3.24 \(\int \frac{1-x+3 x^2}{\sqrt{1-x+x^2} \left (1+x+x^2\right )^2} \, dx\)

Optimal. Leaf size=86 \[ \frac{\sqrt{x^2-x+1} (x+1)}{x^2+x+1}+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{x^2-x+1}}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{2}{3}} (1-x)}{\sqrt{x^2-x+1}}\right )}{\sqrt{6}} \]

[Out]

((1 + x)*Sqrt[1 - x + x^2])/(1 + x + x^2) + Sqrt[2]*ArcTan[(Sqrt[2]*(1 + x))/Sqr
t[1 - x + x^2]] - ArcTanh[(Sqrt[2/3]*(1 - x))/Sqrt[1 - x + x^2]]/Sqrt[6]

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Rubi [A]  time = 0.225655, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{\sqrt{x^2-x+1} (x+1)}{x^2+x+1}+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{x^2-x+1}}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{2}{3}} (1-x)}{\sqrt{x^2-x+1}}\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

((1 + x)*Sqrt[1 - x + x^2])/(1 + x + x^2) + Sqrt[2]*ArcTan[(Sqrt[2]*(1 + x))/Sqr
t[1 - x + x^2]] - ArcTanh[(Sqrt[2/3]*(1 - x))/Sqrt[1 - x + x^2]]/Sqrt[6]

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Rubi in Sympy [A]  time = 18.81, size = 87, normalized size = 1.01 \[ \frac{\left (12 x + 12\right ) \sqrt{x^{2} - x + 1}}{12 \left (x^{2} + x + 1\right )} + \sqrt{2} \operatorname{atan}{\left (- \frac{\sqrt{2} \left (- 144 x - 144\right )}{144 \sqrt{x^{2} - x + 1}} \right )} + \frac{\sqrt{6} \operatorname{atanh}{\left (\frac{\sqrt{6} \left (24 x - 24\right )}{72 \sqrt{x^{2} - x + 1}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2-x+1)/(x**2+x+1)**2/(x**2-x+1)**(1/2),x)

[Out]

(12*x + 12)*sqrt(x**2 - x + 1)/(12*(x**2 + x + 1)) + sqrt(2)*atan(-sqrt(2)*(-144
*x - 144)/(144*sqrt(x**2 - x + 1))) + sqrt(6)*atanh(sqrt(6)*(24*x - 24)/(72*sqrt
(x**2 - x + 1)))/6

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Mathematica [C]  time = 5.47027, size = 961, normalized size = 11.17 \[ \frac{\sqrt{x^2-x+1} (x+1)}{x^2+x+1}+\frac{\left (7-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{3 \left (\left (-21-4 i \sqrt{3}\right ) x^4+14 \left (7-2 i \sqrt{3}\right ) x^3+\left (-103-36 i \sqrt{3}\right ) x^2+\left (94+32 i \sqrt{3}\right ) x-64 i \sqrt{3}-17\right )}{\left (84 i-113 \sqrt{3}\right ) x^4+2 \left (52 \sqrt{3-3 i \sqrt{3}} \sqrt{x^2-x+1}+21 \sqrt{3}+138 i\right ) x^3+\left (52 \sqrt{3-3 i \sqrt{3}} \sqrt{x^2-x+1}-59 \sqrt{3}-180 i\right ) x^2+2 \left (26 \sqrt{3-3 i \sqrt{3}} \sqrt{x^2-x+1}-69 \sqrt{3}+132 i\right ) x-52 \sqrt{3-3 i \sqrt{3}} \sqrt{x^2-x+1}+67 \sqrt{3}+96 i}\right )}{4 \sqrt{3-3 i \sqrt{3}}}-\frac{i \left (-7 i+\sqrt{3}\right ) \tan ^{-1}\left (\frac{3 \left (\left (-21+4 i \sqrt{3}\right ) x^4+14 \left (7+2 i \sqrt{3}\right ) x^3+\left (-103+36 i \sqrt{3}\right ) x^2+\left (94-32 i \sqrt{3}\right ) x+64 i \sqrt{3}-17\right )}{\left (84 i+113 \sqrt{3}\right ) x^4-2 \left (52 \sqrt{3+3 i \sqrt{3}} \sqrt{x^2-x+1}+21 \sqrt{3}-138 i\right ) x^3+\left (-52 \sqrt{3+3 i \sqrt{3}} \sqrt{x^2-x+1}+59 \sqrt{3}-180 i\right ) x^2+\left (-52 \sqrt{3+3 i \sqrt{3}} \sqrt{x^2-x+1}+138 \sqrt{3}+264 i\right ) x+52 \sqrt{3+3 i \sqrt{3}} \sqrt{x^2-x+1}-67 \sqrt{3}+96 i}\right )}{4 \sqrt{3+3 i \sqrt{3}}}-\frac{\left (7 i+\sqrt{3}\right ) \log \left (16 \left (x^2+x+1\right )^2\right )}{8 \sqrt{3-3 i \sqrt{3}}}-\frac{\left (-7 i+\sqrt{3}\right ) \log \left (16 \left (x^2+x+1\right )^2\right )}{8 \sqrt{3+3 i \sqrt{3}}}+\frac{\left (7 i+\sqrt{3}\right ) \log \left (\left (x^2+x+1\right ) \left (\left (11 i+4 \sqrt{3}\right ) x^2-\left (8 i \sqrt{1-i \sqrt{3}} \sqrt{x^2-x+1}+4 \sqrt{3}+17 i\right ) x+10 i \sqrt{1-i \sqrt{3}} \sqrt{x^2-x+1}+4 \sqrt{3}+11 i\right )\right )}{8 \sqrt{3-3 i \sqrt{3}}}+\frac{\left (-7 i+\sqrt{3}\right ) \log \left (\left (x^2+x+1\right ) \left (\left (-11 i+4 \sqrt{3}\right ) x^2+\left (8 i \sqrt{1+i \sqrt{3}} \sqrt{x^2-x+1}-4 \sqrt{3}+17 i\right ) x-10 i \sqrt{1+i \sqrt{3}} \sqrt{x^2-x+1}+4 \sqrt{3}-11 i\right )\right )}{8 \sqrt{3+3 i \sqrt{3}}} \]

