Optimal. Leaf size=181 \[ \frac{a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B+6 C)+\frac{(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac{(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d} \]
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Rubi [A] time = 0.335672, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {3045, 2968, 3023, 2751, 2644} \[ \frac{a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B+6 C)+\frac{(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac{(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2968
Rule 3023
Rule 2751
Rule 2644
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \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+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{\int \cos (c+d x) (a+a \cos (c+d x))^2 (a (5 A+2 C)+a (5 B+2 C) \cos (c+d x)) \, dx}{5 a}\\ &=\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{\int (a+a \cos (c+d x))^2 \left (a (5 A+2 C) \cos (c+d x)+a (5 B+2 C) \cos ^2(c+d x)\right ) \, dx}{5 a}\\ &=\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{(5 B+2 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 a d}+\frac{\int (a+a \cos (c+d x))^2 \left (3 a^2 (5 B+2 C)+a^2 (20 A-5 B+6 C) \cos (c+d x)\right ) \, dx}{20 a^2}\\ &=\frac{(20 A-5 B+6 C) (a+a \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{(5 B+2 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 a d}+\frac{1}{12} (8 A+7 B+6 C) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac{1}{8} a^2 (8 A+7 B+6 C) x+\frac{a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(20 A-5 B+6 C) (a+a \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac{(5 B+2 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 a d}\\ \end{align*}
Mathematica [A] time = 0.613454, size = 132, normalized size = 0.73 \[ \frac{a^2 (60 (14 A+12 B+11 C) \sin (c+d x)+240 (A+B+C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+480 A d x+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B c+420 B d x+90 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+240 c C+360 C d x)}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 247, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( 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}}+2\,A{a}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{2\,{a}^{2}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{2}C \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 ) +{\frac{A{a}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{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{{a}^{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 ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01848, size = 319, normalized size = 1.76 \begin{align*} -\frac{160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{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 a^{2} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{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 a^{2} + 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 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^{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.
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Fricas [A] time = 2.05661, size = 305, normalized size = 1.69 \begin{align*} \frac{15 \,{\left (8 \, A + 7 \, B + 6 \, C\right )} a^{2} d x +{\left (24 \, C a^{2} \cos \left (d x + c\right )^{4} + 30 \,{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, A + 10 \, B + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (8 \, A + 7 \, B + 6 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \,{\left (25 \, A + 20 \, B + 18 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.38498, size = 570, normalized size = 3.15 \begin{align*} \begin{cases} A a^{2} x \sin ^{2}{\left (c + d x \right )} + A a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac{2 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{A a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{A a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{A a^{2} \sin{\left (c + d x \right )}}{d} + \frac{3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{B a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{B a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{4 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 B a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{2 B a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac{3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac{8 C a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 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 ^{4}{\left (c + d x \right )}}{d} + \frac{5 C a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac{C a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\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.
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Giac [A] time = 1.21977, size = 216, normalized size = 1.19 \begin{align*} \frac{C a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (8 \, A a^{2} + 7 \, B a^{2} + 6 \, C a^{2}\right )} x + \frac{{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (4 \, A a^{2} + 8 \, B a^{2} + 9 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (A a^{2} + B a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac{{\left (14 \, A a^{2} + 12 \, B a^{2} + 11 \, C a^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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