Integrand size = 29, antiderivative size = 163 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{16} \left (2 a^2+b^2\right ) x-\frac {2 a b \cos (c+d x)}{5 d}+\frac {2 a b \cos ^3(c+d x)}{15 d}-\frac {\left (2 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a b \cos (c+d x) \sin ^4(c+d x)}{15 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d} \]
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Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3129, 3112, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (2 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {\left (2 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (2 a^2+b^2\right )+\frac {2 a b \cos ^3(c+d x)}{15 d}-\frac {2 a b \cos (c+d x)}{5 d}+\frac {a b \sin ^4(c+d x) \cos (c+d x)}{15 d}+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 2968
Rule 3102
Rule 3112
Rule 3129
Rubi steps \begin{align*} \text {integral}& = \int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (1-\sin ^2(c+d x)\right ) \, dx \\ & = \frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{6} \int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (3 a+b \sin (c+d x)-2 a \sin ^2(c+d x)\right ) \, dx \\ & = \frac {a b \cos (c+d x) \sin ^4(c+d x)}{15 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{30} \int \sin ^2(c+d x) \left (15 a^2+12 a b \sin (c+d x)-5 \left (2 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx \\ & = \frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a b \cos (c+d x) \sin ^4(c+d x)}{15 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{120} \int \sin ^2(c+d x) \left (15 \left (2 a^2+b^2\right )+48 a b \sin (c+d x)\right ) \, dx \\ & = \frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a b \cos (c+d x) \sin ^4(c+d x)}{15 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{5} (2 a b) \int \sin ^3(c+d x) \, dx+\frac {1}{8} \left (2 a^2+b^2\right ) \int \sin ^2(c+d x) \, dx \\ & = -\frac {\left (2 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a b \cos (c+d x) \sin ^4(c+d x)}{15 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{16} \left (2 a^2+b^2\right ) \int 1 \, dx-\frac {(2 a b) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{5 d} \\ & = \frac {1}{16} \left (2 a^2+b^2\right ) x-\frac {2 a b \cos (c+d x)}{5 d}+\frac {2 a b \cos ^3(c+d x)}{15 d}-\frac {\left (2 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a b \cos (c+d x) \sin ^4(c+d x)}{15 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{6 d} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {120 a^2 c+60 b^2 c+120 a^2 d x+60 b^2 d x-240 a b \cos (c+d x)-40 a b \cos (3 (c+d x))+24 a b \cos (5 (c+d x))-15 b^2 \sin (2 (c+d x))-30 a^2 \sin (4 (c+d x))-15 b^2 \sin (4 (c+d x))+5 b^2 \sin (6 (c+d x))}{960 d} \]
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Time = 0.48 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {120 a^{2} d x +60 b^{2} d x +5 b^{2} \sin \left (6 d x +6 c \right )+24 a b \cos \left (5 d x +5 c \right )-30 a^{2} \sin \left (4 d x +4 c \right )-15 b^{2} \sin \left (4 d x +4 c \right )-40 a b \cos \left (3 d x +3 c \right )-15 b^{2} \sin \left (2 d x +2 c \right )-240 a b \cos \left (d x +c \right )-256 a b}{960 d}\) | \(117\) |
risch | \(\frac {a^{2} x}{8}+\frac {b^{2} x}{16}-\frac {a b \cos \left (d x +c \right )}{4 d}+\frac {b^{2} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a b \cos \left (5 d x +5 c \right )}{40 d}-\frac {\sin \left (4 d x +4 c \right ) a^{2}}{32 d}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{64 d}-\frac {a b \cos \left (3 d x +3 c \right )}{24 d}-\frac {b^{2} \sin \left (2 d x +2 c \right )}{64 d}\) | \(127\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+2 a b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(141\) |
default | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+2 a b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(141\) |
norman | \(\frac {\left (\frac {a^{2}}{8}+\frac {b^{2}}{16}\right ) x +\left (\frac {a^{2}}{8}+\frac {b^{2}}{16}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3 a^{2}}{4}+\frac {3 b^{2}}{8}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3 a^{2}}{4}+\frac {3 b^{2}}{8}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15 a^{2}}{8}+\frac {15 b^{2}}{16}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15 a^{2}}{8}+\frac {15 b^{2}}{16}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8 a b}{15 d}-\frac {\left (2 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (2 a^{2}+b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {\left (6 a^{2}+19 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (6 a^{2}+19 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (30 a^{2}-17 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {\left (30 a^{2}-17 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {16 a b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {8 a b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(392\) |
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.63 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {96 \, a b \cos \left (d x + c\right )^{5} - 160 \, a b \cos \left (d x + c\right )^{3} + 15 \, {\left (2 \, a^{2} + b^{2}\right )} d x + 5 \, {\left (8 \, b^{2} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (148) = 296\).
Time = 0.35 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.90 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 a b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a b \cos ^{5}{\left (c + d x \right )}}{15 d} + \frac {b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{2}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.56 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {30 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 128 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a b - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{960 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.71 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{16} \, {\left (2 \, a^{2} + b^{2}\right )} x + \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {a b \cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac {a b \cos \left (d x + c\right )}{4 \, d} + \frac {b^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {b^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} - \frac {{\left (2 \, a^{2} + b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \]
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Time = 11.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.69 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\frac {15\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{2}+\frac {15\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {15\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{4}-\frac {5\,b^2\,\sin \left (6\,c+6\,d\,x\right )}{4}+60\,a\,b\,\cos \left (c+d\,x\right )+10\,a\,b\,\cos \left (3\,c+3\,d\,x\right )-6\,a\,b\,\cos \left (5\,c+5\,d\,x\right )-30\,a^2\,d\,x-15\,b^2\,d\,x}{240\,d} \]
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