3.246 \(\int \frac{3+x}{(1+x^2) \sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=56 \[ \sqrt{2} \tanh ^{-1}\left (\frac{x+1}{\sqrt{2} \sqrt{x^2+x+1}}\right )-2 \sqrt{2} \tan ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{x^2+x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0466675, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1036, 1030, 207, 203} \[ \sqrt{2} \tanh ^{-1}\left (\frac{x+1}{\sqrt{2} \sqrt{x^2+x+1}}\right )-2 \sqrt{2} \tan ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + x)/((1 + x^2)*Sqrt[1 + x + x^2]),x]

[Out]

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

Rule 1036

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-(a*c)]

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 207

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{3+x}{\left (1+x^2\right ) \sqrt{1+x+x^2}} \, dx &=-\left (\frac{1}{2} \int \frac{-4-4 x}{\left (1+x^2\right ) \sqrt{1+x+x^2}} \, dx\right )+\frac{1}{2} \int \frac{2-2 x}{\left (1+x^2\right ) \sqrt{1+x+x^2}} \, dx\\ &=4 \operatorname{Subst}\left (\int \frac{1}{-8+x^2} \, dx,x,\frac{-2-2 x}{\sqrt{1+x+x^2}}\right )+16 \operatorname{Subst}\left (\int \frac{1}{32+x^2} \, dx,x,\frac{-4+4 x}{\sqrt{1+x+x^2}}\right )\\ &=-2 \sqrt{2} \tan ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{1+x+x^2}}\right )+\sqrt{2} \tanh ^{-1}\left (\frac{1+x}{\sqrt{2} \sqrt{1+x+x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0251623, size = 80, normalized size = 1.43 \[ \left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left ((2+i) \tan ^{-1}\left (\frac{\sqrt [4]{-1} ((2+i) x+(1+2 i))}{2 \sqrt{x^2+x+1}}\right )+(1+2 i) \tanh ^{-1}\left (\frac{(-1)^{3/4} ((1+2 i) x+(2+i))}{2 \sqrt{x^2+x+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/((1 + x^2)*Sqrt[1 + x + x^2]),x]

[Out]

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

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Maple [B]  time = 0.016, size = 128, normalized size = 2.3 \begin{align*}{\sqrt{2}\sqrt{{\frac{ \left ( -1+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+3} \left ({\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{{\frac{ \left ( -1+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+3}} \right ) -2\,\arctan \left ({\frac{\sqrt{2} \left ( -1+x \right ) }{-1-x}{\frac{1}{\sqrt{{\frac{ \left ( -1+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+3}}}} \right ) \right ){\frac{1}{\sqrt{{ \left ({\frac{ \left ( -1+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+3 \right ) \left ( 1+{\frac{-1+x}{-1-x}} \right ) ^{-2}}}}} \left ( 1+{\frac{-1+x}{-1-x}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+x)/(x^2+1)/(x^2+x+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 3}{\sqrt{x^{2} + x + 1}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x + 3)/(sqrt(x^2 + x + 1)*(x^2 + 1)), x)

