Integrand size = 22, antiderivative size = 22 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {2^{-3-m} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-3-m} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )}{b}+\text {Int}\left ((c+d x)^m \cot (a+b x),x\right ) \]
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Not integrable
Time = 0.21 (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 = {} \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int (c+d x)^m \cot (a+b x) \, dx-\int (c+d x)^m \cos (a+b x) \sin (a+b x) \, dx \\ & = \int (c+d x)^m \cot (a+b x) \, dx-\int \frac {1}{2} (c+d x)^m \sin (2 a+2 b x) \, dx \\ & = -\left (\frac {1}{2} \int (c+d x)^m \sin (2 a+2 b x) \, dx\right )+\int (c+d x)^m \cot (a+b x) \, dx \\ & = -\left (\frac {1}{4} i \int e^{-i (2 a+2 b x)} (c+d x)^m \, dx\right )+\frac {1}{4} i \int e^{i (2 a+2 b x)} (c+d x)^m \, dx+\int (c+d x)^m \cot (a+b x) \, dx \\ & = \frac {2^{-3-m} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-3-m} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )}{b}+\int (c+d x)^m \cot (a+b x) \, dx \\ \end{align*}
Not integrable
Time = 12.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \left (d x +c \right )^{m} \cos \left (x b +a \right )^{2} \cot \left (x b +a \right )d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{2} \cot \left (b x + a\right ) \,d x } \]
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Not integrable
Time = 24.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int \left (c + d x\right )^{m} \cos ^{2}{\left (a + b x \right )} \cot {\left (a + b x \right )}\, dx \]
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Not integrable
Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{2} \cot \left (b x + a\right ) \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{2} \cot \left (b x + a\right ) \,d x } \]
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Not integrable
Time = 25.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,\mathrm {cot}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^m \,d x \]
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