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)

________________________________________________________________________________________

Rubi [A]  time = 0.0232248, 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, 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*}

Mathematica [A]  time = 0.0041617, 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)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 13, normalized size = 1.1 \begin{align*}{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{1+n}}{1+n}} \end{align*}

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)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 2.07399, size = 42, normalized size = 3.5 \begin{align*} \frac{\arctan \left (x\right )^{n} \arctan \left (x\right )}{n + 1} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 4.10939, size = 15, normalized size = 1.25 \begin{align*} \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{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.11214, size = 16, normalized size = 1.33 \begin{align*} \frac{\arctan \left (x\right )^{n + 1}}{n + 1} \end{align*}

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)