Integrand size = 25, antiderivative size = 45 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {(a+a \sin (c+d x))^4}{4 a d}+\frac {(a+a \sin (c+d x))^5}{5 a^2 d} \]
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Time = 0.03 (sec) , antiderivative size = 45, 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} \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^4}{4 a d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x (a+x)^3}{a} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int x (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (-a (a+x)^3+(a+x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {(a+a \sin (c+d x))^4}{4 a d}+\frac {(a+a \sin (c+d x))^5}{5 a^2 d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^4 (-1+4 \sin (c+d x))}{20 d} \]
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Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+a^{3} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}}{d}\) | \(57\) |
default | \(\frac {\frac {a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+a^{3} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}}{d}\) | \(57\) |
parallelrisch | \(\frac {a^{3} \left (-50 \sin \left (3 d x +3 c \right )+140 \sin \left (d x +c \right )+2 \sin \left (5 d x +5 c \right )+15 \cos \left (4 d x +4 c \right )+85-100 \cos \left (2 d x +2 c \right )\right )}{160 d}\) | \(63\) |
risch | \(\frac {7 a^{3} \sin \left (d x +c \right )}{8 d}+\frac {a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} \cos \left (4 d x +4 c \right )}{32 d}-\frac {5 a^{3} \sin \left (3 d x +3 c \right )}{16 d}-\frac {5 a^{3} \cos \left (2 d x +2 c \right )}{8 d}\) | \(84\) |
norman | \(\frac {\frac {2 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {112 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {18 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {18 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(151\) |
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Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.58 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15 \, a^{3} \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 7 \, a^{3} \cos \left (d x + c\right )^{2} + 6 \, a^{3}\right )} \sin \left (d x + c\right )}{20 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.69 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {a^{3} \sin ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {4 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} + 10 \, a^{3} \sin \left (d x + c\right )^{2}}{20 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {4 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} + 10 \, a^{3} \sin \left (d x + c\right )^{2}}{20 \, d} \]
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Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.24 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^4}{4}+a^3\,{\sin \left (c+d\,x\right )}^3+\frac {a^3\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
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