Integrand size = 19, antiderivative size = 69 \[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=-\frac {(b \csc (e+f x))^m}{f m}+\frac {2 (b \csc (e+f x))^{2+m}}{b^2 f (2+m)}-\frac {(b \csc (e+f x))^{4+m}}{b^4 f (4+m)} \]
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Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2686, 276} \[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=-\frac {(b \csc (e+f x))^{m+4}}{b^4 f (m+4)}+\frac {2 (b \csc (e+f x))^{m+2}}{b^2 f (m+2)}-\frac {(b \csc (e+f x))^m}{f m} \]
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Rule 276
Rule 2686
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int (b x)^{-1+m} \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{f} \\ & = -\frac {b \text {Subst}\left (\int \left ((b x)^{-1+m}-\frac {2 (b x)^{1+m}}{b^2}+\frac {(b x)^{3+m}}{b^4}\right ) \, dx,x,\csc (e+f x)\right )}{f} \\ & = -\frac {(b \csc (e+f x))^m}{f m}+\frac {2 (b \csc (e+f x))^{2+m}}{b^2 f (2+m)}-\frac {(b \csc (e+f x))^{4+m}}{b^4 f (4+m)} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=-\frac {(b \csc (e+f x))^m \left (8+6 m+m^2-2 m (4+m) \csc ^2(e+f x)+m (2+m) \csc ^4(e+f x)\right )}{f m (2+m) (4+m)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.14 (sec) , antiderivative size = 8846, normalized size of antiderivative = 128.20
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Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.67 \[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=-\frac {{\left ({\left (m^{2} + 6 \, m + 8\right )} \cos \left (f x + e\right )^{4} - 4 \, {\left (m + 4\right )} \cos \left (f x + e\right )^{2} + 8\right )} \left (\frac {b}{\sin \left (f x + e\right )}\right )^{m}}{{\left (f m^{3} + 6 \, f m^{2} + 8 \, f m\right )} \cos \left (f x + e\right )^{4} + f m^{3} + 6 \, f m^{2} - 2 \, {\left (f m^{3} + 6 \, f m^{2} + 8 \, f m\right )} \cos \left (f x + e\right )^{2} + 8 \, f m} \]
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\[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=\begin {cases} x \left (b \csc {\left (e \right )}\right )^{m} \cot ^{5}{\left (e \right )} & \text {for}\: f = 0 \\\frac {\int \frac {\cot ^{5}{\left (e + f x \right )}}{\csc ^{4}{\left (e + f x \right )}}\, dx}{b^{4}} & \text {for}\: m = -4 \\\frac {\int \frac {\cot ^{5}{\left (e + f x \right )}}{\csc ^{2}{\left (e + f x \right )}}\, dx}{b^{2}} & \text {for}\: m = -2 \\- \frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {\log {\left (\tan {\left (e + f x \right )} \right )}}{f} + \frac {1}{2 f \tan ^{2}{\left (e + f x \right )}} - \frac {1}{4 f \tan ^{4}{\left (e + f x \right )}} & \text {for}\: m = 0 \\- \frac {m^{2} \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}}{f m^{3} + 6 f m^{2} + 8 f m} - \frac {2 m \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}}{f m^{3} + 6 f m^{2} + 8 f m} + \frac {4 m \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{2}{\left (e + f x \right )}}{f m^{3} + 6 f m^{2} + 8 f m} - \frac {8 \left (b \csc {\left (e + f x \right )}\right )^{m}}{f m^{3} + 6 f m^{2} + 8 f m} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=-\frac {\frac {b^{m} \sin \left (f x + e\right )^{-m}}{m} - \frac {2 \, b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \sin \left (f x + e\right )^{2}} + \frac {b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 4\right )} \sin \left (f x + e\right )^{4}}}{f} \]
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\[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5} \,d x } \]
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Time = 8.93 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.22 \[ \int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx=-\frac {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {2\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-1\right )\,\left (-2\,{\sin \left (2\,e+2\,f\,x\right )}^2+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+1\right )}{f\,m}-\frac {\left (-2\,{\sin \left (2\,e+2\,f\,x\right )}^2+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+1\right )\,\left (6\,m^2+4\,m+48\right )}{f\,m\,\left (m^2+6\,m+8\right )}+\frac {2\,\left (2\,{\sin \left (e+f\,x\right )}^2-1\right )\,\left (-2\,{\sin \left (2\,e+2\,f\,x\right )}^2+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+1\right )\,\left (4\,m^2+8\,m-32\right )}{f\,m\,\left (m^2+6\,m+8\right )}\right )}{16\,{\sin \left (e+f\,x\right )}^4} \]
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