3.1.8 \(\int \frac {1}{(4+2 x+x^2) \sqrt {5+2 x+x^2}} \, dx\) [8]

Optimal. Leaf size=28 \[ \frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3} \sqrt {5+2 x+x^2}}\right )}{\sqrt {3}} \]

[Out]

1/3*arctan(1/3*(1+x)*3^(1/2)/(x^2+2*x+5)^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {996, 210} \begin {gather*} \frac {\text {ArcTan}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

ArcTan[(1 + x)/(Sqrt[3]*Sqrt[5 + 2*x + x^2])]/Sqrt[3]

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 996

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (4+2 x+x^2\right ) \sqrt {5+2 x+x^2}} \, dx &=-\left (4 \text {Subst}\left (\int \frac {1}{-24-2 x^2} \, dx,x,\frac {2+2 x}{\sqrt {5+2 x+x^2}}\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3} \sqrt {5+2 x+x^2}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {4+2 x+x^2-(1+x) \sqrt {5+2 x+x^2}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

-(ArcTan[(4 + 2*x + x^2 - (1 + x)*Sqrt[5 + 2*x + x^2])/Sqrt[3]]/Sqrt[3])

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Maple [A]
time = 0.28, size = 27, normalized size = 0.96

method	result	size
default	\(\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +2\right )}{6 \sqrt {x^{2}+2 x +5}}\right )}{3}\)	\(27\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+3 \sqrt {x^{2}+2 x +5}\, x +2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x +3 \sqrt {x^{2}+2 x +5}+7 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{2}+2 x +4}\right )}{6}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*3^(1/2)*arctan(1/6*3^(1/2)/(x^2+2*x+5)^(1/2)*(2*x+2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)), x)

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Fricas [A]
time = 1.56, size = 38, normalized size = 1.36 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{2} + 2 \, x + 5} {\left (x + 1\right )} - \frac {1}{3} \, \sqrt {3} {\left (x^{2} + 2 \, x + 4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^2 + 2*x + 5)*(x + 1) - 1/3*sqrt(3)*(x^2 + 2*x + 4))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x^{2} + 2 x + 4\right ) \sqrt {x^{2} + 2 x + 5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+2*x+4)/(x**2+2*x+5)**(1/2),x)

[Out]

Integral(1/((x**2 + 2*x + 4)*sqrt(x**2 + 2*x + 5)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
time = 6.76, size = 52, normalized size = 1.86 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5} + 2\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5) + 2)) + 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(x - sqrt(x^
2 + 2*x + 5)))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (x^2+2\,x+4\right )\,\sqrt {x^2+2\,x+5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x + x^2 + 4)*(2*x + x^2 + 5)^(1/2)),x)

[Out]

int(1/((2*x + x^2 + 4)*(2*x + x^2 + 5)^(1/2)), x)

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