Optimal. Leaf size=256 \[ \frac {b \left (-23 a^2 C+35 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac {b^2 \left (-106 a^3 C+130 a^2 b B+71 a b^2 C+45 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac {b \left (-83 a^4 C+95 a^3 b B+32 a^2 b^2 C+80 a b^3 B+16 b^4 C\right ) \sin (c+d x)}{30 d}+\frac {1}{8} x \left (-8 a^5 C+8 a^4 b B-8 a^3 b^2 C+24 a^2 b^3 B+9 a b^4 C+3 b^5 B\right )+\frac {b (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac {b C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]
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Rubi [A] time = 0.55, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3015, 2753, 2734} \[ \frac {b \left (32 a^2 b^2 C+95 a^3 b B-83 a^4 C+80 a b^3 B+16 b^4 C\right ) \sin (c+d x)}{30 d}+\frac {b \left (-23 a^2 C+35 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac {b^2 \left (130 a^2 b B-106 a^3 C+71 a b^2 C+45 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac {1}{8} x \left (24 a^2 b^3 B-8 a^3 b^2 C+8 a^4 b B-8 a^5 C+9 a b^4 C+3 b^5 B\right )+\frac {b (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac {b C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 3015
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx &=\frac {\int (a+b \cos (c+d x))^4 \left (b^2 (b B-a C)+b^3 C \cos (c+d x)\right ) \, dx}{b^2}\\ &=\frac {b C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+b \cos (c+d x))^3 \left (b^2 \left (4 b^2 C+5 a (b B-a C)\right )+b^3 (5 b B-a C) \cos (c+d x)\right ) \, dx}{5 b^2}\\ &=\frac {b (5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {b C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+b \cos (c+d x))^2 \left (b^2 \left (20 a^2 b B+15 b^3 B-20 a^3 C+13 a b^2 C\right )+b^3 \left (35 a b B-23 a^2 C+16 b^2 C\right ) \cos (c+d x)\right ) \, dx}{20 b^2}\\ &=\frac {b \left (35 a b B-23 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac {b (5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {b C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+b \cos (c+d x)) \left (b^2 \left (60 a^3 b B+115 a b^3 B-60 a^4 C-7 a^2 b^2 C+32 b^4 C\right )+b^3 \left (130 a^2 b B+45 b^3 B-106 a^3 C+71 a b^2 C\right ) \cos (c+d x)\right ) \, dx}{60 b^2}\\ &=\frac {1}{8} \left (8 a^4 b B+24 a^2 b^3 B+3 b^5 B-8 a^5 C-8 a^3 b^2 C+9 a b^4 C\right ) x+\frac {b \left (95 a^3 b B+80 a b^3 B-83 a^4 C+32 a^2 b^2 C+16 b^4 C\right ) \sin (c+d x)}{30 d}+\frac {b^2 \left (130 a^2 b B+45 b^3 B-106 a^3 C+71 a b^2 C\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {b \left (35 a b B-23 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac {b (5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {b C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 1.13, size = 287, normalized size = 1.12 \[ \frac {-480 a^5 c C-480 a^5 C d x+480 a^4 b B c+480 a^4 b B d x-480 a^3 b^2 c C-480 a^3 b^2 C d x+1440 a^2 b^3 B c+1440 a^2 b^3 B d x+80 a^2 b^3 C \sin (3 (c+d x))+120 b^2 \left (-2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right ) \sin (2 (c+d x))+60 b \left (-24 a^4 C+32 a^3 b B+12 a^2 b^2 C+24 a b^3 B+5 b^4 C\right ) \sin (c+d x)+160 a b^4 B \sin (3 (c+d x))+45 a b^4 C \sin (4 (c+d x))+540 a b^4 c C+540 a b^4 C d x+15 b^5 B \sin (4 (c+d x))+180 b^5 B c+180 b^5 B d x+50 b^5 C \sin (3 (c+d x))+6 b^5 C \sin (5 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 212, normalized size = 0.83 \[ -\frac {15 \, {\left (8 \, C a^{5} - 8 \, B a^{4} b + 8 \, C a^{3} b^{2} - 24 \, B a^{2} b^{3} - 9 \, C a b^{4} - 3 \, B b^{5}\right )} d x - {\left (24 \, C b^{5} \cos \left (d x + c\right )^{4} - 360 \, C a^{4} b + 480 \, B a^{3} b^{2} + 160 \, C a^{2} b^{3} + 320 \, B a b^{4} + 64 \, C b^{5} + 30 \, {\left (3 \, C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (5 \, C a^{2} b^{3} + 10 \, B a b^{4} + 2 \, C b^{5}\right )} \cos \left (d x + c\right )^{2} - 15 \, {\left (8 \, C a^{3} b^{2} - 24 \, B a^{2} b^{3} - 9 \, C a b^{4} - 3 \, B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 227, normalized size = 0.89 \[ \frac {C b^{5} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {1}{8} \, {\left (8 \, C a^{5} - 8 \, B a^{4} b + 8 \, C a^{3} b^{2} - 24 \, B a^{2} b^{3} - 9 \, C a b^{4} - 3 \, B b^{5}\right )} x + \frac {{\left (3 \, C a