Antiderivative was successfully verified.

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

[Out]

((1 + x)*Sqrt[1 - x + x^2])/(1 + x + x^2) + ((7 - I*Sqrt[3])*ArcTan[(3*(-17 - (6
4*I)*Sqrt[3] + (94 + (32*I)*Sqrt[3])*x + (-103 - (36*I)*Sqrt[3])*x^2 + 14*(7 - (
2*I)*Sqrt[3])*x^3 + (-21 - (4*I)*Sqrt[3])*x^4))/(96*I + 67*Sqrt[3] + (84*I - 113
*Sqrt[3])*x^4 - 52*Sqrt[3 - (3*I)*Sqrt[3]]*Sqrt[1 - x + x^2] + 2*x*(132*I - 69*S
qrt[3] + 26*Sqrt[3 - (3*I)*Sqrt[3]]*Sqrt[1 - x + x^2]) + x^2*(-180*I - 59*Sqrt[3
] + 52*Sqrt[3 - (3*I)*Sqrt[3]]*Sqrt[1 - x + x^2]) + 2*x^3*(138*I + 21*Sqrt[3] +
52*Sqrt[3 - (3*I)*Sqrt[3]]*Sqrt[1 - x + x^2]))])/(4*Sqrt[3 - (3*I)*Sqrt[3]]) - (
(I/4)*(-7*I + Sqrt[3])*ArcTan[(3*(-17 + (64*I)*Sqrt[3] + (94 - (32*I)*Sqrt[3])*x
+ (-103 + (36*I)*Sqrt[3])*x^2 + 14*(7 + (2*I)*Sqrt[3])*x^3 + (-21 + (4*I)*Sqrt[
3])*x^4))/(96*I - 67*Sqrt[3] + (84*I + 113*Sqrt[3])*x^4 + 52*Sqrt[3 + (3*I)*Sqrt
[3]]*Sqrt[1 - x + x^2] + x^2*(-180*I + 59*Sqrt[3] - 52*Sqrt[3 + (3*I)*Sqrt[3]]*S
qrt[1 - x + x^2]) + x*(264*I + 138*Sqrt[3] - 52*Sqrt[3 + (3*I)*Sqrt[3]]*Sqrt[1 -
x + x^2]) - 2*x^3*(-138*I + 21*Sqrt[3] + 52*Sqrt[3 + (3*I)*Sqrt[3]]*Sqrt[1 - x
+ x^2]))])/Sqrt[3 + (3*I)*Sqrt[3]] - ((-7*I + Sqrt[3])*Log[16*(1 + x + x^2)^2])/
(8*Sqrt[3 + (3*I)*Sqrt[3]]) - ((7*I + Sqrt[3])*Log[16*(1 + x + x^2)^2])/(8*Sqrt[
3 - (3*I)*Sqrt[3]]) + ((7*I + Sqrt[3])*Log[(1 + x + x^2)*(11*I + 4*Sqrt[3] + (11
*I + 4*Sqrt[3])*x^2 + (10*I)*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 - x + x^2] - x*(17*I + 4
*Sqrt[3] + (8*I)*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 - x + x^2]))])/(8*Sqrt[3 - (3*I)*Sqr
t[3]]) + ((-7*I + Sqrt[3])*Log[(1 + x + x^2)*(-11*I + 4*Sqrt[3] + (-11*I + 4*Sqr
t[3])*x^2 - (10*I)*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 - x + x^2] + x*(17*I - 4*Sqrt[3] +
(8*I)*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 - x + x^2]))])/(8*Sqrt[3 + (3*I)*Sqrt[3]])

