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

Optimal. Leaf size=27 \[ -\frac{2 \tanh ^{-1}\left (\frac{\pi -6 x}{\sqrt{12+\pi ^2}}\right )}{\sqrt{12+\pi ^2}} \]

[Out]

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

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Rubi [A]  time = 0.0393691, antiderivative size = 27, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\pi -6 x}{\sqrt{12+\pi ^2}}\right )}{\sqrt{12+\pi ^2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + Pi*x - 3*x^2)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.92276, size = 26, normalized size = 0.96 \[ - \frac{2 \operatorname{atanh}{\left (\frac{- 6 x + \pi }{\sqrt{\pi ^{2} + 12}} \right )}}{\sqrt{\pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0141199, size = 29, normalized size = 1.07 \[ \frac{2 \tanh ^{-1}\left (\frac{6 x-\pi }{\sqrt{12+\pi ^2}}\right )}{\sqrt{12+\pi ^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Pi*x - 3*x^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.006, size = 26, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{{\pi }^{2}+12}}{\it Artanh} \left ({\frac{6\,x-\pi }{\sqrt{{\pi }^{2}+12}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(Pi*x-3*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.715809, size = 53, normalized size = 1.96 \[ -\frac{\log \left (\frac{\pi - 6 \, x + \sqrt{\pi ^{2} + 12}}{\pi - 6 \, x - \sqrt{\pi ^{2} + 12}}\right )}{\sqrt{\pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.223624, size = 81, normalized size = 3. \[ \frac{\log \left (\frac{12 \, \pi + \pi ^{3} - 6 \,{\left (\pi ^{2} + 12\right )} x -{\left (\pi ^{2} - 6 \, \pi x + 18 \, x^{2} + 6\right )} \sqrt{\pi ^{2} + 12}}{\pi x - 3 \, x^{2} + 1}\right )}{\sqrt{\pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log((12*pi + pi^3 - 6*(pi^2 + 12)*x - (pi^2 - 6*pi*x + 18*x^2 + 6)*sqrt(pi^2 + 1
2))/(pi*x - 3*x^2 + 1))/sqrt(pi^2 + 12)

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Sympy [A]  time = 0.584178, size = 76, normalized size = 2.81 \[ \frac{\log{\left (x - \frac{\pi }{6} + \frac{\pi ^{2}}{6 \sqrt{\pi ^{2} + 12}} + \frac{2}{\sqrt{\pi ^{2} + 12}} \right )}}{\sqrt{\pi ^{2} + 12}} - \frac{\log{\left (x - \frac{\pi }{6} - \frac{2}{\sqrt{\pi ^{2} + 12}} - \frac{\pi ^{2}}{6 \sqrt{\pi ^{2} + 12}} \right )}}{\sqrt{\pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(pi*x-3*x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209133, size = 61, normalized size = 2.26 \[ -\frac{{\rm ln}\left (\frac{{\left | -\pi + 6 \, x - \sqrt{\pi ^{2} + 12} \right |}}{{\left | -\pi + 6 \, x + \sqrt{\pi ^{2} + 12} \right |}}\right )}{\sqrt{\pi ^{2} + 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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