Integrand size = 21, antiderivative size = 32 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d} \]
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Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2746} \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d} \]
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Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int (a-x) \, dx,x,a \sin (c+d x))}{a^3 d} \\ & = \frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {(-2+\sin (c+d x)) \sin (c+d x)}{2 a d} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a}\) | \(25\) |
default | \(\frac {-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a}\) | \(25\) |
parallelrisch | \(\frac {4 \sin \left (d x +c \right )-1+\cos \left (2 d x +2 c \right )}{4 d a}\) | \(28\) |
risch | \(\frac {\sin \left (d x +c \right )}{a d}+\frac {\cos \left (2 d x +2 c \right )}{4 a d}\) | \(32\) |
norman | \(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(105\) |
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (22) = 44\).
Time = 1.87 (sec) , antiderivative size = 158, normalized size of antiderivative = 4.94 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} \frac {2 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {2 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
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Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
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Time = 2.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin \left (c+d\,x\right )\,\left (\sin \left (c+d\,x\right )-2\right )}{2\,a\,d} \]
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