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Maple [B]  time = 0.057, size = 455, normalized size = 5.3 \[ -{\frac{1}{6} \left ( 9\,{\frac{\sqrt{2} \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}\arctan \left ( 2\,{\frac{ \left ( 1+x \right ) \sqrt{2}}{1-x}{\frac{1}{\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}}}} \right ) \sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}}-6\,{\frac{\sqrt{6} \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}{\it Artanh} \left ( 1/4\,\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}\sqrt{6} \right ) \sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}}+3\,\sqrt{2}\arctan \left ( 2\,{\frac{ \left ( 1+x \right ) \sqrt{2}}{1-x}{\frac{1}{\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}}}} \right ) \sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}-2\,\sqrt{6}{\it Artanh} \left ( 1/4\,\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}\sqrt{6} \right ) \sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}-12\,{\frac{ \left ( 1+x \right ) ^{3}}{ \left ( 1-x \right ) ^{3}}}-36\,{\frac{1+x}{1-x}} \right ){\frac{1}{\sqrt{{1 \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} \left ( 3\,{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+1 \right ) ^{-1}}+{\frac{1}{2}\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3} \left ( 3\,\sqrt{2}\arctan \left ( 2\,{\frac{ \left ( 1+x \right ) \sqrt{2}}{1-x}{\frac{1}{\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}}}} \right ) -\sqrt{6}{\it Artanh} \left ({\frac{\sqrt{6}}{4}\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+3}} \right ) \right ){\frac{1}{\sqrt{{1 \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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2-x+1)/(x^2+x+1)^2/(x^2-x+1)^(1/2),x)

[Out]

