Integrand size = 22, antiderivative size = 22 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\text {Int}\left (\frac {\cot (a+b x)}{c+d x},x\right ) \]
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Not integrable
Time = 0.17 (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 = {} \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot (a+b x)}{c+d x} \, dx-\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx \\ & = \int \frac {\cot (a+b x)}{c+d x} \, dx-\int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx\right )+\int \frac {\cot (a+b x)}{c+d x} \, dx \\ & = -\left (\frac {1}{2} \cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx\right )-\frac {1}{2} \sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\int \frac {\cot (a+b x)}{c+d x} \, dx \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\int \frac {\cot (a+b x)}{c+d x} \, dx \\ \end{align*}
Not integrable
Time = 0.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \]
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Not integrable
Time = 0.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {\cos \left (x b +a \right )^{2} \cot \left (x b +a \right )}{d x +c}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c} \,d x } \]
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Not integrable
Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos ^{2}{\left (a + b x \right )} \cot {\left (a + b x \right )}}{c + d x}\, dx \]
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Not integrable
Time = 0.54 (sec) , antiderivative size = 231, normalized size of antiderivative = 10.50 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c} \,d x } \]
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Not integrable
Time = 26.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,\mathrm {cot}\left (a+b\,x\right )}{c+d\,x} \,d x \]
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