Integrand size = 24, antiderivative size = 28 \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {\cot (c+d x)}{a d}+\frac {\tan (c+d x)}{a d} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2700, 14} \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\tan (c+d x)}{a d}-\frac {\cot (c+d x)}{a d} \]
[In]
[Out]
Rule 14
Rule 2700
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^2(c+d x) \sec ^2(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {\cot (c+d x)}{a d}+\frac {\tan (c+d x)}{a d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57 \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {2 \cot (2 (c+d x))}{a d} \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right )-\frac {1}{\tan \left (d x +c \right )}}{d a}\) | \(25\) |
default | \(\frac {\tan \left (d x +c \right )-\frac {1}{\tan \left (d x +c \right )}}{d a}\) | \(25\) |
risch | \(-\frac {4 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}\) | \(36\) |
parallelrisch | \(\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(56\) |
norman | \(\frac {\frac {1}{2 a d}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(75\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {2 \, \cos \left (d x + c\right )^{2} - 1}{a d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
[In]
[Out]
\[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=- \frac {\int \frac {\csc ^{2}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\frac {\tan \left (d x + c\right )}{a} - \frac {1}{a \tan \left (d x + c\right )}}{d} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {2}{a d \tan \left (2 \, d x + 2 \, c\right )} \]
[In]
[Out]
Time = 14.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {2\,\mathrm {cot}\left (2\,c+2\,d\,x\right )}{a\,d} \]
[In]
[Out]