Optimal. Leaf size=50 \[ \frac{1}{2} x \left (2 a^2+b^2\right )+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0149788, antiderivative size = 50, 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 = {2644} \[ \frac{1}{2} x \left (2 a^2+b^2\right )+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2644
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \, dx &=\frac{1}{2} \left (2 a^2+b^2\right ) x+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0716666, size = 46, normalized size = 0.92 \[ \frac{2 \left (2 a^2+b^2\right ) (c+d x)+8 a b \sin (c+d x)+b^2 \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 51, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,ab\sin \left ( dx+c \right ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.954685, size = 59, normalized size = 1.18 \begin{align*} a^{2} x + \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} b^{2}}{4 \, d} + \frac{2 \, a b \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80501, size = 93, normalized size = 1.86 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} d x +{\left (b^{2} \cos \left (d x + c\right ) + 4 \, a b\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.281447, size = 78, normalized size = 1.56 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b \sin{\left (c + d x \right )}}{d} + \frac{b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.52084, size = 58, normalized size = 1.16 \begin{align*} \frac{1}{2} \,{\left (2 \, a^{2} + b^{2}\right )} x + \frac{b^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{2 \, a b \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]