Integrand size = 8, antiderivative size = 11 \[ \int \frac {1}{1-\sin (x)} \, dx=\frac {\cos (x)}{1-\sin (x)} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2727} \[ \int \frac {1}{1-\sin (x)} \, dx=\frac {\cos (x)}{1-\sin (x)} \]
[In]
[Out]
Rule 2727
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x)}{1-\sin (x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(11)=22\).
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int \frac {1}{1-\sin (x)} \, dx=\frac {2 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
norman | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
parallelrisch | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
risch | \(\frac {2}{{\mathrm e}^{i x}-i}\) | \(13\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {1}{1-\sin (x)} \, dx=\frac {\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {1}{1-\sin (x)} \, dx=- \frac {2}{\tan {\left (\frac {x}{2} \right )} - 1} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {1}{1-\sin (x)} \, dx=-\frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1-\sin (x)} \, dx=-\frac {2}{\tan \left (\frac {1}{2} \, x\right ) - 1} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1-\sin (x)} \, dx=-\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )-1} \]
[In]
[Out]