3.1.86 \(\int \frac {1}{1+\pi x+3 x^2} \, dx\) [86]

Optimal. Leaf size=31 \[ \frac {2 \tan ^{-1}\left (\frac {\pi +6 x}{\sqrt {12-\pi ^2}}\right )}{\sqrt {12-\pi ^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, 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 = {632, 210} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {6 x+\pi }{\sqrt {12-\pi ^2}}\right )}{\sqrt {12-\pi ^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Pi*x + 3*x^2)^(-1),x]

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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+3 x^2} \, dx &=-\left (2 \text {Subst}\left (\int \frac {1}{-12+\pi ^2-x^2} \, dx,x,\pi +6 x\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\pi +6 x}{\sqrt {12-\pi ^2}}\right )}{\sqrt {12-\pi ^2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\pi +6 x}{\sqrt {12-\pi ^2}}\right )}{\sqrt {12-\pi ^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Pi*x + 3*x^2)^(-1),x]

[Out]

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

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Maple [A]
time = 1.00, size = 28, normalized size = 0.90

method	result	size
default	\(\frac {2 \arctan \left (\frac {\pi +6 x}{\sqrt {-\pi ^{2}+12}}\right )}{\sqrt {-\pi ^{2}+12}}\)	\(28\)
risch	\(\frac {\ln \left (-\pi ^{2}+\pi \sqrt {\pi ^{2}-12}+6 x \sqrt {\pi ^{2}-12}+12\right )}{\sqrt {\pi ^{2}-12}}-\frac {\ln \left (\pi ^{2}+\pi \sqrt {\pi ^{2}-12}+6 x \sqrt {\pi ^{2}-12}-12\right )}{\sqrt {\pi ^{2}-12}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(Pi*x+3*x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 27, normalized size = 0.87 \begin {gather*} \frac {2 \, \arctan \left (\frac {\pi + 6 \, x}{\sqrt {-\pi ^{2} + 12}}\right )}{\sqrt {-\pi ^{2} + 12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.79, size = 41, normalized size = 1.32 \begin {gather*} \frac {2 \, \sqrt {-\pi ^{2} + 12} \arctan \left (\frac {{\left (\pi + 6 \, x\right )} \sqrt {-\pi ^{2} + 12}}{\pi ^{2} - 12}\right )}{\pi ^{2} - 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.08, size = 87, normalized size = 2.81 \begin {gather*} - \frac {i \log {\left (x + \frac {\pi }{6} - \frac {2 i}{\sqrt {12 - \pi ^{2}}} + \frac {i \pi ^{2}}{6 \sqrt {12 - \pi ^{2}}} \right )}}{\sqrt {12 - \pi ^{2}}} + \frac {i \log {\left (x + \frac {\pi }{6} - \frac {i \pi ^{2}}{6 \sqrt {12 - \pi ^{2}}} + \frac {2 i}{\sqrt {12 - \pi ^{2}}} \right )}}{\sqrt {12 - \pi ^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(pi*x+3*x**2+1),x)

[Out]

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

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Giac [A]
time = 0.82, size = 27, normalized size = 0.87 \begin {gather*} \frac {2 \, \arctan \left (\frac {\pi + 6 \, x}{\sqrt {-\pi ^{2} + 12}}\right )}{\sqrt {-\pi ^{2} + 12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.38, size = 23, normalized size = 0.74 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {\Pi +6\,x}{\sqrt {\Pi ^2-12}}\right )}{\sqrt {\Pi ^2-12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(Pi*x + 3*x^2 + 1),x)

[Out]

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

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