Integrand size = 22, antiderivative size = 22 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2},x\right ) \]
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Not integrable
Time = 0.18 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \\ \end{align*}
Not integrable
Time = 19.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]
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Not integrable
Time = 0.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {\csc \left (x b +a \right ) \sec \left (x b +a \right )^{2}}{\left (d x +c \right )^{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 1.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
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Not integrable
Time = 1.35 (sec) , antiderivative size = 931, normalized size of antiderivative = 42.32 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 4.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 26.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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