Integrand size = 19, antiderivative size = 32 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3154} \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
[In]
[Out]
Rule 3154
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(-\frac {1}{d b \left (a +b \tan \left (d x +c \right )\right )}\) | \(21\) |
default | \(-\frac {1}{d b \left (a +b \tan \left (d x +c \right )\right )}\) | \(21\) |
risch | \(\frac {2 i}{d \left (-i b +a \right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )} a +i b +a \right )}\) | \(47\) |
parallelrisch | \(\frac {1-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}\) | \(54\) |
norman | \(\frac {\frac {1}{b d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}\) | \(60\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{{\left (a^{3} + a b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{2} b + b^{3}\right )} d \sin \left (d x + c\right )} \]
[In]
[Out]
\[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\int \frac {1}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {1}{{\left (b^{2} \tan \left (d x + c\right ) + a b\right )} d} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )} b d} \]
[In]
[Out]
Time = 28.65 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]
[In]
[Out]