Integrand size = 29, antiderivative size = 187 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 a b x}{128}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {b^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d} \]
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Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2990, 2648, 2715, 8, 3280, 457, 78} \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac {a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac {3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac {a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac {a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a b x}{128}-\frac {b^2 \cos ^{11}(c+d x)}{11 d} \]
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Rule 8
Rule 78
Rule 457
Rule 2648
Rule 2715
Rule 2990
Rule 3280
Rubi steps \begin{align*} \text {integral}& = (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 ) \text {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 ) \text {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 ) \text {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*}
Time = 0.64 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.05 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {83160 a b c+83160 a b d x-6930 \left (12 a^2+5 b^2\right ) \cos (c+d x)-2310 \left (16 a^2+5 b^2\right ) \cos (3 (c+d x))+3465 b^2 \cos (5 (c+d x))+5940 a^2 \cos (7 (c+d x))+2475 b^2 \cos (7 (c+d x))+1540 a^2 \cos (9 (c+d x))-385 b^2 \cos (9 (c+d x))-315 b^2 \cos (11 (c+d x))+13860 a b \sin (2 (c+d x))-27720 a b \sin (4 (c+d x))-6930 a b \sin (6 (c+d x))+3465 a b \sin (8 (c+d x))+1386 a b \sin (10 (c+d x))}{3548160 d} \]
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Time = 1.60 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )}{d}\) | \(171\) |
default | \(\frac {a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )}{d}\) | \(171\) |
parallelrisch | \(\frac {\left (-36960 a^{2}-11550 b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (5940 a^{2}+2475 b^{2}\right ) \cos \left (7 d x +7 c \right )+\left (1540 a^{2}-385 b^{2}\right ) \cos \left (9 d x +9 c \right )-315 \cos \left (11 d x +11 c \right ) b^{2}+1386 a b \sin \left (10 d x +10 c \right )+3465 \cos \left (5 d x +5 c \right ) b^{2}+13860 a b \sin \left (2 d x +2 c \right )-27720 a b \sin \left (4 d x +4 c \right )-6930 a b \sin \left (6 d x +6 c \right )+3465 a b \sin \left (8 d x +8 c \right )+\left (-83160 a^{2}-34650 b^{2}\right ) \cos \left (d x +c \right )+83160 a b x d -112640 a^{2}-40960 b^{2}}{3548160 d}\) | \(197\) |
risch | \(-\frac {\cos \left (11 d x +11 c \right ) b^{2}}{11264 d}+\frac {3 a b x}{128}-\frac {3 a^{2} \cos \left (d x +c \right )}{128 d}-\frac {5 b^{2} \cos \left (d x +c \right )}{512 d}+\frac {a b \sin \left (10 d x +10 c \right )}{2560 d}+\frac {\cos \left (9 d x +9 c \right ) a^{2}}{2304 d}-\frac {\cos \left (9 d x +9 c \right ) b^{2}}{9216 d}+\frac {a b \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 \cos \left (7 d x +7 c \right ) a^{2}}{1792 d}+\frac {5 \cos \left (7 d x +7 c \right ) b^{2}}{7168 d}-\frac {a b \sin \left (6 d x +6 c \right )}{512 d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{1024 d}-\frac {a b \sin \left (4 d x +4 c \right )}{128 d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{96 d}-\frac {5 \cos \left (3 d x +3 c \right ) b^{2}}{1536 d}+\frac {a b \sin \left (2 d x +2 c \right )}{256 d}\) | \(251\) |
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Time = 0.38 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {40320 \, b^{2} \cos \left (d x + c\right )^{11} - 49280 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 63360 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} - 10395 \, a b d x - 693 \, {\left (128 \, a b \cos \left (d x + c\right )^{9} - 176 \, a b \cos \left (d x + c\right )^{7} + 8 \, a b \cos \left (d x + c\right )^{5} + 10 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (178) = 356\).
Time = 1.86 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.05 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {3 a b x \sin ^{10}{\left (c + d x \right )}}{128} + \frac {15 a b x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{128} + \frac {15 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{64} + \frac {15 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a b x \cos ^{10}{\left (c + d x \right )}}{128} + \frac {3 a b \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {7 a b \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {a b \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a b \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {3 a b \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{128 d} - \frac {b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {8 b^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {56320 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} + 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 5120 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{3548160 \, d} \]
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Time = 0.49 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.16 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3}{128} \, a b x - \frac {b^{2} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {b^{2} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} + \frac {a b \sin \left (10 \, d x + 10 \, c\right )}{2560 \, d} + \frac {a b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{512 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{256 \, d} + \frac {{\left (4 \, a^{2} - b^{2}\right )} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {{\left (16 \, a^{2} + 5 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )}{512 \, d} \]
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Time = 15.17 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.06 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3\,a\,b\,x}{128}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {4\,a^2}{3}+\frac {32\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (8\,a^2-48\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {64\,a^2}{7}-\frac {48\,b^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {32\,a^2}{3}-\frac {80\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {44\,a^2}{63}+\frac {16\,b^2}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {32\,a^2}{63}-\frac {80\,b^2}{63}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {64\,a^2}{3}+\frac {176\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {72\,a^2}{7}+\frac {240\,b^2}{7}\right )+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+\frac {4\,a^2}{63}+\frac {16\,b^2}{693}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {3323\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {108\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {841\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}+\frac {841\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {108\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {3323\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}+\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]
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