Optimal. Leaf size=12 \[ \frac {\tan ^{-1}(x)^{n+1}}{n+1} \]
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Rubi [A] time = 0.02, antiderivative size = 12, 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 = {4884} \[ \frac {\tan ^{-1}(x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(x)^n}{1+x^2} \, dx &=\frac {\tan ^{-1}(x)^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {\tan ^{-1}(x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(x)^n}{1+x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.95, size = 12, normalized size = 1.00 \[ \frac {\arctan \relax (x)^{n} \arctan \relax (x)}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 12, normalized size = 1.00 \[ \frac {\arctan \relax (x)^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 13, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\arctan \relax (x )^{1+n}}{1+n}\) | \(13\) |
default | \(\frac {\arctan \relax (x )^{1+n}}{1+n}\) | \(13\) |
risch | \(-\frac {i \left (-\ln \left (-i \left (x +i\right )\right )+\ln \left (-i \left (-x +i\right )\right )\right ) \left (-\frac {i \left (-\ln \left (-i \left (x +i\right )\right )+\ln \left (-i \left (-x +i\right )\right )\right )}{2}\right )^{n}}{2 \left (1+n \right )}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 12, normalized size = 1.00 \[ \frac {\arctan \relax (x)^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 12, normalized size = 1.00 \[ \frac {{\mathrm {atan}\relax (x)}^{n+1}}{n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.79, size = 15, normalized size = 1.25 \[ \begin {cases} \frac {\operatorname {atan}^{n + 1}{\relax (x )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {atan}{\relax (x )} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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