Integrand size = 11, antiderivative size = 19 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {(a+b \sin (x))^{1+n}}{b (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2747, 32} \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {(a+b \sin (x))^{n+1}}{b (n+1)} \]
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Rule 32
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^n \, dx,x,b \sin (x)\right )}{b} \\ & = \frac {(a+b \sin (x))^{1+n}}{b (1+n)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {(a+b \sin (x))^{1+n}}{b (1+n)} \]
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Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {\left (a +b \sin \left (x \right )\right )^{n +1}}{b \left (n +1\right )}\) | \(20\) |
default | \(\frac {\left (a +b \sin \left (x \right )\right )^{n +1}}{b \left (n +1\right )}\) | \(20\) |
parallelrisch | \(\frac {\left (a +b \sin \left (x \right )\right )^{n +1}}{b \left (n +1\right )}\) | \(20\) |
norman | \(\frac {\frac {a \,{\mathrm e}^{n \ln \left (a +\frac {2 b \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )}}{b \left (n +1\right )}+\frac {a \tan \left (\frac {x}{2}\right )^{2} {\mathrm e}^{n \ln \left (a +\frac {2 b \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )}}{b \left (n +1\right )}+\frac {2 \tan \left (\frac {x}{2}\right ) {\mathrm e}^{n \ln \left (a +\frac {2 b \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )}}{n +1}}{1+\tan \left (\frac {x}{2}\right )^{2}}\) | \(119\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {{\left (b \sin \left (x\right ) + a\right )} {\left (b \sin \left (x\right ) + a\right )}^{n}}{b n + b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (14) = 28\).
Time = 0.50 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.95 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\begin {cases} \frac {\sin {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = -1 \\a^{n} \sin {\left (x \right )} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + \sin {\left (x \right )} \right )}}{b} & \text {for}\: n = -1 \\\frac {a \left (a + b \sin {\left (x \right )}\right )^{n}}{b n + b} + \frac {b \left (a + b \sin {\left (x \right )}\right )^{n} \sin {\left (x \right )}}{b n + b} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {{\left (b \sin \left (x\right ) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {{\left (b \sin \left (x\right ) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
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Time = 26.72 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos (x) (a+b \sin (x))^n \, dx=\frac {{\left (a+b\,\sin \left (x\right )\right )}^{n+1}}{b\,\left (n+1\right )} \]
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