Integrand size = 20, antiderivative size = 20 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=2 \text {Int}\left (\frac {\csc (2 a+2 b x)}{(c+d x)^2},x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {\csc (2 a+2 b x)}{(c+d x)^2} \, dx \\ \end{align*}
Not integrable
Time = 6.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]
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Not integrable
Time = 0.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {\csc \left (x b +a \right ) \sec \left (x b +a \right )}{\left (d x +c \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 1.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc {\left (a + b x \right )} \sec {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
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Not integrable
Time = 0.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 4.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 24.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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