Optimal. Leaf size=224 \[ \frac{\sin (c+d x) \left (5 a^2 (3 A+2 C)+20 a b B+2 b^2 (5 A+4 C)\right )}{15 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (2 a^2 C+10 a b B+5 A b^2+4 b^2 C\right )}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^2 B+8 a A b+6 a b C+3 b^2 B\right )}{8 d}+\frac{1}{8} x \left (4 a^2 B+8 a A b+6 a b C+3 b^2 B\right )+\frac{b (2 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.325013, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3049, 3033, 3023, 2734} \[ \frac{\sin (c+d x) \left (5 a^2 (3 A+2 C)+20 a b B+2 b^2 (5 A+4 C)\right )}{15 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (2 a^2 C+10 a b B+5 A b^2+4 b^2 C\right )}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^2 B+8 a A b+6 a b C+3 b^2 B\right )}{8 d}+\frac{1}{8} x \left (4 a^2 B+8 a A b+6 a b C+3 b^2 B\right )+\frac{b (2 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3049
Rule 3033
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos (c+d x) (a+b \cos (c+d x)) \left (a (5 A+2 C)+(5 A b+5 a B+4 b C) \cos (c+d x)+(5 b B+2 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b (5 b B+2 a C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{1}{20} \int \cos (c+d x) \left (4 a^2 (5 A+2 C)+5 \left (8 a A b+4 a^2 B+3 b^2 B+6 a b C\right ) \cos (c+d x)+4 \left (5 A b^2+10 a b B+2 a^2 C+4 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{\left (5 A b^2+10 a b B+2 a^2 C+4 b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac{b (5 b B+2 a C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{1}{60} \int \cos (c+d x) \left (4 \left (20 a b B+5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right )+15 \left (8 a A b+4 a^2 B+3 b^2 B+6 a b C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{1}{8} \left (8 a A b+4 a^2 B+3 b^2 B+6 a b C\right ) x+\frac{\left (20 a b B+5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac{\left (8 a A b+4 a^2 B+3 b^2 B+6 a b C\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{\left (5 A b^2+10 a b B+2 a^2 C+4 b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac{b (5 b B+2 a C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.813885, size = 169, normalized size = 0.75 \[ \frac{60 (c+d x) \left (4 a^2 B+8 a A b+6 a b C+3 b^2 B\right )+60 \sin (c+d x) \left (a^2 (8 A+6 C)+12 a b B+b^2 (6 A+5 C)\right )+120 \sin (2 (c+d x)) \left (a^2 B+2 a b (A+C)+b^2 B\right )+10 \sin (3 (c+d x)) \left (4 a^2 C+8 a b B+4 A b^2+5 b^2 C\right )+15 b (2 a C+b B) \sin (4 (c+d x))+6 b^2 C \sin (5 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 244, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{A{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{b}^{2}B \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{{b}^{2}C\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+2\,aAb \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{2\,abB \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,abC \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +A{a}^{2}\sin \left ( dx+c \right ) +{a}^{2}B \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{\frac{{a}^{2}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00349, size = 315, normalized size = 1.41 \begin{align*} \frac{120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b - 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b + 30 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{2} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{2} + 32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{2} + 480 \, A a^{2} \sin \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7603, size = 414, normalized size = 1.85 \begin{align*} \frac{15 \,{\left (4 \, B a^{2} + 2 \,{\left (4 \, A + 3 \, C\right )} a b + 3 \, B b^{2}\right )} d x +{\left (24 \, C b^{2} \cos \left (d x + c\right )^{4} + 30 \,{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + 40 \,{\left (3 \, A + 2 \, C\right )} a^{2} + 160 \, B a b + 16 \,{\left (5 \, A + 4 \, C\right )} b^{2} + 8 \,{\left (5 \, C a^{2} + 10 \, B a b +{\left (5 \, A + 4 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (4 \, B a^{2} + 2 \,{\left (4 \, A + 3 \, C\right )} a b + 3 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.50372, size = 570, normalized size = 2.54 \begin{align*} \begin{cases} \frac{A a^{2} \sin{\left (c + d x \right )}}{d} + A a b x \sin ^{2}{\left (c + d x \right )} + A a b x \cos ^{2}{\left (c + d x \right )} + \frac{A a b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{2 A b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{A b^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{B a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{4 B a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{2 B a b \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 B b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 B b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 B b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 B b^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 B b^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{2 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{C a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 C a b x \sin ^{4}{\left (c + d x \right )}}{4} + \frac{3 C a b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 C a b x \cos ^{4}{\left (c + d x \right )}}{4} + \frac{3 C a b \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} + \frac{5 C a b \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac{8 C b^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{C b^{2} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{2} \left (A + B \cos{\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21233, size = 248, normalized size = 1.11 \begin{align*} \frac{C b^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (4 \, B a^{2} + 8 \, A a b + 6 \, C a b + 3 \, B b^{2}\right )} x + \frac{{\left (2 \, C a b + B b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (4 \, C a^{2} + 8 \, B a b + 4 \, A b^{2} + 5 \, C b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (B a^{2} + 2 \, A a b + 2 \, C a b + B b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (8 \, A a^{2} + 6 \, C a^{2} + 12 \, B a b + 6 \, A b^{2} + 5 \, C b^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]