Integrand size = 21, antiderivative size = 45 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 (a+a \sin (c+d x))^5}{5 a^2 d}-\frac {(a+a \sin (c+d x))^6}{6 a^3 d} \]
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Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45} \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^6}{6 a^3 d} \]
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Rule 45
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x) (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (2 a (a+x)^4-(a+x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {2 (a+a \sin (c+d x))^5}{5 a^2 d}-\frac {(a+a \sin (c+d x))^6}{6 a^3 d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{10} (-7+5 \sin (c+d x))}{30 d} \]
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Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.51
method | result | size |
derivativedivides | \(-\frac {a^{3} \left (\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )\right )}{d}\) | \(68\) |
default | \(-\frac {a^{3} \left (\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )\right )}{d}\) | \(68\) |
parallelrisch | \(-\frac {a^{3} \left (90 \cos \left (4 d x +4 c \right )-5 \cos \left (6 d x +6 c \right )+36 \sin \left (5 d x +5 c \right )-20 \sin \left (3 d x +3 c \right )+405 \cos \left (2 d x +2 c \right )-1080 \sin \left (d x +c \right )-490\right )}{960 d}\) | \(74\) |
risch | \(\frac {9 a^{3} \sin \left (d x +c \right )}{8 d}+\frac {a^{3} \cos \left (6 d x +6 c \right )}{192 d}-\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}-\frac {3 a^{3} \cos \left (4 d x +4 c \right )}{32 d}+\frac {a^{3} \sin \left (3 d x +3 c \right )}{48 d}-\frac {27 a^{3} \cos \left (2 d x +2 c \right )}{64 d}\) | \(101\) |
norman | \(\frac {\frac {16 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {46 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {84 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {84 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {46 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {6 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {6 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {28 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(225\) |
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Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.60 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \cos \left (d x + c\right )^{6} - 30 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, {\left (9 \, a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{30 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (37) = 74\).
Time = 0.36 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.24 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {2 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a^{3} \cos ^{6}{\left (c + d x \right )}}{12 d} - \frac {3 a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.82 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {5 \, a^{3} \sin \left (d x + c\right )^{6} + 18 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 20 \, a^{3} \sin \left (d x + c\right )^{3} - 45 \, a^{3} \sin \left (d x + c\right )^{2} - 30 \, a^{3} \sin \left (d x + c\right )}{30 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.82 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {5 \, a^{3} \sin \left (d x + c\right )^{6} + 18 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 20 \, a^{3} \sin \left (d x + c\right )^{3} - 45 \, a^{3} \sin \left (d x + c\right )^{2} - 30 \, a^{3} \sin \left (d x + c\right )}{30 \, d} \]
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Time = 6.00 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.78 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {-\frac {a^3\,{\sin \left (c+d\,x\right )}^6}{6}-\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a^3\,{\sin \left (c+d\,x\right )}^4}{2}+\frac {2\,a^3\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^2}{2}+a^3\,\sin \left (c+d\,x\right )}{d} \]
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