Integrand size = 27, antiderivative size = 106 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a b x}{4}-\frac {\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{4 d}-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d} \]
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Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2941, 2748, 2715, 8} \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac {a b x}{4} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2941
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{5} \int \cos ^2(c+d x) (2 b+2 a \sin (c+d x)) (a+b \sin (c+d x)) \, dx \\ & = -\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{20} \int \cos ^2(c+d x) \left (10 a b+2 \left (a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{2} (a b) \int \cos ^2(c+d x) \, dx \\ & = -\frac {\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{4 d}-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{4} (a b) \int 1 \, dx \\ & = \frac {a b x}{4}-\frac {\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{4 d}-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.73 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-30 \left (2 a^2+b^2\right ) \cos (c+d x)-5 \left (4 a^2+b^2\right ) \cos (3 (c+d x))+3 b (20 a (c+d x)+b \cos (5 (c+d x))-5 a \sin (4 (c+d x)))}{240 d} \]
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Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d}\) | \(94\) |
default | \(\frac {-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d}\) | \(94\) |
parallelrisch | \(\frac {60 a b x d -60 \cos \left (d x +c \right ) a^{2}-30 \cos \left (d x +c \right ) b^{2}+3 \cos \left (5 d x +5 c \right ) b^{2}-15 a b \sin \left (4 d x +4 c \right )-20 \cos \left (3 d x +3 c \right ) a^{2}-5 \cos \left (3 d x +3 c \right ) b^{2}-80 a^{2}-32 b^{2}}{240 d}\) | \(100\) |
risch | \(\frac {a b x}{4}-\frac {a^{2} \cos \left (d x +c \right )}{4 d}-\frac {b^{2} \cos \left (d x +c \right )}{8 d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{80 d}-\frac {a b \sin \left (4 d x +4 c \right )}{16 d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{12 d}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{48 d}\) | \(102\) |
norman | \(\frac {-\frac {10 a^{2}+4 b^{2}}{15 d}+\frac {a b x}{4}-\frac {2 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (2 a^{2}+2 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (4 a^{2}-2 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (4 a^{2}+4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {3 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {5 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(289\) |
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Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.69 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {12 \, b^{2} \cos \left (d x + c\right )^{5} + 15 \, a b d x - 20 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (2 \, a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, d} \]
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Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.62 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {a b x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {a b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a b x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {a b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {a b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} - \frac {b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 b^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin {\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {80 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 16 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} b^{2}}{240 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{4} \, a b x + \frac {b^{2} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{8 \, d} \]
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Time = 13.19 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.70 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a\,b\,x}{4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^2+4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {4\,a^2}{3}+\frac {4\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {8\,a^2}{3}-\frac {4\,b^2}{3}\right )+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {2\,a^2}{3}+\frac {4\,b^2}{15}-3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
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