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Fricas [B]  time = 2.61507, size = 1041, normalized size = 18.59 \begin{align*} \frac{4}{5} \, \sqrt{10} \sqrt{5} \arctan \left (\frac{1}{25} \, \sqrt{5} \sqrt{\sqrt{10} \sqrt{5}{\left (x - 1\right )} + 10 \, x^{2} - \sqrt{x^{2} + x + 1}{\left (\sqrt{10} \sqrt{5} + 10 \, x\right )} + 5 \, x + 15}{\left (\sqrt{10} \sqrt{5} + 10\right )} + \frac{1}{5} \, \sqrt{10} \sqrt{5}{\left (x + 1\right )} - \frac{1}{5} \, \sqrt{x^{2} + x + 1}{\left (\sqrt{10} \sqrt{5} + 10\right )} + 2 \, x + 1\right ) + \frac{4}{5} \, \sqrt{10} \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{10} \sqrt{5}{\left (x + 1\right )} + \frac{1}{50} \, \sqrt{-20 \, \sqrt{10} \sqrt{5}{\left (x - 1\right )} + 200 \, x^{2} + 20 \, \sqrt{x^{2} + x + 1}{\left (\sqrt{10} \sqrt{5} - 10 \, x\right )} + 100 \, x + 300}{\left (\sqrt{10} \sqrt{5} - 10\right )} - \frac{1}{5} \, \sqrt{x^{2} + x + 1}{\left (\sqrt{10} \sqrt{5} - 10\right )} - 2 \, x - 1\right ) - \frac{1}{10} \, \sqrt{10} \sqrt{5} \log \left (20 \, \sqrt{10} \sqrt{5}{\left (x - 1\right )} + 200 \, x^{2} - 20 \, \sqrt{x^{2} + x + 1}{\left (\sqrt{10} \sqrt{5} + 10 \, x\right )} + 100 \, x + 300\right ) + \frac{1}{10} \, \sqrt{10} \sqrt{5} \log \left (-20 \, \sqrt{10} \sqrt{5}{\left (x - 1\right )} + 200 \, x^{2} + 20 \, \sqrt{x^{2} + x + 1}{\left (\sqrt{10} \sqrt{5} - 10 \, x\right )} + 100 \, x + 300\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*sqrt(10)*sqrt(5)*arctan(1/25*sqrt(5)*sqrt(sqrt(10)*sqrt(5)*(x - 1) + 10*x^2 - sqrt(x^2 + x + 1)*(sqrt(10)*
sqrt(5) + 10*x) + 5*x + 15)*(sqrt(10)*sqrt(5) + 10) + 1/5*sqrt(10)*sqrt(5)*(x + 1) - 1/5*sqrt(x^2 + x + 1)*(sq
rt(10)*sqrt(5) + 10) + 2*x + 1) + 4/5*sqrt(10)*sqrt(5)*arctan(1/5*sqrt(10)*sqrt(5)*(x + 1) + 1/50*sqrt(-20*sqr
t(10)*sqrt(5)*(x - 1) + 200*x^2 + 20*sqrt(x^2 + x + 1)*(sqrt(10)*sqrt(5) - 10*x) + 100*x + 300)*(sqrt(10)*sqrt
(5) - 10) - 1/5*sqrt(x^2 + x + 1)*(sqrt(10)*sqrt(5) - 10) - 2*x - 1) - 1/10*sqrt(10)*sqrt(5)*log(20*sqrt(10)*s
qrt(5)*(x - 1) + 200*x^2 - 20*sqrt(x^2 + x + 1)*(sqrt(10)*sqrt(5) + 10*x) + 100*x + 300) + 1/10*sqrt(10)*sqrt(
5)*log(-20*sqrt(10)*sqrt(5)*(x - 1) + 200*x^2 + 20*sqrt(x^2 + x + 1)*(sqrt(10)*sqrt(5) - 10*x) + 100*x + 300)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 3}{\left (x^{2} + 1\right ) \sqrt{x^{2} + x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x + 3)/((x**2 + 1)*sqrt(x**2 + x + 1)), x)

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Giac [C]  time = 1.11181, size = 139, normalized size = 2.48 \begin{align*} -\left (i - \frac{1}{2}\right ) \, \sqrt{2} \log \left (-\left (i + 1\right ) \, x + \sqrt{2} + \left (i + 1\right ) \, \sqrt{x^{2} + x + 1} - i + 1\right ) + \left (i + \frac{1}{2}\right ) \, \sqrt{2} \log \left (-\left (i + 1\right ) \, x + i \, \sqrt{2} + \left (i + 1\right ) \, \sqrt{x^{2} + x + 1} + i - 1\right ) - \left (i + \frac{1}{2}\right ) \, \sqrt{2} \log \left (-\left (i + 1\right ) \, x - i \, \sqrt{2} + \left (i + 1\right ) \, \sqrt{x^{2} + x + 1} + i - 1\right ) + \left (i - \frac{1}{2}\right ) \, \sqrt{2} \log \left (-\left (i + 1\right ) \, x - \sqrt{2} + \left (i + 1\right ) \, \sqrt{x^{2} + x + 1} - i + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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