Integrand size = 21, antiderivative size = 55 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {2 \tan (c+d x)}{3 d \left (a^2+a^2 \sec (c+d x)\right )} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3882, 3879} \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \tan (c+d x)}{3 d \left (a^2 \sec (c+d x)+a^2\right )}-\frac {\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
[In]
[Out]
Rule 3879
Rule 3882
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {2 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a} \\ & = -\frac {\tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {2 \tan (c+d x)}{3 d \left (a^2+a^2 \sec (c+d x)\right )} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {3}{2} (c+d x)\right )\right )}{12 a^2 d} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3\right )}{6 a^{2} d}\) | \(31\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}\) | \(32\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}\) | \(32\) |
risch | \(\frac {2 i \left (1+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(36\) |
norman | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 a d}}{a \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\) | \(76\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
[In]
[Out]
\[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{6 \, a^{2} d} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{6 \, a^{2} d} \]
[In]
[Out]
Time = 13.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.55 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\right )}{6\,a^2\,d} \]
[In]
[Out]