Integrand size = 19, antiderivative size = 52 \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=\frac {b^3 (b \sec (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \sec (e+f x))^{-1+n}}{f (1-n)} \]
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Time = 0.04 (sec) , antiderivative size = 52, 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 = {2702, 14} \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=\frac {b^3 (b \sec (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \sec (e+f x))^{n-1}}{f (1-n)} \]
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Rule 14
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {b^3 \text {Subst}\left (\int x^{-4+n} \left (-1+\frac {x^2}{b^2}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^3 \text {Subst}\left (\int \left (-x^{-4+n}+\frac {x^{-2+n}}{b^2}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^3 (b \sec (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \sec (e+f x))^{-1+n}}{f (1-n)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {b (5-n+(-1+n) \cos (2 (e+f x))) (b \sec (e+f x))^{-1+n}}{2 f (-3+n) (-1+n)} \]
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Time = 0.92 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {\left (\left (1-n \right ) \cos \left (3 f x +3 e \right )+\cos \left (f x +e \right ) \left (n -9\right )\right ) \left (\frac {b}{\cos \left (f x +e \right )}\right )^{n}}{4 \left (-3+n \right ) f \left (-1+n \right )}\) | \(54\) |
default | \(\frac {\cos \left (f x +e \right ) {\mathrm e}^{n \ln \left (\frac {b}{\cos \left (f x +e \right )}\right )}}{f \left (-1+n \right )}-\frac {\left (\cos ^{3}\left (f x +e \right )\right ) {\mathrm e}^{n \ln \left (\frac {b}{\cos \left (f x +e \right )}\right )}}{f \left (-3+n \right )}\) | \(63\) |
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Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.02 \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {{\left ({\left (n - 1\right )} \cos \left (f x + e\right )^{3} - {\left (n - 3\right )} \cos \left (f x + e\right )\right )} \left (\frac {b}{\cos \left (f x + e\right )}\right )^{n}}{f n^{2} - 4 \, f n + 3 \, f} \]
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Timed out. \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {\frac {b^{n} \cos \left (f x + e\right )^{-n} \cos \left (f x + e\right )^{3}}{n - 3} - \frac {b^{n} \cos \left (f x + e\right )^{-n} \cos \left (f x + e\right )}{n - 1}}{f} \]
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\[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
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Time = 0.71 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.29 \[ \int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n\,\left (9\,\cos \left (e+f\,x\right )-\cos \left (3\,e+3\,f\,x\right )-n\,\cos \left (e+f\,x\right )+n\,\cos \left (3\,e+3\,f\,x\right )\right )}{4\,f\,\left (n^2-4\,n+3\right )} \]
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