Optimal. Leaf size=298 \[ \frac{4 a b \left (2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (a^2 (39 A+50 C)+4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac{\left (10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{\left (5 a^2 (5 A+6 C)+12 A b^2\right ) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{120 d}+\frac{1}{16} x \left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right )+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}+\frac{2 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{15 d} \]
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Rubi [A] time = 1.03726, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4095, 4094, 4074, 4047, 2637, 4045, 8} \[ \frac{4 a b \left (2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (a^2 (39 A+50 C)+4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac{\left (10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{\left (5 a^2 (5 A+6 C)+12 A b^2\right ) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{120 d}+\frac{1}{16} x \left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right )+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}+\frac{2 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{15 d} \]
Antiderivative was successfully verified.
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Rule 4095
Rule 4094
Rule 4074
Rule 4047
Rule 2637
Rule 4045
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+a (5 A+6 C) \sec (c+d x)+b (A+6 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{30} \int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left (12 A b^2+5 a^2 (5 A+6 C)+2 a b (23 A+30 C) \sec (c+d x)+3 b^2 (3 A+10 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac{2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{120} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (6 b \left (4 A b^2+a^2 (39 A+50 C)\right )+a \left (15 a^2 (5 A+6 C)+8 b^2 (32 A+45 C)\right ) \sec (c+d x)+b \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac{2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac{1}{360} \int \cos ^2(c+d x) \left (-3 \left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right )-96 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sec (c+d x)-3 b^2 \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac{2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac{1}{360} \int \cos ^2(c+d x) \left (-3 \left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right )-3 b^2 \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx+\frac{1}{15} \left (4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \int \cos (c+d x) \, dx\\ &=\frac{4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sin (c+d x)}{15 d}+\frac{\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac{2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac{1}{16} \left (-8 b^4 (A+2 C)-12 a^2 b^2 (3 A+4 C)-a^4 (5 A+6 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} \left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) x+\frac{4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sin (c+d x)}{15 d}+\frac{\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac{2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.906359, size = 302, normalized size = 1.01 \[ \frac{480 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \sin (c+d x)+15 \left (96 a^2 b^2 (A+C)+a^4 (15 A+16 C)+16 A b^4\right ) \sin (2 (c+d x))+180 a^2 A b^2 \sin (4 (c+d x))+2160 a^2 A b^2 c+2160 a^2 A b^2 d x+400 a^3 A b \sin (3 (c+d x))+48 a^3 A b \sin (5 (c+d x))+45 a^4 A \sin (4 (c+d x))+5 a^4 A \sin (6 (c+d x))+300 a^4 A c+300 a^4 A d x+2880 a^2 b^2 c C+2880 a^2 b^2 C d x+320 a^3 b C \sin (3 (c+d x))+30 a^4 C \sin (4 (c+d x))+360 a^4 c C+360 a^4 C d x+320 a A b^3 \sin (3 (c+d x))+480 A b^4 c+480 A b^4 d x+960 b^4 c C+960 b^4 C d x}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 294, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( A{a}^{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\,A{a}^{3}b\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 ) }+6\,A{a}^{2}{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 ) +{a}^{4}C \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 ) +{\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}}+A{b}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +6\,C{a}^{2}{b}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,Ca{b}^{3}\sin \left ( dx+c \right ) +C{b}^{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 = 1.01971, size = 382, normalized size = 1.28 \begin{align*} -\frac{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 )} A a^{4} - 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 )} C a^{4} - 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 )} A a^{3} b + 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} b - 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 )} A a^{2} b^{2} - 1440 \,{\left (2 \, d x + 2 \, c + \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} - 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 960 \,{\left (d x + c\right )} C b^{4} - 3840 \, C a b^{3} \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 = 0.567601, size = 504, normalized size = 1.69 \begin{align*} \frac{15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \,{\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \,{\left (A + 2 \, C\right )} b^{4}\right )} d x +{\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 192 \, A a^{3} b \cos \left (d x + c\right )^{4} + 128 \,{\left (4 \, A + 5 \, C\right )} a^{3} b + 320 \,{\left (2 \, A + 3 \, C\right )} a b^{3} + 10 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 64 \,{\left ({\left (4 \, A + 5 \, C\right )} a^{3} b + 5 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \,{\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, A 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27712, size = 1396, normalized size = 4.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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