Optimal. Leaf size=264 \[ \frac {a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac {a \left (C \left (a^2+12 b^2\right )+15 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} b x \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d} \]
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Rubi [A] time = 0.54, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3050, 3049, 3033, 3023, 2734} \[ \frac {a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac {a \left (C \left (a^2+12 b^2\right )+15 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} b x \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rule 3033
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a (3 A+C)+b (6 A+5 C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a^2 (15 A+8 C)+a b (60 A+47 C) \cos (c+d x)+\left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos (c+d x) \left (8 a^3 (15 A+8 C)+15 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x)+24 a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{360} \int \cos (c+d x) \left (24 a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right )+45 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {1}{16} b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) x+\frac {a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 252, normalized size = 0.95 \[ \frac {80 a^3 C \sin (3 (c+d x))+120 a \left (a^2 (8 A+6 C)+3 b^2 (6 A+5 C)\right ) \sin (c+d x)+15 b \left (48 a^2 (A+C)+b^2 (16 A+15 C)\right ) \sin (2 (c+d x))+1440 a^2 A b c+1440 a^2 A b d x+90 a^2 b C \sin (4 (c+d x))+1080 a^2 b c C+1080 a^2 b C d x+240 a A b^2 \sin (3 (c+d x))+300 a b^2 C \sin (3 (c+d x))+36 a b^2 C \sin (5 (c+d x))+30 A b^3 \sin (4 (c+d x))+360 A b^3 c+360 A b^3 d x+45 b^3 C \sin (4 (c+d x))+5 b^3 C \sin (6 (c+d x))+300 b^3 c C+300 b^3 C d x}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.83, size = 189, normalized size = 0.72 \[ \frac {15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} d x + {\left (40 \, C b^{3} \cos \left (d x + c\right )^{5} + 144 \, C a b^{2} \cos \left (d x + c\right )^{4} + 80 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 96 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 10 \, {\left (18 \, C a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (5 \, C a^{3} + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.59, size = 216, normalized size = 0.82 \[ \frac {C b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, C a b^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{16} \, {\left (24 \, A a^{2} b + 18 \, C a^{2} b + 6 \, A b^{3} + 5 \, C b^{3}\right )} x + \frac {{\left (6 \, C a^{2} b + 2 \, A b^{3} + 3 \, C b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, C a^{3} + 12 \, A a b^{2} + 15 \, C a b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (48 \, A a^{2} b + 48 \, C a^{2} b + 16 \, A b^{3} + 15 \, C b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (8 \, A a^{3} + 6 \, C a^{3} + 18 \, A a b^{2} + 15 \, C a b^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 249, normalized size = 0.94 \[ \frac {A \,a^{3} \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A a \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b^{3} C \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 243, normalized size = 0.92 \[ -\frac {320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{2} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a b^{2} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} - 960 \, A a^{3} \sin \left (d x + c\right )}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 617, normalized size = 2.34 \[ \frac {\left (2\,A\,a^3-\frac {5\,A\,b^3}{4}+2\,C\,a^3-\frac {11\,C\,b^3}{8}+6\,A\,a\,b^2-3\,A\,a^2\,b+6\,C\,a\,b^2-\frac {15\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (10\,A\,a^3-\frac {7\,A\,b^3}{4}+\frac {22\,C\,a^3}{3}+\frac {5\,C\,b^3}{24}+22\,A\,a\,b^2-9\,A\,a^2\,b+14\,C\,a\,b^2-\frac {21\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (20\,A\,a^3-\frac {A\,b^3}{2}+12\,C\,a^3-\frac {15\,C\,b^3}{4}+36\,A\,a\,b^2-6\,A\,a^2\,b+\frac {156\,C\,a\,b^2}{5}-\frac {3\,C\,a^2\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (20\,A\,a^3+\frac {A\,b^3}{2}+12\,C\,a^3+\frac {15\,C\,b^3}{4}+36\,A\,a\,b^2+6\,A\,a^2\,b+\frac {156\,C\,a\,b^2}{5}+\frac {3\,C\,a^2\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (10\,A\,a^3+\frac {7\,A\,b^3}{4}+\frac {22\,C\,a^3}{3}-\frac {5\,C\,b^3}{24}+22\,A\,a\,b^2+9\,A\,a^2\,b+14\,C\,a\,b^2+\frac {21\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^3+\frac {5\,A\,b^3}{4}+2\,C\,a^3+\frac {11\,C\,b^3}{8}+6\,A\,a\,b^2+3\,A\,a^2\,b+6\,C\,a\,b^2+\frac {15\,C\,a^2\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {b\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,d}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,\left (\frac {3\,A\,b^3}{4}+\frac {5\,C\,b^3}{8}+3\,A\,a^2\,b+\frac {9\,C\,a^2\,b}{4}\right )}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.74, size = 668, normalized size = 2.53 \[ \begin {cases} \frac {A a^{3} \sin {\left (c + d x \right )}}{d} + \frac {3 A a^{2} b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A a b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {3 A a b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A b^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A b^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A b^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 A b^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a^{2} b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 C a^{2} b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{2} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 C a^{2} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C a b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {4 C a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C b^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 C b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 C b^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 C b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 C b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 C b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right )^{3} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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