Optimal. Leaf size=11 \[ \frac{\cos (x)}{1-\sin (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0078634, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2648} \[ \frac{\cos (x)}{1-\sin (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{1-\sin (x)} \, dx &=\frac{\cos (x)}{1-\sin (x)}\\ \end{align*}
Mathematica [B] time = 0.0118904, size = 25, normalized size = 2.27 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 11, normalized size = 1. \begin{align*} -2\, \left ( -1+\tan \left ( x/2 \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.932386, size = 20, normalized size = 1.82 \begin{align*} -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.94193, size = 61, normalized size = 5.55 \begin{align*} \frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.380706, size = 8, normalized size = 0.73 \begin{align*} - \frac{2}{\tan{\left (\frac{x}{2} \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05628, size = 14, normalized size = 1.27 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]