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

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]

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

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Rubi [A]  time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1060, 1035, 1029, 206, 204} \[ \frac {\sqrt {x^2-x+1} (x+1)}{x^2+x+1}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {x^2-x+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} (1-x)}{\sqrt {x^2-x+1}}\right )}{\sqrt {6}} \]

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

Rule 204

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

Rule 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rule 1035

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

Rule 1060

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {1-x+3 x^2}{\sqrt {1-x+x^2} \left (1+x+x^2\right )^2} \, dx &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+\frac {1}{12} \int \frac {18-6 x}{\sqrt {1-x+x^2} \left (1+x+x^2\right )} \, dx\\ &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+\frac {1}{48} \int \frac {24+24 x}{\sqrt {1-x+x^2} \left (1+x+x^2\right )} \, dx-\frac {1}{48} \int \frac {-48+48 x}{\sqrt {1-x+x^2} \left (1+x+x^2\right )} \, dx\\ &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+24 \operatorname {Subst}\left (\int \frac {1}{1728-2 x^2} \, dx,x,\frac {-24+24 x}{\sqrt {1-x+x^2}}\right )+288 \operatorname {Subst}\left (\int \frac {1}{-20736-2 x^2} \, dx,x,\frac {-144-144 x}{\sqrt {1-x+x^2}}\right )\\ &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} (1+x)}{\sqrt {1-x+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} (1-x)}{\sqrt {1-x+x^2}}\right )}{\sqrt {6}}\\ \end {align*}

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Mathematica [C]  time = 2.52, 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 - (64*I)*Sqrt[3] + (94 + (32*I)*Sq
rt[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*S
qrt[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*Sqrt[3] + 2
6*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)*Sqr
t[3]]) - ((I/4)*(-7*I + Sqrt[3])*ArcTan[(3*(-17 + (64*I)*Sqrt[3] + (94 - (32*I)*Sqrt[3])*x + (-103 + (36*I)*Sq
rt[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 + 2
1*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[1
6*(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)*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)*S
qrt[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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fricas [B]  time = 1.01, size = 358, normalized size = 4.16 \[ -\frac {8 \, \sqrt {6} \sqrt {3} {\left (x^{2} + x + 1\right )} \arctan \left (\frac {2}{3} \, \sqrt {6} \sqrt {3} {\left (x - 1\right )} + \frac {2}{3} \, \sqrt {2 \, x^{2} - \sqrt {x^{2} - x + 1} {\left (2 \, x - \sqrt {6} + 1\right )} - \sqrt {6} {\left (x + 1\right )} + 4} {\left (\sqrt {6} \sqrt {3} + 3 \, \sqrt {3}\right )} - \frac {2}{3} \, \sqrt {x^{2} - x + 1} {\left (\sqrt {6} \sqrt {3} + 3 \, \sqrt {3}\right )} + \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 8 \, \sqrt {6} \sqrt {3} {\left (x^{2} + x + 1\right )} \arctan \left (\frac {2}{3} \, \sqrt {6} \sqrt {3} {\left (x - 1\right )} + \frac {2}{3} \, \sqrt {2 \, x^{2} - \sqrt {x^{2} - x + 1} {\left (2 \, x + \sqrt {6} + 1\right )} + \sqrt {6} {\left (x + 1\right )} + 4} {\left (\sqrt {6} \sqrt {3} - 3 \, \sqrt {3}\right )} - \frac {2}{3} \, \sqrt {x^{2} - x + 1} {\left (\sqrt {6} \sqrt {3} - 3 \, \sqrt {3}\right )} - \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \sqrt {6} {\left (x^{2} + x + 1\right )} \log \left (12168 \, x^{2} - 6084 \, \sqrt {x^{2} - x + 1} {\left (2 \, x + \sqrt {6} + 1\right )} + 6084 \, \sqrt {6} {\left (x + 1\right )} + 24336\right ) + \sqrt {6} {\left (x^{2} + x + 1\right )} \log \left (12168 \, x^{2} - 6084 \, \sqrt {x^{2} - x + 1} {\left (2 \, x - \sqrt {6} + 1\right )} - 6084 \, \sqrt {6} {\left (x + 1\right )} + 24336\right ) - 12 \, x^{2} - 12 \, \sqrt {x^{2} - x + 1} {\left (x + 1\right )} - 12 \, x - 12}{12 \, {\left (x^{2} + x + 1\right )}} \]

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, algorithm="fricas")

[Out]

-1/12*(8*sqrt(6)*sqrt(3)*(x^2 + x + 1)*arctan(2/3*sqrt(6)*sqrt(3)*(x - 1) + 2/3*sqrt(2*x^2 - sqrt(x^2 - x + 1)
*(2*x - sqrt(6) + 1) - sqrt(6)*(x + 1) + 4)*(sqrt(6)*sqrt(3) + 3*sqrt(3)) - 2/3*sqrt(x^2 - x + 1)*(sqrt(6)*sqr
t(3) + 3*sqrt(3)) + sqrt(3)*(2*x - 1)) + 8*sqrt(6)*sqrt(3)*(x^2 + x + 1)*arctan(2/3*sqrt(6)*sqrt(3)*(x - 1) +
2/3*sqrt(2*x^2 - sqrt(x^2 - x + 1)*(2*x + sqrt(6) + 1) + sqrt(6)*(x + 1) + 4)*(sqrt(6)*sqrt(3) - 3*sqrt(3)) -
2/3*sqrt(x^2 - x + 1)*(sqrt(6)*sqrt(3) - 3*sqrt(3)) - sqrt(3)*(2*x - 1)) - sqrt(6)*(x^2 + x + 1)*log(12168*x^2
- 6084*sqrt(x^2 - x + 1)*(2*x + sqrt(6) + 1) + 6084*sqrt(6)*(x + 1) + 24336) + sqrt(6)*(x^2 + x + 1)*log(1216
8*x^2 - 6084*sqrt(x^2 - x + 1)*(2*x - sqrt(6) + 1) - 6084*sqrt(6)*(x + 1) + 24336) - 12*x^2 - 12*sqrt(x^2 - x
+ 1)*(x + 1) - 12*x - 12)/(x^2 + x + 1)

