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 = 0.0182718, 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, Rules used = {618, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{\pi -6 x}{\sqrt{12+\pi ^2}}\right )}{\sqrt{12+\pi ^2}} \]

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]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 \tanh ^{-1}\left (\frac{\pi -6 x}{\sqrt{12+\pi ^2}}\right )}{\sqrt{12+\pi ^2}}\\ \end{align*}

Mathematica [A]  time = 0.0083681, 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 = 0.054, size = 26, normalized size = 1. \begin{align*} 2\,{\frac{1}{\sqrt{{\pi }^{2}+12}}{\it Artanh} \left ({\frac{6\,x-\pi }{\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/(Pi^2+12)^(1/2)*arctanh((6*x-Pi)/(Pi^2+12)^(1/2))

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Maxima [A]  time = 1.21781, size = 53, normalized size = 1.96 \begin{align*} -\frac{\log \left (\frac{\pi - 6 \, x + \sqrt{\pi ^{2} + 12}}{\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]

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

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Fricas [B]  time = 2.35865, size = 135, normalized size = 5. \begin{align*} \frac{\log \left (-\frac{\pi ^{2} - 6 \, \pi x + 18 \, x^{2} -{\left (\pi - 6 \, x\right )} \sqrt{\pi ^{2} + 12} + 6}{\pi x - 3 \, x^{2} + 1}\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="fricas")

[Out]

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

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Sympy [B]  time = 0.291715, size = 76, normalized size = 2.81 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.31224, size = 61, normalized size = 2.26 \begin{align*} -\frac{\log \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}} \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]

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