Optimal. Leaf size=22 \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0123108, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2648} \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{a+a \cos (c+d x)} \, dx &=\frac{\sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0139331, size = 17, normalized size = 0.77 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right )}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.032, size = 17, normalized size = 0.8 \begin{align*}{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.20812, size = 31, normalized size = 1.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56411, size = 53, normalized size = 2.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.597071, size = 20, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x}{a \cos{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.38707, size = 22, normalized size = 1. \begin{align*} \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]