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 = 0.35, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, 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 199
Rule 203
Rule 270
Rule 385
Rule 455
Rule 1157
Rule 2565
Rule 2911
Rule 3200
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*}
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Mathematica [A] time = 1.47, 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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fricas [A] time = 0.78, size = 182, normalized size = 0.73 \[ -\frac {645120 \, a b \cos \left (d x + c\right )^{11} - 1576960 \, a b \cos \left (d x + c\right )^{9} + 1013760 \, a b \cos \left (d x + c\right )^{7} - 3465 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} d x + 231 \, {\left (1280 \, b^{2} \cos \left (d x + c\right )^{11} - 128 \, {\left (12 \, a^{2} + 25 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 48 \, {\left (44 \, a^{2} + 45 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 8 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 226, normalized size = 0.90 \[ \frac {1}{1024} \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} x - \frac {a b \cos \left (11 \, d x + 11 \, c\right )}{5632 \, d} - \frac {a b \cos \left (9 \, d x + 9 \, c\right )}{4608 \, d} + \frac {5 \, a b \cos \left (7 \, d x + 7 \, c\right )}{3584 \, d} + \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{512 \, d} - \frac {5 \, a b \cos \left (3 \, d x + 3 \, c\right )}{768 \, d} - \frac {5 \, a b \cos \left (d x + c\right )}{256 \, d} - \frac {b^{2} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {{\left (32 \, a^{2} + 15 \, b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 237, normalized size = 0.95 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a b \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+b^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 137, normalized size = 0.55 \[ \frac {2772 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} - 81920 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a b + 1155 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{28385280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.55, size = 207, normalized size = 0.83 \[ \frac {6930\,a^2\,\sin \left (2\,c+2\,d\,x\right )-13860\,a^2\,\sin \left (4\,c+4\,d\,x\right )-3465\,a^2\,\sin \left (6\,c+6\,d\,x\right )+\frac {3465\,a^2\,\sin \left (8\,c+8\,d\,x\right )}{2}+693\,a^2\,\sin \left (10\,c+10\,d\,x\right )-\frac {51975\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {10395\,b^2\,\sin \left (8\,c+8\,d\,x\right )}{8}-\frac {1155\,b^2\,\sin \left (12\,c+12\,d\,x\right )}{8}-69300\,a\,b\,\cos \left (c+d\,x\right )-23100\,a\,b\,\cos \left (3\,c+3\,d\,x\right )+6930\,a\,b\,\cos \left (5\,c+5\,d\,x\right )+4950\,a\,b\,\cos \left (7\,c+7\,d\,x\right )-770\,a\,b\,\cos \left (9\,c+9\,d\,x\right )-630\,a\,b\,\cos \left (11\,c+11\,d\,x\right )+41580\,a^2\,d\,x+17325\,b^2\,d\,x}{3548160\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.31, size = 656, normalized size = 2.62 \[ \begin {cases} \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {2 a b \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {8 a b \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {16 a b \cos ^{11}{\left (c + d x \right )}}{693 d} + \frac {5 b^{2} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {15 b^{2} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {75 b^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {25 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {75 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {15 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {5 b^{2} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {5 b^{2} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 b^{2} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {33 b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} - \frac {33 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} - \frac {85 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {5 b^{2} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \sin ^{4}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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