Integrand size = 19, antiderivative size = 45 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2 (a+a \sin (c+d x))^3}{3 a^2 d}-\frac {(a+a \sin (c+d x))^4}{4 a^3 d} \]
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Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2746, 45} \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac {(a \sin (c+d x)+a)^4}{4 a^3 d} \]
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Rule 45
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x) (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (2 a (a+x)^2-(a+x)^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {2 (a+a \sin (c+d x))^3}{3 a^2 d}-\frac {(a+a \sin (c+d x))^4}{4 a^3 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^4(c+d x)}{4 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{3 d} \]
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Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )\right )}{d}\) | \(46\) |
default | \(-\frac {a \left (\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )\right )}{d}\) | \(46\) |
risch | \(\frac {3 a \sin \left (d x +c \right )}{4 d}-\frac {a \cos \left (4 d x +4 c \right )}{32 d}+\frac {a \sin \left (3 d x +3 c \right )}{12 d}-\frac {a \cos \left (2 d x +2 c \right )}{8 d}\) | \(59\) |
parallelrisch | \(\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{3 d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(96\) |
norman | \(\frac {\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {10 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {10 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(118\) |
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 \, a \cos \left (d x + c\right )^{4} - 4 \, {\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.33 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 6 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 6 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+a\,\sin \left (c+d\,x\right )}{d} \]
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