Optimal. Leaf size=6 \[ \frac{1}{\cos (x)+3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0176402, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2668, 32} \[ \frac{1}{\cos (x)+3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \frac{\sin (x)}{(3+\cos (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(3+x)^2} \, dx,x,\cos (x)\right )\\ &=\frac{1}{3+\cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0131301, size = 6, normalized size = 1. \[ \frac{1}{\cos (x)+3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 7, normalized size = 1.2 \begin{align*} \left ( 3+\cos \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.945644, size = 8, normalized size = 1.33 \begin{align*} \frac{1}{\cos \left (x\right ) + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.432652, size = 22, normalized size = 3.67 \begin{align*} \frac{1}{\cos \left (x\right ) + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.413677, size = 5, normalized size = 0.83 \begin{align*} \frac{1}{\cos{\left (x \right )} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07614, size = 8, normalized size = 1.33 \begin{align*} \frac{1}{\cos \left (x\right ) + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]