3.278 \(\int \frac{1}{(1+x) \sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=22 \[ -\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right ) \]

[Out]

-ArcTanh[(1 - x)/(2*Sqrt[1 + x + x^2])]

________________________________________________________________________________________

Rubi [A]  time = 0.0092097, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {724, 206} \[ -\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x)*Sqrt[1 + x + x^2]),x]

[Out]

-ArcTanh[(1 - x)/(2*Sqrt[1 + x + x^2])]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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+x) \sqrt{1+x+x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{1-x}{\sqrt{1+x+x^2}}\right )\right )\\ &=-\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{1+x+x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0031113, size = 22, normalized size = 1. \[ -\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[(1 - x)/(2*Sqrt[1 + x + x^2])]

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 1. \begin{align*} -{\it Artanh} \left ({\frac{1-x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-x}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x)/(x^2+x+1)^(1/2),x)

[Out]

-arctanh(1/2*(1-x)/((1+x)^2-x)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.68206, size = 34, normalized size = 1.55 \begin{align*} \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x + 1 \right |}} - \frac{\sqrt{3}}{3 \,{\left | x + 1 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(1/3*sqrt(3)*x/abs(x + 1) - 1/3*sqrt(3)/abs(x + 1))

________________________________________________________________________________________

Fricas [A]  time = 2.02714, size = 86, normalized size = 3.91 \begin{align*} -\log \left (-x + \sqrt{x^{2} + x + 1}\right ) + \log \left (-x + \sqrt{x^{2} + x + 1} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 + x + 1)) + log(-x + sqrt(x^2 + x + 1) - 2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.09693, size = 43, normalized size = 1.95 \begin{align*} -\log \left ({\left | -x + \sqrt{x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt{x^{2} + x + 1} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-x + sqrt(x^2 + x + 1))) + log(abs(-x + sqrt(x^2 + x + 1) - 2))