Integrand size = 19, antiderivative size = 80 \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=-\frac {b^5 (b \tan (e+f x))^{-5+n}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)} \]
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Time = 0.07 (sec) , antiderivative size = 80, 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 = {2671, 276} \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=-\frac {b^5 (b \tan (e+f x))^{n-5}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
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Rule 276
Rule 2671
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int x^{-6+n} \left (b^2+x^2\right )^2 \, dx,x,b \tan (e+f x)\right )}{f} \\ & = \frac {b \text {Subst}\left (\int \left (b^4 x^{-6+n}+2 b^2 x^{-4+n}+x^{-2+n}\right ) \, dx,x,b \tan (e+f x)\right )}{f} \\ & = -\frac {b^5 (b \tan (e+f x))^{-5+n}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86 \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=\frac {b \left (8-6 n+n^2+2 (-3+n) \cos (2 (e+f x))+\cos (4 (e+f x))\right ) \csc ^4(e+f x) (b \tan (e+f x))^{-1+n}}{f (-5+n) (-3+n) (-1+n)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 151.82 (sec) , antiderivative size = 10580, normalized size of antiderivative = 132.25
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none
Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.80 \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=\frac {{\left (8 \, \cos \left (f x + e\right )^{5} + 4 \, {\left (n - 5\right )} \cos \left (f x + e\right )^{3} + {\left (n^{2} - 8 \, n + 15\right )} \cos \left (f x + e\right )\right )} \left (\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n}}{{\left ({\left (f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f\right )} \cos \left (f x + e\right )^{4} + f n^{3} - 9 \, f n^{2} - 2 \, {\left (f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f\right )} \cos \left (f x + e\right )^{2} + 23 \, f n - 15 \, f\right )} \sin \left (f x + e\right )} \]
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Timed out. \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=\text {Timed out} \]
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none
Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=\frac {\frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 1\right )} \tan \left (f x + e\right )} + \frac {2 \, b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 3\right )} \tan \left (f x + e\right )^{3}} + \frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 5\right )} \tan \left (f x + e\right )^{5}}}{f} \]
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\[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{6} \,d x } \]
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Timed out. \[ \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^6} \,d x \]
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