Optimal. Leaf size=25 \[ -\frac {(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2607, 32} \[ -\frac {(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n \csc ^2(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int (-d x)^n \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac {(d \cot (e+f x))^{1+n}}{d f (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.04 \[ -\frac {\cot (e+f x) (d \cot (e+f x))^n}{f (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 41, normalized size = 1.64 \[ -\frac {\left (\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{{\left (f n + f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 26, normalized size = 1.04 \[ -\frac {\left (d \cot \left (f x +e \right )\right )^{1+n}}{d f \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 1.00 \[ -\frac {\left (d \cot \left (f x + e\right )\right )^{n + 1}}{d f {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 73, normalized size = 2.92 \[ \frac {\left (\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{2\,n+2}-\frac {1}{2\,n+2}\right )\,{\left (-\frac {d\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}{2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}^n}{f\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )} \]
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 \[ \int \left (d \cot {\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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