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

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

[Out]

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

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Rubi [A]  time = 0.0396846, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{4 x+\pi }{\sqrt{\pi ^2-8}}\right )}{\sqrt{\pi ^2-8}} \]

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 = 3.15625, 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 = 0.0138082, size = 27, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{4 x+\pi }{\sqrt{\pi ^2-8}}\right )}{\sqrt{\pi ^2-8}} \]

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 = 0.005, size = 24, normalized size = 0.9 \[ -2\,{\frac{1}{\sqrt{{\pi }^{2}-8}}{\it Artanh} \left ({\frac{\pi +4\,x}{\sqrt{{\pi }^{2}-8}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(Pi*x+2*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.741532, size = 51, normalized size = 1.89 \[ \frac{\log \left (\frac{\pi + 4 \, x - \sqrt{\pi ^{2} - 8}}{\pi + 4 \, x + \sqrt{\pi ^{2} - 8}}\right )}{\sqrt{\pi ^{2} - 8}} \]

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 = 0.244069, size = 82, normalized size = 3.04 \[ \frac{\log \left (\frac{8 \, \pi - \pi ^{3} - 4 \,{\left (\pi ^{2} - 8\right )} x +{\left (\pi ^{2} + 4 \, \pi x + 8 \, x^{2} - 4\right )} \sqrt{\pi ^{2} - 8}}{\pi x + 2 \, x^{2} + 1}\right )}{\sqrt{\pi ^{2} - 8}} \]

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 = 0.525891, size = 76, normalized size = 2.81 \[ \frac{\log{\left (x - \frac{\pi ^{2}}{4 \sqrt{-8 + \pi ^{2}}} + \frac{\pi }{4} + \frac{2}{\sqrt{-8 + \pi ^{2}}} \right )}}{\sqrt{-8 + \pi ^{2}}} - \frac{\log{\left (x - \frac{2}{\sqrt{-8 + \pi ^{2}}} + \frac{\pi }{4} + \frac{\pi ^{2}}{4 \sqrt{-8 + \pi ^{2}}} \right )}}{\sqrt{-8 + \pi ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(pi*x+2*x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209669, size = 54, normalized size = 2. \[ \frac{{\rm ln}\left (\frac{{\left | \pi + 4 \, x - \sqrt{\pi ^{2} - 8} \right |}}{{\left | \pi + 4 \, x + \sqrt{\pi ^{2} - 8} \right |}}\right )}{\sqrt{\pi ^{2} - 8}} \]

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)