Optimal. Leaf size=12 \[ \frac {e^x \sin (x)}{1+\cos (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2326}
\begin {gather*} \frac {e^x \sin (x)}{\cos (x)+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2326
Rubi steps
\begin {align*} \int \frac {e^x (1+\sin (x))}{1+\cos (x)} \, dx &=\frac {e^x \sin (x)}{1+\cos (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 10, normalized size = 0.83 \begin {gather*} e^x \tan \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 8, normalized size = 0.67
method | result | size |
norman | \({\mathrm e}^{x} \tan \left (\frac {x}{2}\right )\) | \(8\) |
risch | \(-i {\mathrm e}^{x}+\frac {2 i {\mathrm e}^{x}}{1+{\mathrm e}^{i x}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.09, size = 22, normalized size = 1.83 \begin {gather*} \frac {2 \, e^{x} \sin \left (x\right )}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.62, size = 11, normalized size = 0.92 \begin {gather*} \frac {e^{x} \sin \left (x\right )}{\cos \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\sin {\left (x \right )} + 1\right ) e^{x}}{\cos {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.88, size = 7, normalized size = 0.58 \begin {gather*} e^{x} \tan \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.37, size = 7, normalized size = 0.58 \begin {gather*} \mathrm {tan}\left (\frac {x}{2}\right )\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________