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 = 0.0461908, 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 \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]

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]

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Rubi in Sympy [A]  time = 2.8625, size = 24, normalized size = 0.77 \[ \frac{2 \operatorname{atan}{\left (\frac{6 x + \pi }{\sqrt{- \pi ^{2} + 12}} \right )}}{\sqrt{- \pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(pi*x+3*x**2+1),x)

[Out]

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

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Mathematica [A]  time = 0.0144901, 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 = 0.004, size = 28, normalized size = 0.9 \[ 2\,{\frac{1}{\sqrt{-{\pi }^{2}+12}}\arctan \left ({\frac{\pi +6\,x}{\sqrt{-{\pi }^{2}+12}}} \right ) } \]

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 = 0.728159, size = 36, normalized size = 1.16 \[ \frac{2 \, \arctan \left (\frac{\pi + 6 \, x}{\sqrt{-\pi ^{2} + 12}}\right )}{\sqrt{-\pi ^{2} + 12}} \]

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 = 0.229734, size = 46, normalized size = 1.48 \[ -\frac{2 \, \arctan \left (\frac{{\left (\pi + 6 \, x\right )} \sqrt{-\pi ^{2} + 12}}{\pi ^{2} - 12}\right )}{\sqrt{-\pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.441801, size = 87, normalized size = 2.81 \[ - \frac{i \log{\left (x + \frac{\pi }{6} - \frac{2 i}{\sqrt{- \pi ^{2} + 12}} + \frac{i \pi ^{2}}{6 \sqrt{- \pi ^{2} + 12}} \right )}}{\sqrt{- \pi ^{2} + 12}} + \frac{i \log{\left (x + \frac{\pi }{6} - \frac{i \pi ^{2}}{6 \sqrt{- \pi ^{2} + 12}} + \frac{2 i}{\sqrt{- \pi ^{2} + 12}} \right )}}{\sqrt{- \pi ^{2} + 12}} \]

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

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GIAC/XCAS [A]  time = 0.207752, size = 36, normalized size = 1.16 \[ \frac{2 \, \arctan \left (\frac{\pi + 6 \, x}{\sqrt{-\pi ^{2} + 12}}\right )}{\sqrt{-\pi ^{2} + 12}} \]

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)