Optimal. Leaf size=76 \[ -\frac {(d \cot (e+f x))^{n+5}}{d^5 f (n+5)}-\frac {2 (d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac {(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 76, 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 = {2607, 270} \[ -\frac {2 (d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac {(d \cot (e+f x))^{n+5}}{d^5 f (n+5)}-\frac {(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2607
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n \csc ^6(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int (-d x)^n \left (1+x^2\right )^2 \, dx,x,-\cot (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left ((-d x)^n+\frac {2 (-d x)^{2+n}}{d^2}+\frac {(-d x)^{4+n}}{d^4}\right ) \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac {(d \cot (e+f x))^{1+n}}{d f (1+n)}-\frac {2 (d \cot (e+f x))^{3+n}}{d^3 f (3+n)}-\frac {(d \cot (e+f x))^{5+n}}{d^5 f (5+n)}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 73, normalized size = 0.96 \[ -\frac {\cot (e+f x) \csc ^4(e+f x) \left (-2 (n+3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2+6 n+8\right ) (d \cot (e+f x))^n}{f (n+1) (n+3) (n+5)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 145, normalized size = 1.91 \[ -\frac {{\left (8 \, \cos \left (f x + e\right )^{5} - 4 \, {\left (n + 5\right )} \cos \left (f x + e\right )^{3} + {\left (n^{2} + 8 \, n + 15\right )} \cos \left (f x + e\right )\right )} \left (\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right )^{n}}{{\left ({\left (f n^{3} + 9 \, f n^{2} + 23 \, f n + 15 \, f\right )} \cos \left (f x + e\right )^{4} + f n^{3} + 9 \, f n^{2} - 2 \, {\left (f n^{3} + 9 \, f n^{2} + 23 \, f n + 15 \, f\right )} \cos \left (f x + e\right )^{2} + 23 \, f n + 15 \, f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.52, size = 21900, normalized size = 288.16 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 84, normalized size = 1.11 \[ -\frac {\frac {\left (\frac {d}{\tan \left (f x + e\right )}\right )^{n + 1}}{d {\left (n + 1\right )}} + \frac {2 \, d^{n} \tan \left (f x + e\right )^{-n}}{{\left (n + 3\right )} \tan \left (f x + e\right )^{3}} + \frac {d^{n} \tan \left (f x + e\right )^{-n}}{{\left (n + 5\right )} \tan \left (f x + e\right )^{5}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 123, normalized size = 1.62 \[ -\frac {{\left (\frac {d\,\cos \left (e+f\,x\right )}{2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}^n\,\left (5\,\cos \left (e+f\,x\right )-\frac {5\,\cos \left (3\,e+3\,f\,x\right )}{2}+\frac {\cos \left (5\,e+5\,f\,x\right )}{2}+5\,n\,\cos \left (e+f\,x\right )-n\,\cos \left (3\,e+3\,f\,x\right )+n^2\,\cos \left (e+f\,x\right )\right )}{f\,{\sin \left (e+f\,x\right )}^5\,\left (n+1\right )\,\left (n+3\right )\,\left (n+5\right )} \]
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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