Integrand size = 21, antiderivative size = 21 \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\text {Int}\left (\cot ^4(c+d x) (a+b \sec (c+d x))^n,x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 21, 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 = {} \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx \\ \end{align*}
Not integrable
Time = 13.64 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx \]
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Not integrable
Time = 1.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
\[\int \cot \left (d x +c \right )^{4} \left (a +b \sec \left (d x +c \right )\right )^{n}d x\]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\text {Timed out} \]
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Not integrable
Time = 18.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \]
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Not integrable
Time = 0.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \]
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Not integrable
Time = 28.69 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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