Optimal. Leaf size=51 \[ -\frac {(d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac {(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 51, 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, 14} \[ -\frac {(d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac {(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n \csc ^4(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int (-d x)^n \left (1+x^2\right ) \, dx,x,-\cot (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left ((-d x)^n+\frac {(-d x)^{2+n}}{d^2}\right ) \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac {(d \cot (e+f x))^{1+n}}{d f (1+n)}-\frac {(d \cot (e+f x))^{3+n}}{d^3 f (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 45, normalized size = 0.88 \[ -\frac {\cot (e+f x) \left ((n+1) \csc ^2(e+f x)+2\right ) (d \cot (e+f x))^n}{f (n+1) (n+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 87, normalized size = 1.71 \[ \frac {{\left (2 \, \cos \left (f x + e\right )^{3} - {\left (n + 3\right )} \cos \left (f x + e\right )\right )} \left (\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right )^{n}}{{\left (f n^{2} - {\left (f n^{2} + 4 \, f n + 3 \, f\right )} \cos \left (f x + e\right )^{2} + 4 \, f n + 3 \, f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.92, size = 10907, normalized size = 213.86 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 56, normalized size = 1.10 \[ -\frac {\frac {\left (\frac {d}{\tan \left (f x + e\right )}\right )^{n + 1}}{d {\left (n + 1\right )}} + \frac {d^{n} \tan \left (f x + e\right )^{-n}}{{\left (n + 3\right )} \tan \left (f x + e\right )^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 84, normalized size = 1.65 \[ -\frac {\left (\frac {3\,\cos \left (e+f\,x\right )}{2}-\frac {\cos \left (3\,e+3\,f\,x\right )}{2}+n\,\cos \left (e+f\,x\right )\right )\,{\left (\frac {d\,\cos \left (e+f\,x\right )}{2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}^n}{f\,{\sin \left (e+f\,x\right )}^3\,\left (n+1\right )\,\left (n+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot {\left (e + f x \right )}\right )^{n} \csc ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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