Optimal. Leaf size=17 \[ \frac {1}{2} i (\cos (x)-i \sin (x))^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4385} \[ \frac {1}{2} i (\cos (x)-i \sin (x))^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4385
Rubi steps
\begin {align*} \int \frac {\cos (x)-i \sin (x)}{\cos (x)+i \sin (x)} \, dx &=\frac {1}{2} i (\cos (x)-i \sin (x))^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 19, normalized size = 1.12 \[ \frac {1}{2} \sin (2 x)+\frac {1}{2} i \cos (2 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 6, normalized size = 0.35 \[ \frac {1}{2} i \, e^{\left (-2 i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 14, normalized size = 0.82 \[ -\frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.20, size = 8, normalized size = 0.47 \[ \frac {1}{\tan \relax (x )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.76, size = 16, normalized size = 0.94 \[ -\frac {\cos \relax (x)}{-\sin \relax (x)+\cos \relax (x)\,1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 8, normalized size = 0.47 \[ \frac {i e^{- 2 i x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________