Optimal. Leaf size=269 \[ -\frac{\left (15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right ) \sin ^3(c+d x)}{15 d}+\frac{\left (15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right ) \sin (c+d x)}{5 d}+\frac{b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right )+\frac{b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d} \]
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Rubi [A] time = 0.507479, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2990, 3033, 3023, 2748, 2635, 8, 2633} \[ -\frac{\left (15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right ) \sin ^3(c+d x)}{15 d}+\frac{\left (15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right ) \sin (c+d x)}{5 d}+\frac{b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right )+\frac{b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 2990
Rule 3033
Rule 3023
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx &=\frac{b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{1}{6} \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (3 a (2 a A+b B)+\left (5 b^2 B+6 a (2 A b+a B)\right ) \cos (c+d x)+2 b (3 A b+4 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 (3 A b+4 a B) \cos ^4(c+d x) \sin (c+d x)}{15 d}+\frac{b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{1}{30} \int \cos ^2(c+d x) \left (15 a^2 (2 a A+b B)+6 \left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \cos (c+d x)+5 b \left (18 a A b+14 a^2 B+5 b^2 B\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b \left (18 a A b+14 a^2 B+5 b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 (3 A b+4 a B) \cos ^4(c+d x) \sin (c+d x)}{15 d}+\frac{b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{1}{120} \int \cos ^2(c+d x) \left (15 \left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right )+24 \left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{b \left (18 a A b+14 a^2 B+5 b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 (3 A b+4 a B) \cos ^4(c+d x) \sin (c+d x)}{15 d}+\frac{b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{1}{5} \left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{8} \left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{\left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{b \left (18 a A b+14 a^2 B+5 b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 (3 A b+4 a B) \cos ^4(c+d x) \sin (c+d x)}{15 d}+\frac{b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{1}{16} \left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) \int 1 \, dx-\frac{\left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{1}{16} \left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) x+\frac{\left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \sin (c+d x)}{5 d}+\frac{\left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{b \left (18 a A b+14 a^2 B+5 b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 (3 A b+4 a B) \cos ^4(c+d x) \sin (c+d x)}{15 d}+\frac{b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}-\frac{\left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.653479, size = 289, normalized size = 1.07 \[ \frac{120 \left (18 a^2 A b+6 a^3 B+15 a b^2 B+5 A b^3\right ) \sin (c+d x)+15 \left (16 a^3 A+48 a^2 b B+48 a A b^2+15 b^3 B\right ) \sin (2 (c+d x))+240 a^2 A b \sin (3 (c+d x))+480 a^3 A c+480 a^3 A d x+90 a^2 b B \sin (4 (c+d x))+1080 a^2 b B c+1080 a^2 b B d x+80 a^3 B \sin (3 (c+d x))+90 a A b^2 \sin (4 (c+d x))+1080 a A b^2 c+1080 a A b^2 d x+300 a b^2 B \sin (3 (c+d x))+36 a b^2 B \sin (5 (c+d x))+100 A b^3 \sin (3 (c+d x))+12 A b^3 \sin (5 (c+d x))+45 b^3 B \sin (4 (c+d x))+5 b^3 B \sin (6 (c+d x))+300 b^3 B c+300 b^3 B d x}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 270, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( A{a}^{3} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{\frac{{a}^{3}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+A{a}^{2}b \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,{a}^{2}bB \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 ) +3\,Aa{b}^{2} \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{3\,Ba{b}^{2}\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 ) }+{\frac{A{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 ) }+B{b}^{3} \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00011, size = 359, normalized size = 1.33 \begin{align*} \frac{240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 90 \,{\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} b + 90 \,{\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 a b^{2} + 192 \,{\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 )} B a b^{2} + 64 \,{\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 )} A b^{3} - 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 )} B b^{3}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58892, size = 517, normalized size = 1.92 \begin{align*} \frac{15 \,{\left (8 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} d x +{\left (40 \, B b^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{4} + 160 \, B a^{3} + 480 \, A a^{2} b + 384 \, B a b^{2} + 128 \, A b^{3} + 10 \,{\left (18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (5 \, B a^{3} + 15 \, A a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (8 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\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 = 7.0304, size = 721, normalized size = 2.68 \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 = 1.30493, size = 311, normalized size = 1.16 \begin{align*} \frac{B b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{1}{16} \,{\left (8 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} x + \frac{{\left (3 \, B a b^{2} + A b^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{3 \,{\left (2 \, B a^{2} b + 2 \, A a b^{2} + B b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (4 \, B a^{3} + 12 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (16 \, A a^{3} + 48 \, B a^{2} b + 48 \, A a b^{2} + 15 \, B b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (6 \, B a^{3} + 18 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\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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