Integrand size = 25, antiderivative size = 65 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {b x}{8}-\frac {a \cos ^3(c+d x)}{3 d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d} \]
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Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2917, 2645, 30, 2648, 2715, 8} \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cos ^3(c+d x)}{3 d}-\frac {b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {b x}{8} \]
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Rule 8
Rule 30
Rule 2645
Rule 2648
Rule 2715
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^2(c+d x) \sin (c+d x) \, dx+b \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx \\ & = -\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} b \int \cos ^2(c+d x) \, dx-\frac {a \text {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^3(c+d x)}{3 d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} b \int 1 \, dx \\ & = \frac {b x}{8}-\frac {a \cos ^3(c+d x)}{3 d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {b x}{8}-\frac {a \cos ^3(c+d x)}{3 d}+\frac {1}{8} b \left (-\frac {\cos (4 d x) \sin (4 c)}{4 d}-\frac {\cos (4 c) \sin (4 d x)}{4 d}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {b x}{8}-\frac {a \cos \left (d x +c \right )}{4 d}-\frac {b \sin \left (4 d x +4 c \right )}{32 d}-\frac {a \cos \left (3 d x +3 c \right )}{12 d}\) | \(48\) |
parallelrisch | \(\frac {12 b x d -24 a \cos \left (d x +c \right )-3 b \sin \left (4 d x +4 c \right )-8 a \cos \left (3 d x +3 c \right )-32 a}{96 d}\) | \(48\) |
derivativedivides | \(\frac {-\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{3}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(57\) |
default | \(\frac {-\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{3}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(57\) |
norman | \(\frac {\frac {b x}{8}-\frac {2 a}{3 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {7 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {7 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(205\) |
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=-\frac {8 \, a \cos \left (d x + c\right )^{3} - 3 \, b d x + 3 \, {\left (2 \, b \cos \left (d x + c\right )^{3} - b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (56) = 112\).
Time = 0.16 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.83 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\begin {cases} - \frac {a \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right ) \sin {\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Timed out. \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {1}{8} \, b x - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {a \cos \left (d x + c\right )}{4 \, d} - \frac {b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} \]
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Time = 13.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.92 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {b\,x}{8}-\frac {-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {2\,a}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
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