Integrand size = 28, antiderivative size = 28 \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]
[Out]
Not integrable
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \\ \end{align*}
Not integrable
Time = 38.91 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
[In]
[Out]
Not integrable
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
\[\int \frac {\csc ^{2}\left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}d x\]
[In]
[Out]
Not integrable
Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\csc \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
[In]
[Out]
Not integrable
Time = 4.49 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\csc ^{2}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]
[In]
[Out]
Exception generated. \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 1.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
[In]
[Out]