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

Optimal. Leaf size=16 \[ \frac{2 \sqrt{\tan ^{-1}(a x)}}{a c} \]

[Out]

(2*Sqrt[ArcTan[a*x]])/(a*c)

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Rubi [A]  time = 0.0243045, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4884} \[ \frac{2 \sqrt{\tan ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]),x]

[Out]

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

Mathematica [A]  time = 0.0031818, size = 16, normalized size = 1. \[ \frac{2 \sqrt{\tan ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]),x]

[Out]

(2*Sqrt[ArcTan[a*x]])/(a*c)

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Maple [A]  time = 0.087, size = 15, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*c*x^2+c)/arctan(a*x)^(1/2),x)

[Out]

2*arctan(a*x)^(1/2)/a/c

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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(1/(a^2*c*x^2+c)/arctan(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.88247, size = 36, normalized size = 2.25 \begin{align*} \frac{2 \, \sqrt{\arctan \left (a x\right )}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)/arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(arctan(a*x))/(a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.1066, size = 19, normalized size = 1.19 \begin{align*} \frac{2 \, \sqrt{\arctan \left (a x\right )}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)/arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(arctan(a*x))/(a*c)