Optimal. Leaf size=78 \[ \frac {b^5 (b \csc (e+f x))^{n-5}}{f (5-n)}-\frac {2 b^3 (b \csc (e+f x))^{n-3}}{f (3-n)}+\frac {b (b \csc (e+f x))^{n-1}}{f (1-n)} \]
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Rubi [A] time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2621, 270} \[ \frac {b^5 (b \csc (e+f x))^{n-5}}{f (5-n)}-\frac {2 b^3 (b \csc (e+f x))^{n-3}}{f (3-n)}+\frac {b (b \csc (e+f x))^{n-1}}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2621
Rubi steps
\begin {align*} \int \cos ^5(e+f x) (b \csc (e+f x))^n \, dx &=-\frac {b^5 \operatorname {Subst}\left (\int x^{-6+n} \left (-1+\frac {x^2}{b^2}\right )^2 \, dx,x,b \csc (e+f x)\right )}{f}\\ &=-\frac {b^5 \operatorname {Subst}\left (\int \left (x^{-6+n}-\frac {2 x^{-4+n}}{b^2}+\frac {x^{-2+n}}{b^4}\right ) \, dx,x,b \csc (e+f x)\right )}{f}\\ &=\frac {b^5 (b \csc (e+f x))^{-5+n}}{f (5-n)}-\frac {2 b^3 (b \csc (e+f x))^{-3+n}}{f (3-n)}+\frac {b (b \csc (e+f x))^{-1+n}}{f (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 81, normalized size = 1.04 \[ -\frac {\sin ^5(e+f x) \left (\left (n^2-8 n+15\right ) \csc ^4(e+f x)-2 \left (n^2-6 n+5\right ) \csc ^2(e+f x)+n^2-4 n+3\right ) (b \csc (e+f x))^n}{f (n-5) (n-3) (n-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 73, normalized size = 0.94 \[ -\frac {{\left ({\left (n^{2} - 4 \, n + 3\right )} \cos \left (f x + e\right )^{4} - 4 \, {\left (n - 1\right )} \cos \left (f x + e\right )^{2} + 8\right )} \left (\frac {b}{\sin \left (f x + e\right )}\right )^{n} \sin \left (f x + e\right )}{f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.43, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 86, normalized size = 1.10 \[ -\frac {\frac {b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )^{5}}{n - 5} - \frac {2 \, b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )^{3}}{n - 3} + \frac {b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )}{n - 1}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 134, normalized size = 1.72 \[ -\frac {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n\,\left (150\,\sin \left (e+f\,x\right )+25\,\sin \left (3\,e+3\,f\,x\right )+3\,\sin \left (5\,e+5\,f\,x\right )+3\,n^2\,\sin \left (3\,e+3\,f\,x\right )+n^2\,\sin \left (5\,e+5\,f\,x\right )-24\,n\,\sin \left (e+f\,x\right )-28\,n\,\sin \left (3\,e+3\,f\,x\right )-4\,n\,\sin \left (5\,e+5\,f\,x\right )+2\,n^2\,\sin \left (e+f\,x\right )\right )}{16\,f\,\left (n^3-9\,n^2+23\,n-15\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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