Optimal. Leaf size=250 \[ \frac{\left (12 a^2+25 b^2\right ) \sin (c+d x) \cos ^9(c+d x)}{120 d}-\frac{\left (44 a^2+45 b^2\right ) \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{x \left (12 a^2+5 b^2\right )}{1024}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^7(c+d x)}{7 d}-\frac{b^2 \sin (c+d x) \cos ^{11}(c+d x)}{12 d} \]
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Rubi [A] time = 0.354725, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2911, 2565, 270, 3200, 455, 1157, 385, 199, 203} \[ \frac{\left (12 a^2+25 b^2\right ) \sin (c+d x) \cos ^9(c+d x)}{120 d}-\frac{\left (44 a^2+45 b^2\right ) \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{x \left (12 a^2+5 b^2\right )}{1024}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^7(c+d x)}{7 d}-\frac{b^2 \sin (c+d x) \cos ^{11}(c+d x)}{12 d} \]
Antiderivative was successfully verified.
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Rule 2911
Rule 2565
Rule 270
Rule 3200
Rule 455
Rule 1157
Rule 385
Rule 199
Rule 203
Rubi steps
\begin{align*} \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\int \cos ^6(c+d x) \sin ^4(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a^2+\left (a^2+b^2\right ) x^2\right )}{\left (1+x^2\right )^7} \, dx,x,\tan (c+d x)\right )}{d}-\frac{(2 a b) \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}-\frac{\operatorname{Subst}\left (\int \frac{-b^2+12 b^2 x^2-12 \left (a^2+b^2\right ) x^4}{\left (1+x^2\right )^6} \, dx,x,\tan (c+d x)\right )}{12 d}-\frac{(2 a b) \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac{\operatorname{Subst}\left (\int \frac{-3 \left (4 a^2+5 b^2\right )+120 \left (a^2+b^2\right ) x^2}{\left (1+x^2\right )^5} \, dx,x,\tan (c+d x)\right )}{120 d}\\ &=-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}-\frac{\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac{\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac{\left (12 a^2+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^4} \, dx,x,\tan (c+d x)\right )}{320 d}\\ &=-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac{\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac{\left (12 a^2+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{384 d}\\ &=-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac{\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac{\left (12 a^2+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{512 d}\\ &=-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{\left (12 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac{\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac{\left (12 a^2+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{1024 d}\\ &=\frac{\left (12 a^2+5 b^2\right ) x}{1024}-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{\left (12 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac{\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac{b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}\\ \end{align*}
Mathematica [A] time = 1.38634, size = 202, normalized size = 0.81 \[ \frac{55440 a^2 \sin (2 (c+d x))-110880 a^2 \sin (4 (c+d x))-27720 a^2 \sin (6 (c+d x))+13860 a^2 \sin (8 (c+d x))+5544 a^2 \sin (10 (c+d x))+332640 a^2 d x-554400 a b \cos (c+d x)-184800 a b \cos (3 (c+d x))+55440 a b \cos (5 (c+d x))+39600 a b \cos (7 (c+d x))-6160 a b \cos (9 (c+d x))-5040 a b \cos (11 (c+d x))-51975 b^2 \sin (4 (c+d x))+10395 b^2 \sin (8 (c+d x))-1155 b^2 \sin (12 (c+d x))+166320 b^2 c+138600 b^2 d x}{28385280 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 237, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10}}-{\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}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +2\,ab \left ( -1/11\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\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 ) +{b}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{12}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{64}}+{\frac{\sin \left ( dx+c \right ) }{384} \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}{1024}}+{\frac{5\,c}{1024}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01978, size = 185, normalized size = 0.74 \begin{align*} \frac{2772 \,{\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^{2} - 81920 \,{\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 )} a b + 1155 \,{\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{28385280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99478, size = 486, normalized size = 1.94 \begin{align*} -\frac{645120 \, a b \cos \left (d x + c\right )^{11} - 1576960 \, a b \cos \left (d x + c\right )^{9} + 1013760 \, a b \cos \left (d x + c\right )^{7} - 3465 \,{\left (12 \, a^{2} + 5 \, b^{2}\right )} d x + 231 \,{\left (1280 \, b^{2} \cos \left (d x + c\right )^{11} - 128 \,{\left (12 \, a^{2} + 25 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 48 \,{\left (44 \, a^{2} + 45 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 8 \,{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \,{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \,{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 90.7959, size = 656, normalized size = 2.62 \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 = 1.30758, size = 305, normalized size = 1.22 \begin{align*} \frac{1}{1024} \,{\left (12 \, a^{2} + 5 \, b^{2}\right )} x - \frac{a b \cos \left (11 \, d x + 11 \, c\right )}{5632 \, d} - \frac{a b \cos \left (9 \, d x + 9 \, c\right )}{4608 \, d} + \frac{5 \, a b \cos \left (7 \, d x + 7 \, c\right )}{3584 \, d} + \frac{a b \cos \left (5 \, d x + 5 \, c\right )}{512 \, d} - \frac{5 \, a b \cos \left (3 \, d x + 3 \, c\right )}{768 \, d} - \frac{5 \, a b \cos \left (d x + c\right )}{256 \, d} - \frac{b^{2} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac{a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac{a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} + \frac{a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac{{\left (4 \, a^{2} + 3 \, b^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac{{\left (32 \, a^{2} + 15 \, b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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