b^{4} + B b^{5}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (8 \, C a^{2} b^{3} + 16 \, B a b^{4} + 5 \, C b^{5}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (2 \, C a^{3} b^{2} - 6 \, B a^{2} b^{3} - 3 \, C a b^{4} - B b^{5}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {{\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 12 \, C a^{2} b^{3} - 24 \, B a b^{4} - 5 \, C b^{5}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 276, normalized size = 1.08 \[ \frac {\frac {C \,b^{5} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,b^{5} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C a \,b^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a \,b^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 C \,a^{2} b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 B \,a^{2} b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-2 a^{3} b^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b^{2} B \sin \left (d x +c \right )-3 C \,a^{4} b \sin \left (d x +c \right )+B \left (d x +c \right ) a^{4} b -a^{5} C \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 263, normalized size = 1.03 \[ -\frac {480 \, {\left (d x + c\right )} C a^{5} - 480 \, {\left (d x + c\right )} B a^{4} b + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{3} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{3} + 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{4} - 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{5} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{5} + 1440 \, C a^{4} b \sin \left (d x + c\right ) - 1920 \, B a^{3} b^{2} \sin \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 325, normalized size = 1.27 \[ \frac {3\,B\,b^5\,x}{8}-C\,a^5\,x+B\,a^4\,b\,x+\frac {9\,C\,a\,b^4\,x}{8}+\frac {5\,C\,b^5\,\sin \left (c+d\,x\right )}{8\,d}+3\,B\,a^2\,b^3\,x-C\,a^3\,b^2\,x+\frac {B\,b^5\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b^5\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,C\,b^5\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {C\,b^5\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,a\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {4\,B\,a^3\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,C\,a\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a^2\,b^3\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}-\frac {C\,a^3\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {3\,B\,a\,b^4\,\sin \left (c+d\,x\right )}{d}-\frac {3\,C\,a^4\,b\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.17, size = 619, normalized size = 2.42 \[ \begin {cases} B a^{4} b x + \frac {4 B a^{3} b^{2} \sin {\left (c + d x \right )}}{d} + 3 B a^{2} b^{3} x \sin ^{2}{\left (c + d x \right )} + 3 B a^{2} b^{3} x \cos ^{2}{\left (c + d x \right )} + \frac {3 B a^{2} b^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {8 B a b^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 B a b^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 B b^{5} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B b^{5} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B b^{5} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B b^{5} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 B b^{5} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - C a^{5} x - \frac {3 C a^{4} b \sin {\left (c + d x \right )}}{d} - C a^{3} b^{2} x \sin ^{2}{\left (c + d x \right )} - C a^{3} b^{2} x \cos ^{2}{\left (c + d x \right )} - \frac {C a^{3} b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 C a^{2} b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a^{2} b^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a b^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 C a b^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a b^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 C a b^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C b^{5} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C b^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C b^{5} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{3} \left (B a b + B b^{2} \cos {\relax (c )} - C a^{2} + C b^{2} \cos ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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