3.391 \(\int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{\tan ^{-1}(a x)^4}{4 a c} \]

[Out]

ArcTan[a*x]^4/(4*a*c)

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Rubi [A]  time = 0.0242376, antiderivative size = 16, 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} \[ \frac{\tan ^{-1}(a x)^4}{4 a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]^3/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^4/(4*a*c)

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)^3}{c+a^2 c x^2} \, dx &=\frac{\tan ^{-1}(a x)^4}{4 a c}\\ \end{align*}

Mathematica [A]  time = 0.0035491, size = 16, normalized size = 1. \[ \frac{\tan ^{-1}(a x)^4}{4 a c} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[a*x]^3/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^4/(4*a*c)

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Maple [A]  time = 0.078, size = 15, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{4}}{4\,ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^3/(a^2*c*x^2+c),x)

[Out]

1/4*arctan(a*x)^4/a/c

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Maxima [A]  time = 1.49116, size = 19, normalized size = 1.19 \begin{align*} \frac{\arctan \left (a x\right )^{4}}{4 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/4*arctan(a*x)^4/(a*c)

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Fricas [A]  time = 1.78814, size = 34, normalized size = 2.12 \begin{align*} \frac{\arctan \left (a x\right )^{4}}{4 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

1/4*arctan(a*x)^4/(a*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{3}{\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)**3/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**3/(a**2*x**2 + 1), x)/c

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Giac [A]  time = 1.17405, size = 19, normalized size = 1.19 \begin{align*} \frac{\arctan \left (a x\right )^{4}}{4 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

1/4*arctan(a*x)^4/(a*c)