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)/(Pi^2+8)^(1/2))/(Pi^2+8)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 \tanh ^{-1}\left (\frac {\pi -4 x}{\sqrt {8+\pi ^2}}\right )}{\sqrt {8+\pi ^2}} \]

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]

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])

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]

Rubi steps

\begin {align*} \int \frac {1}{1+\pi x-2 x^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{8+\pi ^2-x^2} \, dx,x,\pi -4 x\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\pi -4 x}{\sqrt {8+\pi ^2}}\right )}{\sqrt {8+\pi ^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.99, size = 51, normalized size = 1.89 \[ \frac {\log \left (-\frac {\pi ^{2} - 4 \, \pi x + 8 \, x^{2} - {\left (\pi - 4 \, x\right )} \sqrt {\pi ^{2} + 8} + 4}{\pi x - 2 \, x^{2} + 1}\right )}{\sqrt {\pi ^{2} + 8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.38, size = 45, normalized size = 1.67 \[ -\frac {\log \left (\frac {{\left | -\pi + 4 \, x - \sqrt {\pi ^{2} + 8} \right |}}{{\left | -\pi + 4 \, x + \sqrt {\pi ^{2} + 8} \right |}}\right )}{\sqrt {\pi ^{2} + 8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 26, normalized size = 0.96 \[ \frac {2 \arctanh \left (\frac {4 x -\pi }{\sqrt {\pi ^{2}+8}}\right )}{\sqrt {\pi ^{2}+8}} \]

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 = 1.33, size = 39, normalized size = 1.44 \[ -\frac {\log \left (\frac {\pi - 4 \, x + \sqrt {\pi ^{2} + 8}}{\pi - 4 \, x - \sqrt {\pi ^{2} + 8}}\right )}{\sqrt {\pi ^{2} + 8}} \]

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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mupad [B]  time = 0.39, size = 23, normalized size = 0.85 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\Pi -4\,x}{\sqrt {\Pi ^2+8}}\right )}{\sqrt {\Pi ^2+8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(Pi*x - 2*x^2 + 1),x)

[Out]

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

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sympy [B]  time = 0.23, 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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