Optimal. Leaf size=22 \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+b)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0311741, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2754, 8} \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \cos (c+d x)}{(b+a \cos (c+d x))^2} \, dx &=\frac{\sin (c+d x)}{d (b+a \cos (c+d x))}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=\frac{\sin (c+d x)}{d (b+a \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0918501, size = 22, normalized size = 1. \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.052, size = 51, normalized size = 2.3 \begin{align*} -2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a- \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b-a-b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.36325, size = 53, normalized size = 2.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.2601, size = 68, normalized size = 3.09 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a - b\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]