3.1112 \(\int \frac{\tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{\tan ^{-1}(a x)^{n+1}}{a c (n+1)} \]

[Out]

ArcTan[a*x]^(1 + n)/(a*c*(1 + n))

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Rubi [A]  time = 0.0402534, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4884} \[ \frac{\tan ^{-1}(a x)^{n+1}}{a c (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]^n/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^(1 + n)/(a*c*(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}(a x)^n}{c+a^2 c x^2} \, dx &=\frac{\tan ^{-1}(a x)^{1+n}}{a c (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0057055, size = 20, normalized size = 1. \[ \frac{\tan ^{-1}(a x)^{n+1}}{a c (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[a*x]^n/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^(1 + n)/(a*c*(1 + n))

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Maple [A]  time = 0.066, size = 21, normalized size = 1.1 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{1+n}}{ \left ( 1+n \right ) ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^n/(a^2*c*x^2+c),x)

[Out]

arctan(a*x)^(1+n)/a/c/(1+n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^n/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.73808, size = 55, normalized size = 2.75 \begin{align*} \frac{\arctan \left (a x\right )^{n} \arctan \left (a x\right )}{a c n + a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^n/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

arctan(a*x)^n*arctan(a*x)/(a*c*n + a*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**n/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**n/(a**2*x**2 + 1), x)/c

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Giac [A]  time = 1.07713, size = 27, normalized size = 1.35 \begin{align*} \frac{\arctan \left (a x\right )^{n + 1}}{a c{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^n/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

arctan(a*x)^(n + 1)/(a*c*(n + 1))