Optimal. Leaf size=46 \[ -\frac {\cot ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(a+b x)\right )}{b (n+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3476, 364} \[ -\frac {\cot ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rubi steps
\begin {align*} \int \cot ^n(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^n}{1+x^2} \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac {\cot ^{1+n}(a+b x) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(a+b x)\right )}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 1.04 \[ -\frac {\cot ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+1}{2}+1;-\cot ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cot \left (b x + a\right )^{n}, x\right ) \]
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 \[ \int \cot \left (b x + a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.35, size = 0, normalized size = 0.00 \[ \int \cot ^{n}\left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot \left (b x + a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cot}\left (a+b\,x\right )}^n \,d x \]
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 \[ \int \cot ^{n}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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