Optimal. Leaf size=13 \[ -\frac {\cot ^{-1}(x)^{n+1}}{n+1} \]
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Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4885} \[ -\frac {\cot ^{-1}(x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 4885
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx &=-\frac {\cot ^{-1}(x)^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ -\frac {\cot ^{-1}(x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 13, normalized size = 1.00 \[ -\frac {\operatorname {arccot}\relax (x)^{n} \operatorname {arccot}\relax (x)}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 15, normalized size = 1.15 \[ -\frac {\arctan \left (\frac {1}{x}\right )^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 1.08 \[ -\frac {\mathrm {arccot}\relax (x )^{1+n}}{1+n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 13, normalized size = 1.00 \[ -\frac {\operatorname {arccot}\relax (x)^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 13, normalized size = 1.00 \[ -\frac {{\mathrm {acot}\relax (x)}^{n+1}}{n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.45, size = 17, normalized size = 1.31 \[ - \begin {cases} \frac {\operatorname {acot}^{n + 1}{\relax (x )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {acot}{\relax (x )} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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