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

Optimal. Leaf size=27 \[ -\frac{2 \tanh ^{-1}\left (\frac{\pi -4 x}{\sqrt{8+\pi ^2}}\right )}{\sqrt{8+\pi ^2}} \]

[Out]

(-2*ArcTanh[(Pi - 4*x)/Sqrt[8 + Pi^2]])/Sqrt[8 + Pi^2]

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Rubi [A]  time = 0.0405191, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\pi -4 x}{\sqrt{8+\pi ^2}}\right )}{\sqrt{8+\pi ^2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + Pi*x - 2*x^2)^(-1),x]

[Out]

(-2*ArcTanh[(Pi - 4*x)/Sqrt[8 + Pi^2]])/Sqrt[8 + Pi^2]

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Rubi in Sympy [A]  time = 3.08479, size = 26, normalized size = 0.96 \[ - \frac{2 \operatorname{atanh}{\left (\frac{- 4 x + \pi }{\sqrt{8 + \pi ^{2}}} \right )}}{\sqrt{8 + \pi ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh((-4*x + pi)/sqrt(8 + pi**2))/sqrt(8 + pi**2)

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Mathematica [A]  time = 0.0141532, size = 29, normalized size = 1.07 \[ \frac{2 \tanh ^{-1}\left (\frac{4 x-\pi }{\sqrt{8+\pi ^2}}\right )}{\sqrt{8+\pi ^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(-Pi + 4*x)/Sqrt[8 + Pi^2]])/Sqrt[8 + Pi^2]

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Maple [A]  time = 0.004, size = 26, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{{\pi }^{2}+8}}{\it Artanh} \left ({\frac{4\,x-\pi }{\sqrt{{\pi }^{2}+8}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(Pi*x-2*x^2+1),x)

[Out]

2/(Pi^2+8)^(1/2)*arctanh((4*x-Pi)/(Pi^2+8)^(1/2))

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Maxima [A]  time = 0.733548, size = 53, normalized size = 1.96 \[ -\frac{\log \left (\frac{\pi - 4 \, x + \sqrt{\pi ^{2} + 8}}{\pi - 4 \, x - \sqrt{\pi ^{2} + 8}}\right )}{\sqrt{\pi ^{2} + 8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log((pi - 4*x + sqrt(pi^2 + 8))/(pi - 4*x - sqrt(pi^2 + 8)))/sqrt(pi^2 + 8)

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Fricas [A]  time = 0.218713, size = 81, normalized size = 3. \[ \frac{\log \left (\frac{8 \, \pi + \pi ^{3} - 4 \,{\left (\pi ^{2} + 8\right )} x -{\left (\pi ^{2} - 4 \, \pi x + 8 \, x^{2} + 4\right )} \sqrt{\pi ^{2} + 8}}{\pi x - 2 \, x^{2} + 1}\right )}{\sqrt{\pi ^{2} + 8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log((8*pi + pi^3 - 4*(pi^2 + 8)*x - (pi^2 - 4*pi*x + 8*x^2 + 4)*sqrt(pi^2 + 8))/
(pi*x - 2*x^2 + 1))/sqrt(pi^2 + 8)

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Sympy [A]  time = 0.565906, size = 76, normalized size = 2.81 \[ - \frac{\log{\left (x - \frac{\pi }{4} - \frac{\pi ^{2}}{4 \sqrt{8 + \pi ^{2}}} - \frac{2}{\sqrt{8 + \pi ^{2}}} \right )}}{\sqrt{8 + \pi ^{2}}} + \frac{\log{\left (x - \frac{\pi }{4} + \frac{2}{\sqrt{8 + \pi ^{2}}} + \frac{\pi ^{2}}{4 \sqrt{8 + \pi ^{2}}} \right )}}{\sqrt{8 + \pi ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(pi*x-2*x**2+1),x)

[Out]

-log(x - pi/4 - pi**2/(4*sqrt(8 + pi**2)) - 2/sqrt(8 + pi**2))/sqrt(8 + pi**2) +
log(x - pi/4 + 2/sqrt(8 + pi**2) + pi**2/(4*sqrt(8 + pi**2)))/sqrt(8 + pi**2)

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GIAC/XCAS [A]  time = 0.211031, size = 61, normalized size = 2.26 \[ -\frac{{\rm ln}\left (\frac{{\left | -\pi + 4 \, x - \sqrt{\pi ^{2} + 8} \right |}}{{\left | -\pi + 4 \, x + \sqrt{\pi ^{2} + 8} \right |}}\right )}{\sqrt{\pi ^{2} + 8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(-pi + 4*x - sqrt(pi^2 + 8))/abs(-pi + 4*x + sqrt(pi^2 + 8)))/sqrt(pi^2 +
8)