Integrand size = 26, antiderivative size = 44 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {b \cos ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{3 d} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3169, 2713, 2645, 30} \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {b \cos ^3(c+d x)}{3 d} \]
[In]
[Out]
Rule 30
Rule 2645
Rule 2713
Rule 3169
Rubi steps \begin{align*} \text {integral}& = \int \left (a \cos ^3(c+d x)+b \cos ^2(c+d x) \sin (c+d x)\right ) \, dx \\ & = a \int \cos ^3(c+d x) \, dx+b \int \cos ^2(c+d x) \sin (c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {b \cos ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {b \cos ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{3 d} \]
[In]
[Out]
Time = 0.61 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}-\frac {\cos \left (d x +c \right )^{3} b}{3}}{d}\) | \(36\) |
default | \(\frac {\frac {a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}-\frac {\cos \left (d x +c \right )^{3} b}{3}}{d}\) | \(36\) |
parts | \(\frac {a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}-\frac {b \cos \left (d x +c \right )^{3}}{3 d}\) | \(38\) |
risch | \(-\frac {b \cos \left (d x +c \right )}{4 d}+\frac {3 a \sin \left (d x +c \right )}{4 d}-\frac {b \cos \left (3 d x +3 c \right )}{12 d}+\frac {a \sin \left (3 d x +3 c \right )}{12 d}\) | \(56\) |
parallelrisch | \(\frac {6 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+6 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{3 d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(79\) |
norman | \(\frac {-\frac {2 b}{3 d}+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(90\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {b \cos \left (d x + c\right )^{3} - {\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right )}{3 \, d} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.43 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\begin {cases} \frac {2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {b \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {b \cos \left (d x + c\right )^{3} + {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a}{3 \, d} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.25 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {b \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {b \cos \left (d x + c\right )}{4 \, d} + \frac {a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
Time = 21.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.07 \[ \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {2\,a\,\sin \left (c+d\,x\right )}{3\,d}-\frac {b\,{\cos \left (c+d\,x\right )}^3}{3\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]
[In]
[Out]