Integrand size = 25, antiderivative size = 33 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^4(c+d x)}{4 d} \]
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Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 45} \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{3 d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 (a+x)}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int x^2 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^4(c+d x)}{4 d} \]
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Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(28\) |
default | \(\frac {\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(28\) |
parallelrisch | \(\frac {a \left (-12 \cos \left (2 d x +2 c \right )+3 \cos \left (4 d x +4 c \right )+9-8 \sin \left (3 d x +3 c \right )+24 \sin \left (d x +c \right )\right )}{96 d}\) | \(50\) |
risch | \(\frac {a \sin \left (d x +c \right )}{4 d}+\frac {a \cos \left (4 d x +4 c \right )}{32 d}-\frac {a \sin \left (3 d x +3 c \right )}{12 d}-\frac {a \cos \left (2 d x +2 c \right )}{8 d}\) | \(59\) |
norman | \(\frac {\frac {8 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(69\) |
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \cos \left (d x + c\right )^{4} - 6 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {a \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {a \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3}}{12 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3}}{12 \, d} \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\sin \left (c+d\,x\right )}^3\,\left (3\,\sin \left (c+d\,x\right )+4\right )}{12\,d} \]
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