \right ) \tan \left (\frac {x}{2}\right )}{3}+\frac {\tan \left (x \right )}{3}-\frac {2 \tan \left (\frac {x}{2}\right )}{3}+i}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (\tan \left (x \right )^{2}+1\right )}\) | \(78\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=\frac {1}{12} \, {\left (3 i \, e^{\left (4 i \, x\right )} + 6 i \, e^{\left (2 i \, x\right )} - i\right )} e^{\left (-3 i \, x\right )} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=\frac {i e^{i x}}{4} + \frac {i e^{- i x}}{2} - \frac {i e^{- 3 i x}}{12} \]
[In]
[Out]
Exception generated. \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=-\frac {1}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}} + \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 i \, \tan \left (\frac {1}{2} \, x\right ) - 5}{6 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}^{3}} \]
[In]
[Out]
Time = 12.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {\sin (x)}{i+\cot (x)} \, dx=-\frac {\frac {4}{3}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,8{}\mathrm {i}}{3}}{{\left (1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )} \]
[In]
[Out]