Integrand size = 19, antiderivative size = 16 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (1+\sin (c+d x))}{a d} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2746, 31} \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (\sin (c+d x)+1)}{a d} \]
[In]
[Out]
Rule 31
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\log (1+\sin (c+d x))}{a d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (1+\sin (c+d x))}{a d} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\ln \left (a +a \sin \left (d x +c \right )\right )}{d a}\) | \(19\) |
default | \(\frac {\ln \left (a +a \sin \left (d x +c \right )\right )}{d a}\) | \(19\) |
parallelrisch | \(\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(37\) |
risch | \(-\frac {i x}{a}-\frac {2 i c}{a d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}\) | \(40\) |
norman | \(\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(44\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log \left (a \sin \left (d x + c\right ) + a\right )}{a d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log \left ({\left | a \sin \left (d x + c\right ) + a \right |}\right )}{a d} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a\,d} \]
[In]
[Out]