Integrand size = 16, antiderivative size = 20 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=a x+\frac {b \sin ^2(c+d x)}{2 d} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2644, 30} \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=a x+\frac {b \sin ^2(c+d x)}{2 d} \]
[In]
[Out]
Rule 30
Rule 2644
Rubi steps \begin{align*} \text {integral}& = a x+b \int \cos (c+d x) \sin (c+d x) \, dx \\ & = a x+\frac {b \text {Subst}(\int x \, dx,x,\sin (c+d x))}{d} \\ & = a x+\frac {b \sin ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=a x-\frac {b \cos (2 c) \cos (2 d x)}{4 d}+\frac {b \sin (2 c) \sin (2 d x)}{4 d} \]
[In]
[Out]
Time = 0.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(a x +\frac {b \sin \left (d x +c \right )^{2}}{2 d}\) | \(19\) |
parts | \(a x +\frac {b \sin \left (d x +c \right )^{2}}{2 d}\) | \(19\) |
risch | \(a x -\frac {b \cos \left (2 d x +2 c \right )}{4 d}\) | \(20\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +\frac {b \sin \left (d x +c \right )^{2}}{2}}{d}\) | \(24\) |
parallelrisch | \(\frac {b \left (1-\cos \left (2 d x +2 c \right )\right )}{4 d}+a x\) | \(24\) |
norman | \(\frac {a x +a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}\) | \(67\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=\frac {2 \, a d x - b \cos \left (d x + c\right )^{2}}{2 \, d} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=a x + b \left (\begin {cases} - \frac {\cos ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \sin {\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=a x - \frac {b \cos \left (d x + c\right )^{2}}{2 \, d} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=a x + \frac {b \sin \left (d x + c\right )^{2}}{2 \, d} \]
[In]
[Out]
Time = 26.77 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (a+b \cos (c+d x) \sin (c+d x)) \, dx=-\frac {\frac {b\,{\cos \left (c+d\,x\right )}^2}{2}-a\,d\,x}{d} \]
[In]
[Out]