Optimal. Leaf size=301 \[ -\frac{a \left (-a^2 b^2 (190 A+121 C)+4 a^4 C-32 b^4 (5 A+4 C)\right ) \sin (c+d x)}{60 b d}-\frac{\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{a \left (-4 a^2 C+70 A b^2+53 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{\left (-2 a^2 b^2 (130 A+89 C)+8 a^4 C-15 b^4 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left (12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+b^4 (6 A+5 C)\right )+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d} \]
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Rubi [A] time = 0.528148, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3024, 2753, 2734} \[ -\frac{a \left (-a^2 b^2 (190 A+121 C)+4 a^4 C-32 b^4 (5 A+4 C)\right ) \sin (c+d x)}{60 b d}-\frac{\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{a \left (-4 a^2 C+70 A b^2+53 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{\left (-2 a^2 b^2 (130 A+89 C)+8 a^4 C-15 b^4 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left (12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+b^4 (6 A+5 C)\right )+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x))^4 (b (6 A+5 C)-a C \cos (c+d x)) \, dx}{6 b}\\ &=-\frac{a C (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{C (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x))^3 \left (3 a b (10 A+7 C)-\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) \cos (c+d x)\right ) \, dx}{30 b}\\ &=-\frac{\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}-\frac{a C (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{C (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x))^2 \left (3 b \left (8 a^2 (5 A+3 C)+5 b^2 (6 A+5 C)\right )+3 a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) \cos (c+d x)\right ) \, dx}{120 b}\\ &=\frac{a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{120 b d}-\frac{\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}-\frac{a C (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{C (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x)) \left (3 a b \left (8 a^2 (15 A+8 C)+b^2 (230 A+181 C)\right )-3 \left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \cos (c+d x)\right ) \, dx}{360 b}\\ &=\frac{1}{16} \left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) x-\frac{a \left (4 a^4 C-32 b^4 (5 A+4 C)-a^2 b^2 (190 A+121 C)\right ) \sin (c+d x)}{60 b d}-\frac{\left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac{a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{120 b d}-\frac{\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}-\frac{a C (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{C (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}\\ \end{align*}
Mathematica [A] time = 0.841072, size = 301, normalized size = 1. \[ \frac{480 a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x)+15 \left (96 a^2 b^2 (A+C)+16 a^4 C+b^4 (16 A+15 C)\right ) \sin (2 (c+d x))+2880 a^2 A b^2 c+2880 a^2 A b^2 d x+960 a^4 A c+960 a^4 A d x+180 a^2 b^2 C \sin (4 (c+d x))+2160 a^2 b^2 c C+2160 a^2 b^2 C d x+320 a^3 b C \sin (3 (c+d x))+480 a^4 c C+480 a^4 C d x+320 a A b^3 \sin (3 (c+d x))+400 a b^3 C \sin (3 (c+d x))+48 a b^3 C \sin (5 (c+d x))+30 A b^4 \sin (4 (c+d x))+360 A b^4 c+360 A b^4 d x+45 b^4 C \sin (4 (c+d x))+5 b^4 C \sin (6 (c+d x))+300 b^4 c C+300 b^4 C d x}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 294, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( C{b}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{4\,Ca{b}^{3}\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 ) }+A{b}^{4} \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 ) +6\,{a}^{2}{b}^{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{4\,aA{b}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{4\,{a}^{3}bC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+6\,{a}^{2}A{b}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{4}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +4\,A{a}^{3}b\sin \left ( dx+c \right ) +A{a}^{4} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01342, size = 382, normalized size = 1.27 \begin{align*} \frac{960 \,{\left (d x + c\right )} A a^{4} + 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} b + 1440 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 180 \,{\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} b^{2} - 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} + 256 \,{\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 b^{3} + 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 )} A b^{4} - 5 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} + 3840 \, A a^{3} b \sin \left (d x + c\right )}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63747, size = 504, normalized size = 1.67 \begin{align*} \frac{15 \,{\left (8 \,{\left (2 \, A + C\right )} a^{4} + 12 \,{\left (4 \, A + 3 \, C\right )} a^{2} b^{2} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} d x +{\left (40 \, C b^{4} \cos \left (d x + c\right )^{5} + 192 \, C a b^{3} \cos \left (d x + c\right )^{4} + 320 \,{\left (3 \, A + 2 \, C\right )} a^{3} b + 128 \,{\left (5 \, A + 4 \, C\right )} a b^{3} + 10 \,{\left (36 \, C a^{2} b^{2} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} + 64 \,{\left (5 \, C a^{3} b +{\left (5 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (8 \, C a^{4} + 12 \,{\left (4 \, A + 3 \, C\right )} a^{2} b^{2} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.08065, size = 748, normalized size = 2.49 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.68198, size = 333, normalized size = 1.11 \begin{align*} \frac{C b^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{C a b^{3} \sin \left (5 \, d x + 5 \, c\right )}{20 \, d} + \frac{1}{16} \,{\left (16 \, A a^{4} + 8 \, C a^{4} + 48 \, A a^{2} b^{2} + 36 \, C a^{2} b^{2} + 6 \, A b^{4} + 5 \, C b^{4}\right )} x + \frac{{\left (12 \, C a^{2} b^{2} + 2 \, A b^{4} + 3 \, C b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (4 \, C a^{3} b + 4 \, A a b^{3} + 5 \, C a b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (16 \, C a^{4} + 96 \, A a^{2} b^{2} + 96 \, C a^{2} b^{2} + 16 \, A b^{4} + 15 \, C b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (8 \, A a^{3} b + 6 \, C a^{3} b + 6 \, A a b^{3} + 5 \, C a b^{3}\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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