-1/6*(9*2^(1/2)*arctan(2*2^(1/2)/((1+x)^2/(1-x)^2+3)^(1/2)*(1+x)/(1-x))*((1+x)^2
/(1-x)^2+3)^(1/2)*(1+x)^2/(1-x)^2-6*6^(1/2)*arctanh(1/4*((1+x)^2/(1-x)^2+3)^(1/2
)*6^(1/2))*((1+x)^2/(1-x)^2+3)^(1/2)*(1+x)^2/(1-x)^2+3*2^(1/2)*arctan(2*2^(1/2)/
((1+x)^2/(1-x)^2+3)^(1/2)*(1+x)/(1-x))*((1+x)^2/(1-x)^2+3)^(1/2)-2*6^(1/2)*arcta
nh(1/4*((1+x)^2/(1-x)^2+3)^(1/2)*6^(1/2))*((1+x)^2/(1-x)^2+3)^(1/2)-12*(1+x)^3/(
1-x)^3-36*(1+x)/(1-x))/(((1+x)^2/(1-x)^2+3)/(1+(1+x)/(1-x))^2)^(1/2)/(1+(1+x)/(1
-x))/(3*(1+x)^2/(1-x)^2+1)+1/2*((1+x)^2/(1-x)^2+3)^(1/2)*(3*2^(1/2)*arctan(2*2^(
1/2)/((1+x)^2/(1-x)^2+3)^(1/2)*(1+x)/(1-x))-6^(1/2)*arctanh(1/4*((1+x)^2/(1-x)^2
+3)^(1/2)*6^(1/2)))/(1+(1+x)/(1-x))/(((1+x)^2/(1-x)^2+3)/(1+(1+x)/(1-x))^2)^(1/2
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 - x + 1)/((x^2 + x + 1)^2*sqrt(x^2 - x + 1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.25221, size = 720, normalized size = 8.37 \[ \frac{2 \, \sqrt{6} \sqrt{3}{\left (4 \, x^{2} + 7 \, x - 9\right )} \sqrt{x^{2} - x + 1} - 2 \, \sqrt{6} \sqrt{3}{\left (4 \, x^{3} + 5 \, x^{2} - 11 \, x + 9\right )} - 24 \,{\left (8 \, x^{4} + 5 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2} + x - 1\right )} \sqrt{x^{2} - x + 1} - 3 \, x + 5\right )} \arctan \left (-\frac{\sqrt{6} \sqrt{3} + 2 \, \sqrt{3}}{\sqrt{6}{\left (2 \, x + 1\right )} - 2 \, \sqrt{6} \sqrt{2 \, x^{2} - \sqrt{x^{2} - x + 1}{\left (2 \, x + \sqrt{6} + 1\right )} + \sqrt{6}{\left (x + 1\right )} + 4} - 2 \, \sqrt{6} \sqrt{x^{2} - x + 1} + 6}\right ) + 24 \,{\left (8 \, x^{4} + 5 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2} + x - 1\right )} \sqrt{x^{2} - x + 1} - 3 \, x + 5\right )} \arctan \left (-\frac{\sqrt{6} \sqrt{3} - 2 \, \sqrt{3}}{\sqrt{6}{\left (2 \, x + 1\right )} - 2 \, \sqrt{6} \sqrt{2 \, x^{2} - \sqrt{x^{2} - x + 1}{\left (2 \, x - \sqrt{6} + 1\right )} - \sqrt{6}{\left (x + 1\right )} + 4} - 2 \, \sqrt{6} \sqrt{x^{2} - x + 1} - 6}\right ) +{\left (4 \, \sqrt{3}{\left (2 \, x^{3} + x^{2} + x - 1\right )} \sqrt{x^{2} - x + 1} - \sqrt{3}{\left (8 \, x^{4} + 5 \, x^{2} - 3 \, x + 5\right )}\right )} \log \left (4056 \, x^{2} - 2028 \, \sqrt{x^{2} - x + 1}{\left (2 \, x + \sqrt{6} + 1\right )} + 2028 \, \sqrt{6}{\left (x + 1\right )} + 8112\right ) -{\left (4 \, \sqrt{3}{\left (2 \, x^{3} + x^{2} + x - 1\right )} \sqrt{x^{2} - x + 1} - \sqrt{3}{\left (8 \, x^{4} + 5 \, x^{2} - 3 \, x + 5\right )}\right )} \log \left (4056 \, x^{2} - 2028 \, \sqrt{x^{2} - x + 1}{\left (2 \, x - \sqrt{6} + 1\right )} - 2028 \, \sqrt{6}{\left (x + 1\right )} + 8112\right )}{2 \,{\left (4 \, \sqrt{6} \sqrt{3}{\left (2 \, x^{3} + x^{2} + x - 1\right )} \sqrt{x^{2} - x + 1} - \sqrt{6} \sqrt{3}{\left (8 \, x^{4} + 5 \, x^{2} - 3 \, x + 5\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 - x + 1)/((x^2 + x + 1)^2*sqrt(x^2 - x + 1)),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(6)*sqrt(3)*(4*x^2 + 7*x - 9)*sqrt(x^2 - x + 1) - 2*sqrt(6)*sqrt(3)*(
4*x^3 + 5*x^2 - 11*x + 9) - 24*(8*x^4 + 5*x^2 - 4*(2*x^3 + x^2 + x - 1)*sqrt(x^2
- x + 1) - 3*x + 5)*arctan(-(sqrt(6)*sqrt(3) + 2*sqrt(3))/(sqrt(6)*(2*x + 1) -
2*sqrt(6)*sqrt(2*x^2 - sqrt(x^2 - x + 1)*(2*x + sqrt(6) + 1) + sqrt(6)*(x + 1) +
4) - 2*sqrt(6)*sqrt(x^2 - x + 1) + 6)) + 24*(8*x^4 + 5*x^2 - 4*(2*x^3 + x^2 + x
- 1)*sqrt(x^2 - x + 1) - 3*x + 5)*arctan(-(sqrt(6)*sqrt(3) - 2*sqrt(3))/(sqrt(6
)*(2*x + 1) - 2*sqrt(6)*sqrt(2*x^2 - sqrt(x^2 - x + 1)*(2*x - sqrt(6) + 1) - sqr
t(6)*(x + 1) + 4) - 2*sqrt(6)*sqrt(x^2 - x + 1) - 6)) + (4*sqrt(3)*(2*x^3 + x^2
+ x - 1)*sqrt(x^2 - x + 1) - sqrt(3)*(8*x^4 + 5*x^2 - 3*x + 5))*log(4056*x^2 - 2
028*sqrt(x^2 - x + 1)*(2*x + sqrt(6) + 1) + 2028*sqrt(6)*(x + 1) + 8112) - (4*sq
rt(3)*(2*x^3 + x^2 + x - 1)*sqrt(x^2 - x + 1) - sqrt(3)*(8*x^4 + 5*x^2 - 3*x + 5
))*log(4056*x^2 - 2028*sqrt(x^2 - x + 1)*(2*x - sqrt(6) + 1) - 2028*sqrt(6)*(x +
1) + 8112))/(4*sqrt(6)*sqrt(3)*(2*x^3 + x^2 + x - 1)*sqrt(x^2 - x + 1) - sqrt(6
)*sqrt(3)*(8*x^4 + 5*x^2 - 3*x + 5))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2-x+1)/(x**2+x+1)**2/(x**2-x+1)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, x^{2} - x + 1}{{\left (x^{2} + x + 1\right )}^{2} \sqrt{x^{2} - x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 - x + 1)/((x^2 + x + 1)^2*sqrt(x^2 - x + 1)),x, algorithm="giac")

[Out]

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