3.1.68 \(\int \frac {\tan ^{-1}(x)^n}{1+x^2} \, dx\) [68]

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

[Out]

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

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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 = {5004} \begin {gather*} \frac {\text {ArcTan}(x)^{n+1}}{n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

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 = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}(x)^{1+n}}{1+n} \end {gather*}

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 = 0.14, size = 13, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {\arctan \left (x \right )^{1+n}}{1+n}\)	\(13\)
default	\(\frac {\arctan \left (x \right )^{1+n}}{1+n}\)	\(13\)
risch	\(\frac {i \left (\ln \left (-i x +1\right )-\ln \left (i x +1\right )\right ) \left (\frac {i \left (\ln \left (-i x +1\right )-\ln \left (i x +1\right )\right )}{2}\right )^{n}}{2+2 n}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x)^n/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 12, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (x\right )^{n + 1}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)^(n + 1)/(n + 1)

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Fricas [A]
time = 0.70, size = 12, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (x\right )^{n} \arctan \left (x\right )}{n + 1} \end {gather*}

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 = 0.88, size = 15, normalized size = 1.25 \begin {gather*} \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} \end {gather*}

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 [A]
time = 0.74, size = 12, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (x\right )^{n + 1}}{n + 1} \end {gather*}

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)

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Mupad [B]
time = 0.23, size = 12, normalized size = 1.00 \begin {gather*} \frac {{\mathrm {atan}\left (x\right )}^{n+1}}{n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(x)^n/(x^2 + 1),x)

[Out]

atan(x)^(n + 1)/(n + 1)

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