Integrand size = 27, antiderivative size = 129 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a b x}{8}-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d} \]
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Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2941, 2748, 2715, 8} \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}+\frac {a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a b x}{8} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2941
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{7} \int \cos ^4(c+d x) (2 b+2 a \sin (c+d x)) (a+b \sin (c+d x)) \, dx \\ & = -\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{42} \int \cos ^4(c+d x) \left (14 a b+2 \left (a^2+6 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{3} (a b) \int \cos ^4(c+d x) \, dx \\ & = -\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{4} (a b) \int \cos ^2(c+d x) \, dx \\ & = -\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{8} (a b) \int 1 \, dx \\ & = \frac {a b x}{8}-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {840 a b c+840 a b d x-105 \left (8 a^2+3 b^2\right ) \cos (c+d x)-105 \left (4 a^2+b^2\right ) \cos (3 (c+d x))-84 a^2 \cos (5 (c+d x))+21 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))+210 a b \sin (2 (c+d x))-210 a b \sin (4 (c+d x))-70 a b \sin (6 (c+d x))}{6720 d} \]
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Time = 0.64 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(105\) |
default | \(\frac {-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(105\) |
parallelrisch | \(\frac {\left (-420 a^{2}-105 b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (-84 a^{2}+21 b^{2}\right ) \cos \left (5 d x +5 c \right )+15 b^{2} \cos \left (7 d x +7 c \right )+210 a b \sin \left (2 d x +2 c \right )-210 a b \sin \left (4 d x +4 c \right )-70 a b \sin \left (6 d x +6 c \right )+\left (-840 a^{2}-315 b^{2}\right ) \cos \left (d x +c \right )+840 a b x d -1344 a^{2}-384 b^{2}}{6720 d}\) | \(136\) |
risch | \(\frac {a b x}{8}-\frac {a^{2} \cos \left (d x +c \right )}{8 d}-\frac {3 b^{2} \cos \left (d x +c \right )}{64 d}+\frac {\cos \left (7 d x +7 c \right ) b^{2}}{448 d}-\frac {a b \sin \left (6 d x +6 c \right )}{96 d}-\frac {\cos \left (5 d x +5 c \right ) a^{2}}{80 d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{320 d}-\frac {a b \sin \left (4 d x +4 c \right )}{32 d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{16 d}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{64 d}+\frac {a b \sin \left (2 d x +2 c \right )}{32 d}\) | \(168\) |
norman | \(\frac {-\frac {14 a^{2}+4 b^{2}}{35 d}+\frac {a b x}{8}-\frac {2 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (4 a^{2}+4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (4 a^{2}+4 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (6 a^{2}-4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (8 a^{2}+8 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (22 a^{2}-8 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {11 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {31 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {31 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {11 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {7 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a b x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a b x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(411\) |
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Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {120 \, b^{2} \cos \left (d x + c\right )^{7} - 168 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, a b d x - 35 \, {\left (8 \, a b \cos \left (d x + c\right )^{5} - 2 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Time = 0.47 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.73 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {a b x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a b x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {a b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {a b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 b^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin {\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.63 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {672 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 96 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} b^{2}}{3360 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.09 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{8} \, a b x + \frac {b^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} - \frac {{\left (4 \, a^{2} - b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (8 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )}{64 \, d} \]
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Time = 14.49 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.98 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a\,b\,x}{8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a^2-4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (4\,a^2+4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {4\,a^2}{5}+\frac {4\,b^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,a^2+8\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {22\,a^2}{5}-\frac {8\,b^2}{5}\right )+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {2\,a^2}{5}+\frac {4\,b^2}{35}-\frac {11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {31\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}-\frac {31\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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