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

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.

[In]

Int[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

Rule 4884

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 = 0.01, size = 12, normalized size = 1.00 \[ \frac {\tan ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

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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.

[In]

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

[Out]

Could not integrate

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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.

[In]

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

[Out]

arctan(x)^n*arctan(x)/(n + 1)

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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.

[In]

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

[Out]

arctan(x)^(n + 1)/(n + 1)

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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.

[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 = 0.43, size = 12, normalized size = 1.00 \[ \frac {\arctan \relax (x)^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)^(n + 1)/(n + 1)

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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.

[In]

int(atan(x)^n/(x^2 + 1),x)

[Out]

atan(x)^(n + 1)/(n + 1)

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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.

[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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