3.86 \(\int \frac{1}{1+\pi x+3 x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]

[Out]

(2*ArcTan[(Pi + 6*x)/Sqrt[12 - Pi^2]])/Sqrt[12 - Pi^2]

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Rubi [A]  time = 0.0202124, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Pi*x + 3*x^2)^(-1),x]

[Out]

(2*ArcTan[(Pi + 6*x)/Sqrt[12 - Pi^2]])/Sqrt[12 - Pi^2]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{1+\pi x+3 x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-12+\pi ^2-x^2} \, dx,x,\pi +6 x\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\pi +6 x}{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}}\\ \end{align*}

Mathematica [A]  time = 0.0098903, size = 31, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Pi*x + 3*x^2)^(-1),x]

[Out]

(2*ArcTan[(Pi + 6*x)/Sqrt[12 - Pi^2]])/Sqrt[12 - Pi^2]

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Maple [A]  time = 0.045, size = 28, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{\sqrt{-{\pi }^{2}+12}}\arctan \left ({\frac{\pi +6\,x}{\sqrt{-{\pi }^{2}+12}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(Pi*x+3*x^2+1),x)

[Out]

2*arctan((Pi+6*x)/(-Pi^2+12)^(1/2))/(-Pi^2+12)^(1/2)

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Maxima [A]  time = 1.10936, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, \arctan \left (\frac{\pi + 6 \, x}{\sqrt{-\pi ^{2} + 12}}\right )}{\sqrt{-\pi ^{2} + 12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(pi*x+3*x^2+1),x, algorithm="maxima")

[Out]

2*arctan((pi + 6*x)/sqrt(-pi^2 + 12))/sqrt(-pi^2 + 12)

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Fricas [A]  time = 2.31212, size = 108, normalized size = 3.48 \begin{align*} \frac{2 \, \sqrt{-\pi ^{2} + 12} \arctan \left (\frac{{\left (\pi + 6 \, x\right )} \sqrt{-\pi ^{2} + 12}}{\pi ^{2} - 12}\right )}{\pi ^{2} - 12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(pi*x+3*x^2+1),x, algorithm="fricas")

[Out]

2*sqrt(-pi^2 + 12)*arctan((pi + 6*x)*sqrt(-pi^2 + 12)/(pi^2 - 12))/(pi^2 - 12)

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Sympy [C]  time = 0.305933, size = 87, normalized size = 2.81 \begin{align*} - \frac{i \log{\left (x + \frac{\pi }{6} - \frac{2 i}{\sqrt{12 - \pi ^{2}}} + \frac{i \pi ^{2}}{6 \sqrt{12 - \pi ^{2}}} \right )}}{\sqrt{12 - \pi ^{2}}} + \frac{i \log{\left (x + \frac{\pi }{6} - \frac{i \pi ^{2}}{6 \sqrt{12 - \pi ^{2}}} + \frac{2 i}{\sqrt{12 - \pi ^{2}}} \right )}}{\sqrt{12 - \pi ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(pi*x+3*x**2+1),x)

[Out]

-I*log(x + pi/6 - 2*I/sqrt(12 - pi**2) + I*pi**2/(6*sqrt(12 - pi**2)))/sqrt(12 - pi**2) + I*log(x + pi/6 - I*p
i**2/(6*sqrt(12 - pi**2)) + 2*I/sqrt(12 - pi**2))/sqrt(12 - pi**2)

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Giac [A]  time = 1.24485, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, \arctan \left (\frac{\pi + 6 \, x}{\sqrt{-\pi ^{2} + 12}}\right )}{\sqrt{-\pi ^{2} + 12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(pi*x+3*x^2+1),x, algorithm="giac")

[Out]

2*arctan((pi + 6*x)/sqrt(-pi^2 + 12))/sqrt(-pi^2 + 12)