Optimal. Leaf size=187 \[ \frac{\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac{\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac{a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac{3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a b x}{128}-\frac{b^2 \cos ^{11}(c+d x)}{11 d} \]
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Rubi [A] time = 0.314548, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2911, 2568, 2635, 8, 3201, 446, 77} \[ \frac{\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac{\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac{a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac{3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a b x}{128}-\frac{b^2 \cos ^{11}(c+d x)}{11 d} \]
Antiderivative was successfully verified.
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Rule 2911
Rule 2568
Rule 2635
Rule 8
Rule 3201
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\int \cos ^6(c+d x) \sin ^3(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac{1}{5} (3 a b) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac{\left (\sqrt{\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname{Subst}\left (\int x^3 \left (1-x^2\right )^{5/2} \left (a^2+b^2 x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac{a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac{1}{40} (3 a b) \int \cos ^6(c+d x) \, dx+\frac{\left (\sqrt{\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname{Subst}\left (\int (1-x)^{5/2} x \left (a^2+b^2 x\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac{3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac{a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac{1}{16} (a b) \int \cos ^4(c+d x) \, dx+\frac{\left (\sqrt{\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \left (\left (a^2+b^2\right ) (1-x)^{5/2}+\left (-a^2-2 b^2\right ) (1-x)^{7/2}+b^2 (1-x)^{9/2}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac{\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac{b^2 \cos ^{11}(c+d x)}{11 d}+\frac{a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac{a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac{3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac{a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac{1}{64} (3 a b) \int \cos ^2(c+d x) \, dx\\ &=-\frac{\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac{\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac{b^2 \cos ^{11}(c+d x)}{11 d}+\frac{3 a b \cos (c+d x) \sin (c+d x)}{128 d}+\frac{a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac{a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac{3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac{a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac{1}{128} (3 a b) \int 1 \, dx\\ &=\frac{3 a b x}{128}-\frac{\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac{\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac{b^2 \cos ^{11}(c+d x)}{11 d}+\frac{3 a b \cos (c+d x) \sin (c+d x)}{128 d}+\frac{a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac{a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac{3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac{a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 1.08595, size = 197, normalized size = 1.05 \[ \frac{-6930 \left (12 a^2+5 b^2\right ) \cos (c+d x)-2310 \left (16 a^2+5 b^2\right ) \cos (3 (c+d x))+5940 a^2 \cos (7 (c+d x))+1540 a^2 \cos (9 (c+d x))+13860 a b \sin (2 (c+d x))-27720 a b \sin (4 (c+d x))-6930 a b \sin (6 (c+d x))+3465 a b \sin (8 (c+d x))+1386 a b \sin (10 (c+d x))+83160 a b c+83160 a b d x+3465 b^2 \cos (5 (c+d x))+2475 b^2 \cos (7 (c+d x))-385 b^2 \cos (9 (c+d x))-315 b^2 \cos (11 (c+d x))}{3548160 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 171, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) +2\,ab \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \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 ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +{b}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997068, size = 155, normalized size = 0.83 \begin{align*} \frac{56320 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} + 693 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 5120 \,{\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{3548160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9666, size = 360, normalized size = 1.93 \begin{align*} -\frac{40320 \, b^{2} \cos \left (d x + c\right )^{11} - 49280 \,{\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 63360 \,{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} - 10395 \, a b d x - 693 \,{\left (128 \, a b \cos \left (d x + c\right )^{9} - 176 \, a b \cos \left (d x + c\right )^{7} + 8 \, a b \cos \left (d x + c\right )^{5} + 10 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 61.7796, size = 384, normalized size = 2.05 \begin{align*} \begin{cases} - \frac{a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac{3 a b x \sin ^{10}{\left (c + d x \right )}}{128} + \frac{15 a b x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{128} + \frac{15 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{15 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{64} + \frac{15 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{128} + \frac{3 a b x \cos ^{10}{\left (c + d x \right )}}{128} + \frac{3 a b \sin ^{9}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{7 a b \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac{a b \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{7 a b \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac{3 a b \sin{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{128 d} - \frac{b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac{8 b^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25825, size = 293, normalized size = 1.57 \begin{align*} \frac{3}{128} \, a b x - \frac{b^{2} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac{b^{2} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} + \frac{a b \sin \left (10 \, d x + 10 \, c\right )}{2560 \, d} + \frac{a b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{a b \sin \left (6 \, d x + 6 \, c\right )}{512 \, d} - \frac{a b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{a b \sin \left (2 \, d x + 2 \, c\right )}{256 \, d} + \frac{{\left (4 \, a^{2} - b^{2}\right )} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac{{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac{{\left (16 \, a^{2} + 5 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac{{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )}{512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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