Integrand size = 16, antiderivative size = 16 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\csc ^2(a+b x)}{(c+d x)^2},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx \\ \end{align*}
Not integrable
Time = 4.99 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int \frac {\csc ^{2}\left (b x +a \right )}{\left (d x +c \right )^{2}}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.55 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
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Not integrable
Time = 1.02 (sec) , antiderivative size = 718, normalized size of antiderivative = 44.88 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.77 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]
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