Optimal. Leaf size=65 \[ \frac{(A+2 B) \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}+\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0537473, antiderivative size = 65, 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 = {2750, 2648} \[ \frac{(A+2 B) \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}+\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx &=\frac{(A-B) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(A+2 B) \int \frac{1}{a+a \cos (c+d x)} \, dx}{3 a}\\ &=\frac{(A-B) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(A+2 B) \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.168725, size = 76, normalized size = 1.17 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left ((A+2 B) \sin \left (c+\frac{3 d x}{2}\right )+3 (A+B) \sin \left (\frac{d x}{2}\right )-3 B \sin \left (c+\frac{d x}{2}\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{a}^{2}d} \left ({\frac{A}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{B}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+A\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +B\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.2343, size = 126, normalized size = 1.94 \begin{align*} \frac{\frac{A{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac{B{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.36321, size = 144, normalized size = 2.22 \begin{align*} \frac{{\left ({\left (A + 2 \, B\right )} \cos \left (d x + c\right ) + 2 \, A + B\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.85249, size = 94, normalized size = 1.45 \begin{align*} \begin{cases} \frac{A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d} + \frac{A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a^{2} d} - \frac{B \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d} + \frac{B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + B \cos{\left (c \right )}\right )}{\left (a \cos{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21363, size = 81, normalized size = 1.25 \begin{align*} \frac{A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{6 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]