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giac [B]  time = 1.06, size = 304, normalized size = 3.53 \[ -\frac {1}{3} \, \sqrt {6} \sqrt {3} \arctan \left (-\frac {2 \, x + \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1}{\sqrt {3} + \sqrt {2}}\right ) + \frac {1}{3} \, \sqrt {6} \sqrt {3} \arctan \left (-\frac {2 \, x - \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1}{\sqrt {3} - \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \log \left (4 \, {\left (\sqrt {6} \sqrt {3} + 3 \, \sqrt {3}\right )}^{2} + 36 \, {\left (2 \, x + \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1\right )}^{2}\right ) - \frac {1}{12} \, \sqrt {6} \log \left (4 \, {\left (\sqrt {6} \sqrt {3} - 3 \, \sqrt {3}\right )}^{2} + 36 \, {\left (2 \, x - \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1\right )}^{2}\right ) + \frac {{\left (x - \sqrt {x^{2} - x + 1}\right )}^{3} + 4 \, {\left (x - \sqrt {x^{2} - x + 1}\right )}^{2} - 10 \, x + 10 \, \sqrt {x^{2} - x + 1} + 5}{{\left (x - \sqrt {x^{2} - x + 1}\right )}^{4} + 2 \, {\left (x - \sqrt {x^{2} - x + 1}\right )}^{3} + {\left (x - \sqrt {x^{2} - x + 1}\right )}^{2} - 6 \, x + 6 \, \sqrt {x^{2} - x + 1} + 3} \]

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, algorithm="giac")

[Out]

-1/3*sqrt(6)*sqrt(3)*arctan(-(2*x + sqrt(6) - 2*sqrt(x^2 - x + 1) + 1)/(sqrt(3) + sqrt(2))) + 1/3*sqrt(6)*sqrt
(3)*arctan(-(2*x - sqrt(6) - 2*sqrt(x^2 - x + 1) + 1)/(sqrt(3) - sqrt(2))) + 1/12*sqrt(6)*log(4*(sqrt(6)*sqrt(
3) + 3*sqrt(3))^2 + 36*(2*x + sqrt(6) - 2*sqrt(x^2 - x + 1) + 1)^2) - 1/12*sqrt(6)*log(4*(sqrt(6)*sqrt(3) - 3*
sqrt(3))^2 + 36*(2*x - sqrt(6) - 2*sqrt(x^2 - x + 1) + 1)^2) + ((x - sqrt(x^2 - x + 1))^3 + 4*(x - sqrt(x^2 -
x + 1))^2 - 10*x + 10*sqrt(x^2 - x + 1) + 5)/((x - sqrt(x^2 - x + 1))^4 + 2*(x - sqrt(x^2 - x + 1))^3 + (x - s
qrt(x^2 - x + 1))^2 - 6*x + 6*sqrt(x^2 - x + 1) + 3)

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maple [B]  time = 0.04, size = 455, normalized size = 5.29 \[ -\frac {-\frac {6 \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}\, \left (x +1\right )^{2} \arctanh \left (\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}}{4}\right )}{\left (-x +1\right )^{2}}-2 \sqrt {6}\, \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \arctanh \left (\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}}{4}\right )+\frac {9 \sqrt {2}\, \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (x +1\right )^{2} \arctan \left (\frac {2 \sqrt {2}\, \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-x +1\right )}\right )}{\left (-x +1\right )^{2}}+3 \sqrt {2}\, \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \arctan \left (\frac {2 \sqrt {2}\, \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-x +1\right )}\right )-\frac {12 \left (x +1\right )^{3}}{\left (-x +1\right )^{3}}-\frac {36 \left (x +1\right )}{-x +1}}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}{\left (\frac {x +1}{-x +1}+1\right )^{2}}}\, \left (\frac {x +1}{-x +1}+1\right ) \left (\frac {3 \left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+1\right )}+\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-\sqrt {6}\, \arctanh \left (\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}}{4}\right )+3 \sqrt {2}\, \arctan \left (\frac {2 \sqrt {2}\, \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-x +1\right )}\right )\right )}{2 \left (\frac {x +1}{-x +1}+1\right ) \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}{\left (\frac {x +1}{-x +1}+1\right )^{2}}}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 \, x^{2} - x + 1}{{\left (x^{2} + x + 1\right )}^{2} \sqrt {x^{2} - x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+1)/(x^2+x+1)^2/(x^2-x+1)^(1/2),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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {3\,x^2-x+1}{\sqrt {x^2-x+1}\,{\left (x^2+x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 x^{2} - x + 1}{\sqrt {x^{2} - x + 1} \left (x^{2} + x + 1\right )^{2}}\, dx \]

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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