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

Optimal. Leaf size=45 \[ \sqrt{x^2+x+1}+2 \log \left (\sqrt{x^2+x+1}+x\right )-x-\frac{3}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

-x + Sqrt[1 + x + x^2] - (3*ArcSinh[(1 + 2*x)/Sqrt[3]])/2 + 2*Log[x + Sqrt[1 + x + x^2]]

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Rubi [A]  time = 0.0324864, antiderivative size = 59, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2116, 893} \[ \frac{3}{2 \left (2 \left (\sqrt{x^2+x+1}+x\right )+1\right )}+2 \log \left (\sqrt{x^2+x+1}+x\right )-\frac{3}{2} \log \left (2 \left (\sqrt{x^2+x+1}+x\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[1 + x + x^2])^(-1),x]

[Out]

3/(2*(1 + 2*(x + Sqrt[1 + x + x^2]))) + 2*Log[x + Sqrt[1 + x + x^2]] - (3*Log[1 + 2*(x + Sqrt[1 + x + x^2])])/
2

Rule 2116

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol]
:> Dist[2, Subst[Int[((g + h*x^n)^p*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2))/(-2*d*e + b*f^2 +
2*e*x)^2, x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && EqQ[e^2 -
c*f^2, 0] && IntegerQ[p]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{1}{x+\sqrt{1+x+x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1+x+x^2}{x (1+2 x)^2} \, dx,x,x+\sqrt{1+x+x^2}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{x}-\frac{3}{2 (1+2 x)^2}-\frac{3}{2 (1+2 x)}\right ) \, dx,x,x+\sqrt{1+x+x^2}\right )\\ &=\frac{3}{2 \left (1+2 \left (x+\sqrt{1+x+x^2}\right )\right )}+2 \log \left (x+\sqrt{1+x+x^2}\right )-\frac{3}{2} \log \left (1+2 \left (x+\sqrt{1+x+x^2}\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0296762, size = 59, normalized size = 1.31 \[ \frac{3}{2 \left (2 \left (\sqrt{x^2+x+1}+x\right )+1\right )}+2 \log \left (\sqrt{x^2+x+1}+x\right )-\frac{3}{2} \log \left (2 \left (\sqrt{x^2+x+1}+x\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3/(2*(1 + 2*(x + Sqrt[1 + x + x^2]))) + 2*Log[x + Sqrt[1 + x + x^2]] - (3*Log[1 + 2*(x + Sqrt[1 + x + x^2])])/
2

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Maple [A]  time = 0.005, size = 52, normalized size = 1.2 \begin{align*} \sqrt{ \left ( 1+x \right ) ^{2}-x}-{\frac{1}{2}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) }-{\it Artanh} \left ({\frac{1-x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-x}}}} \right ) -x+\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1+x)^2-x)^(1/2)-1/2*arcsinh(2/3*3^(1/2)*(x+1/2))-arctanh(1/2*(1-x)/((1+x)^2-x)^(1/2))-x+ln(1+x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x + \sqrt{x^{2} + x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x + sqrt(x^2 + x + 1)), x)

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Fricas [A]  time = 1.72745, size = 193, normalized size = 4.29 \begin{align*} -x + \sqrt{x^{2} + x + 1} + \log \left (x + 1\right ) - \log \left (-x + \sqrt{x^{2} + x + 1}\right ) + \log \left (-x + \sqrt{x^{2} + x + 1} - 2\right ) + \frac{1}{2} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09049, size = 89, normalized size = 1.98 \begin{align*} -x + \sqrt{x^{2} + x + 1} + \frac{1}{2} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) + \log \left ({\left | x + 1 \right |}\right ) - \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/(x+(x^2+x+1)^(1/2)),x, algorithm="giac")

[Out]

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