Optimal. Leaf size=345 \[ \frac{\left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac{\left (2 a^2 C+b^2 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{a b \left (6 a^2 C+126 A b^2+103 b^2 C\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac{\left (3 a^2 b^2 (63 A+50 C)+4 a^4 C+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac{a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right )+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d} \]
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Rubi [A] time = 0.863321, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac{\left (2 a^2 C+b^2 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{a b \left (6 a^2 C+126 A b^2+103 b^2 C\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac{\left (3 a^2 b^2 (63 A+50 C)+4 a^4 C+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac{a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right )+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d} \]
Antiderivative was successfully verified.
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Rule 3050
Rule 3049
Rule 3033
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{7} \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (a (7 A+2 C)+b (7 A+6 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{42} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a^2 (21 A+10 C)+4 a b (21 A+17 C) \cos (c+d x)+6 \left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac{2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{210} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a \left (6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+2 b \left (12 b^2 (7 A+6 C)+a^2 (315 A+244 C)\right ) \cos (c+d x)+4 a \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac{\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac{2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{840} \int \cos (c+d x) \left (8 a^2 \left (6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+420 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x)+24 \left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{\left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac{a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac{\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac{2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos (c+d x) \left (24 \left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right )+1260 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x)\right ) \, dx}{2520}\\ &=\frac{1}{4} a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) x+\frac{\left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac{a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac{\left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac{a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac{\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac{2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.842598, size = 351, normalized size = 1.02 \[ \frac{420 a b \left (16 a^2 (A+C)+b^2 (16 A+15 C)\right ) \sin (2 (c+d x))+105 \left (48 a^2 b^2 (6 A+5 C)+16 a^4 (4 A+3 C)+5 b^4 (8 A+7 C)\right ) \sin (c+d x)+3360 a^2 A b^2 \sin (3 (c+d x))+13440 a^3 A b c+13440 a^3 A b d x+4200 a^2 b^2 C \sin (3 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+560 a^4 C \sin (3 (c+d x))+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+1260 a b^3 C \sin (4 (c+d x))+140 a b^3 C \sin (6 (c+d x))+8400 a b^3 c C+8400 a b^3 C d x+700 A b^4 \sin (3 (c+d x))+84 A b^4 \sin (5 (c+d x))+735 b^4 C \sin (3 (c+d x))+147 b^4 C \sin (5 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 332, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{A{b}^{4}\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{C{b}^{4}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+4\,aA{b}^{3} \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 ) +4\,Ca{b}^{3} \left ( 1/6\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +2\,{a}^{2}A{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{6\,{a}^{2}{b}^{2}C\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 ) }+4\,A{a}^{3}b \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,{a}^{3}bC \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 ) +A{a}^{4}\sin \left ( dx+c \right ) +{\frac{{a}^{4}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998762, size = 444, normalized size = 1.29 \begin{align*} -\frac{560 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 210 \,{\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^{3} b + 3360 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} - 672 \,{\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^{2} b^{2} - 210 \,{\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^{3} + 35 \,{\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 a b^{3} - 112 \,{\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^{4} + 48 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C b^{4} - 1680 \, A a^{4} \sin \left (d x + c\right )}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62317, size = 595, normalized size = 1.72 \begin{align*} \frac{105 \,{\left (2 \,{\left (4 \, A + 3 \, C\right )} a^{3} b +{\left (6 \, A + 5 \, C\right )} a b^{3}\right )} d x +{\left (60 \, C b^{4} \cos \left (d x + c\right )^{6} + 280 \, C a b^{3} \cos \left (d x + c\right )^{5} + 140 \,{\left (3 \, A + 2 \, C\right )} a^{4} + 336 \,{\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \,{\left (7 \, A + 6 \, C\right )} b^{4} + 12 \,{\left (42 \, C a^{2} b^{2} +{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \,{\left (6 \, C a^{3} b +{\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (35 \, C a^{4} + 42 \,{\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 4 \,{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \,{\left (2 \,{\left (4 \, A + 3 \, C\right )} a^{3} b +{\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.1851, size = 850, normalized size = 2.46 \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 = 1.26344, size = 392, normalized size = 1.14 \begin{align*} \frac{C b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{C a b^{3} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac{1}{4} \,{\left (8 \, A a^{3} b + 6 \, C a^{3} b + 6 \, A a b^{3} + 5 \, C a b^{3}\right )} x + \frac{{\left (24 \, C a^{2} b^{2} + 4 \, A b^{4} + 7 \, C b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (2 \, C a^{3} b + 2 \, A a b^{3} + 3 \, C a b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac{{\left (16 \, C a^{4} + 96 \, A a^{2} b^{2} + 120 \, C a^{2} b^{2} + 20 \, A b^{4} + 21 \, C b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (16 \, A a^{3} b + 16 \, C a^{3} b + 16 \, A a b^{3} + 15 \, C a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac{{\left (64 \, A a^{4} + 48 \, C a^{4} + 288 \, A a^{2} b^{2} + 240 \, C a^{2} b^{2} + 40 \, A b^{4} + 35 \, C b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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