Integrand size = 19, antiderivative size = 54 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=\frac {a x}{2}+\frac {a \sin (c+d x)}{d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{3 d} \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=-\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^2(c+d x) \, dx+a \int \cos ^3(c+d x) \, dx \\ & = \frac {a \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a \int 1 \, dx-\frac {a \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {a x}{2}+\frac {a \sin (c+d x)}{d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=\frac {a (c+d x)}{2 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin (2 (c+d x))}{4 d} \]
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Time = 1.66 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {a \left (6 d x +\sin \left (3 d x +3 c \right )+3 \sin \left (2 d x +2 c \right )+9 \sin \left (d x +c \right )\right )}{12 d}\) | \(40\) |
risch | \(\frac {a x}{2}+\frac {3 a \sin \left (d x +c \right )}{4 d}+\frac {a \sin \left (3 d x +3 c \right )}{12 d}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(48\) |
derivativedivides | \(\frac {\frac {a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(49\) |
default | \(\frac {\frac {a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(49\) |
parts | \(\frac {a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(51\) |
norman | \(\frac {\frac {a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a x}{2}+\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {3 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(115\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=\frac {3 \, a d x + {\left (2 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + 4 \, a\right )} \sin \left (d x + c\right )}{6 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.70 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=\begin {cases} \frac {a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a}{12 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=\frac {1}{2} \, a x + \frac {a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{4 \, d} \]
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Time = 14.56 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx=\frac {a\,x}{2}+\frac {2\,a\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]
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