Optimal. Leaf size=13 \[ -\frac {\cot ^{-1}(x)^{1+n}}{1+n} \]
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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 = {5005}
\begin {gather*} -\frac {\cot ^{-1}(x)^{n+1}}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 5005
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 \begin {gather*} -\frac {\cot ^{-1}(x)^{1+n}}{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 14, normalized size = 1.08
method | result | size |
derivativedivides | \(-\frac {\mathrm {arccot}\left (x \right )^{1+n}}{1+n}\) | \(14\) |
default | \(-\frac {\mathrm {arccot}\left (x \right )^{1+n}}{1+n}\) | \(14\) |
risch | \(-\frac {\left (\pi -i \ln \left (-i \left (i+x \right )\right )+i \ln \left (-i \left (i-x \right )\right )\right ) \left (\frac {1}{2}\right )^{n} \left (\pi -i \ln \left (-i \left (i+x \right )\right )+i \ln \left (-i \left (i-x \right )\right )\right )^{n}}{2 \left (1+n \right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\operatorname {arccot}\left (x\right )^{n + 1}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\operatorname {arccot}\left (x\right )^{n} \operatorname {arccot}\left (x\right )}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.92, size = 17, normalized size = 1.31 \begin {gather*} - \begin {cases} \frac {\operatorname {acot}^{n + 1}{\left (x \right )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {acot}{\left (x \right )} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 1.15 \begin {gather*} -\frac {\arctan \left (\frac {1}{x}\right )^{n + 1}}{n + 1} \end {gather*}
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 \begin {gather*} -\frac {{\mathrm {acot}\left (x\right )}^{n+1}}{n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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