3.68 \(\int \frac{\tan ^{-1}(x)^n}{1+x^2} \, dx\)

Optimal. Leaf size=12 \[ \frac{\tan ^{-1}(x)^{n+1}}{n+1} \]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0414055, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\tan ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

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Rubi in Sympy [A]  time = 3.10798, size = 8, normalized size = 0.67 \[ \frac{\operatorname{atan}^{n + 1}{\left (x \right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(atan(x)**n/(x**2+1),x)

[Out]

atan(x)**(n + 1)/(n + 1)

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Mathematica [A]  time = 0.00460808, size = 12, normalized size = 1. \[ \frac{\tan ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

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Maple [A]  time = 0.001, size = 13, normalized size = 1.1 \[{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{1+n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x)^n/(x^2+1),x)

[Out]

arctan(x)^(1+n)/(1+n)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x)^n/(x^2 + 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235106, size = 16, normalized size = 1.33 \[ \frac{\arctan \left (x\right )^{n} \arctan \left (x\right )}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x)^n/(x^2 + 1),x, algorithm="fricas")

[Out]

arctan(x)^n*arctan(x)/(n + 1)

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Sympy [A]  time = 9.21979, size = 15, normalized size = 1.25 \[ \begin{cases} \frac{\operatorname{atan}^{n + 1}{\left (x \right )}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (\operatorname{atan}{\left (x \right )} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(atan(x)**n/(x**2+1),x)

[Out]

Piecewise((atan(x)**(n + 1)/(n + 1), Ne(n, -1)), (log(atan(x)), True))

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GIAC/XCAS [A]  time = 0.202165, size = 16, normalized size = 1.33 \[ \frac{\arctan \left (x\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x)^n/(x^2 + 1),x, algorithm="giac")

[Out]

arctan(x)^(n + 1)/(n + 1)