Integrand size = 19, antiderivative size = 25 \[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)} \]
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Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2671, 30} \[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
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Rule 30
Rule 2671
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int x^{-2+n} \, dx,x,b \tan (e+f x)\right )}{f} \\ & = -\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=\frac {b (b \tan (e+f x))^{-1+n}}{f (-1+n)} \]
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Time = 1.82 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {{\mathrm e}^{n \ln \left (b \tan \left (f x +e \right )\right )}}{f \left (-1+n \right ) \tan \left (f x +e \right )}\) | \(30\) |
default | \(\frac {{\mathrm e}^{n \ln \left (b \tan \left (f x +e \right )\right )}}{f \left (-1+n \right ) \tan \left (f x +e \right )}\) | \(30\) |
risch | \(\text {Expression too large to display}\) | \(1750\) |
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=\frac {\left (\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{{\left (f n - f\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=\int \left (b \tan {\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=\frac {b^{n} \tan \left (f x + e\right )^{n}}{f {\left (n - 1\right )} \tan \left (f x + e\right )} \]
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\[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \]
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Time = 3.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx=-\frac {\sin \left (2\,e+2\,f\,x\right )\,{\left (\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{2\,{\cos \left (e+f\,x\right )}^2}\right )}^n}{2\,f\,\left ({\cos \left (e+f\,x\right )}^2-1\right )\,\left (n-1\right )} \]
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