Optimal. Leaf size=91 \[ \frac{a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+B+C)+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0831877, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {3023, 2734} \[ \frac{a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+B+C)+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}+\frac{\int (a+a \cos (c+d x)) (a (3 A+2 C)+a (3 B-C) \cos (c+d x)) \, dx}{3 a}\\ &=\frac{1}{2} a (2 A+B+C) x+\frac{a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac{a (3 B-C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.228186, size = 65, normalized size = 0.71 \[ \frac{a (3 (4 A+4 B+3 C) \sin (c+d x)+12 A d x+3 (B+C) \sin (2 (c+d x))+6 B d x+C \sin (3 (c+d x))+6 C d x)}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.022, size = 102, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{aC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Ba \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aA\sin \left ( dx+c \right ) +Ba\sin \left ( dx+c \right ) +aA \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.988821, size = 132, normalized size = 1.45 \begin{align*} \frac{12 \,{\left (d x + c\right )} A a + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 12 \, A a \sin \left (d x + c\right ) + 12 \, B a \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.11899, size = 162, normalized size = 1.78 \begin{align*} \frac{3 \,{\left (2 \, A + B + C\right )} a d x +{\left (2 \, C a \cos \left (d x + c\right )^{2} + 3 \,{\left (B + C\right )} a \cos \left (d x + c\right ) + 2 \,{\left (3 \, A + 3 \, B + 2 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.703342, size = 189, normalized size = 2.08 \begin{align*} \begin{cases} A a x + \frac{A a \sin{\left (c + d x \right )}}{d} + \frac{B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{B a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{B a \sin{\left (c + d x \right )}}{d} + \frac{C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{C a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{C a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right ) \left (A + B \cos{\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19219, size = 103, normalized size = 1.13 \begin{align*} \frac{1}{2} \,{\left (2 \, A a + B a + C a\right )} x + \frac{C a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (B a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, A a + 4 \, B a + 3 \, C a\right )} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]