Optimal. Leaf size=161 \[ \frac {a \left (a^2 (-C)+12 A b^2+8 b^2 C\right ) \sin (c+d x)}{6 b d}-\frac {\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d} \]
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Rubi [A] time = 0.21, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3024, 2753, 2734} \[ \frac {a \left (a^2 (-C)+12 A b^2+8 b^2 C\right ) \sin (c+d x)}{6 b d}-\frac {\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 3024
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}+\frac {\int (a+b \cos (c+d x))^2 (b (4 A+3 C)-a C \cos (c+d x)) \, dx}{4 b}\\ &=-\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}+\frac {\int (a+b \cos (c+d x)) \left (a b (12 A+7 C)-\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \cos (c+d x)\right ) \, dx}{12 b}\\ &=\frac {1}{8} \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac {a \left (12 A b^2-a^2 C+8 b^2 C\right ) \sin (c+d x)}{6 b d}-\frac {\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 106, normalized size = 0.66 \[ \frac {12 (c+d x) \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+24 \left (C \left (a^2+b^2\right )+A b^2\right ) \sin (2 (c+d x))+48 a b (4 A+3 C) \sin (c+d x)+16 a b C \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 104, normalized size = 0.65 \[ \frac {3 \, {\left (4 \, {\left (2 \, A + C\right )} a^{2} + {\left (4 \, A + 3 \, C\right )} b^{2}\right )} d x + {\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 16 \, C a b \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, A + 2 \, C\right )} a b + 3 \, {\left (4 \, C a^{2} + {\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 116, normalized size = 0.72 \[ \frac {C b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {C a b \sin \left (3 \, d x + 3 \, c\right )}{6 \, d} + \frac {1}{8} \, {\left (8 \, A a^{2} + 4 \, C a^{2} + 4 \, A b^{2} + 3 \, C b^{2}\right )} x + \frac {{\left (C a^{2} + A b^{2} + C b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a b + 3 \, C a b\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 140, normalized size = 0.87 \[ \frac {b^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 C a b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A a b \sin \left (d x +c \right )+a^{2} A \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 130, normalized size = 0.81 \[ \frac {96 \, {\left (d x + c\right )} A a^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 192 \, A a b \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 145, normalized size = 0.90 \[ A\,a^2\,x+\frac {A\,b^2\,x}{2}+\frac {C\,a^2\,x}{2}+\frac {3\,C\,b^2\,x}{8}+\frac {A\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {2\,A\,a\,b\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {C\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{6\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 309, normalized size = 1.92 \[ \begin {cases} A a^{2} x + \frac {2 A a b \sin {\left (c + d x \right )}}{d} + \frac {A b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 C a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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