\(\int \frac {1+x}{(-2+2 x+x^2) \sqrt {1+x^3}} \, dx\) [1458]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{6} \sqrt {-3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6860, 2160, 224, 2165, 212, 209} \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{2 \sqrt {3 \left (2 \sqrt {3}-3\right )}} \]

[In]

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

[Out]

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

Rule 209

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

Rule 212

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

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 2160

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-6*a*(d^3/(c*(b*c^3 - 28*a*d^3))), In
t[1/Sqrt[a + b*x^3], x], x] + Dist[1/(c*(b*c^3 - 28*a*d^3)), Int[Simp[c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x, x]/((c
+ d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]

Rule 2165

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[(1 + k)*(e/d), Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*(x/c))/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {1}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx \\ & = \int \frac {1}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\int \frac {1}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx \\ & = -\frac {\int \frac {96 \left (1-\sqrt {3}\right )+96 x}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{192 \sqrt {3}}+\frac {\int \frac {96 \left (1+\sqrt {3}\right )+96 x}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{192 \sqrt {3}} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{2-2 \sqrt {3}}}{\sqrt {1+x^3}}\right )}{2 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{2+2 \sqrt {3}}}{\sqrt {1+x^3}}\right )}{2 \sqrt {3}} \\ & = \frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )}{2 \sqrt {3 \left (-3+2 \sqrt {3}\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{6} \sqrt {-3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]

[In]

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

[Out]

(Sqrt[-3 + 2*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6 - (Sqrt[3 + 2*Sqrt[3]]*ArcT
anh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.61 (sec) , antiderivative size = 694, normalized size of antiderivative = 6.74

method	result	size
default	\(-\frac {\sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {\sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}-\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\)	\(694\)
elliptic	\(-\frac {\sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {\sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}-\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\)	\(694\)

[In]

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

[Out]

-1/2*(1/(3/2-1/2*I*3^(1/2))+1/(3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/2/(-3/2-1/2*I*3^(1/2))-
1/2*I/(-3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/2/(-3/2+1/2*I*3^(1/2))+1/2*I/(-3/2+1/2*I
*3^(1/2))*3^(1/2))^(1/2)/(x^3+1)^(1/2)*3^(1/2)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),-1/3*(-3/2+1/2*I*3
^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+1/2*I*(1/(3/2-1/2*I*3^(1/2))+1/(3/2-1/2*I*3
^(1/2))*x)^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/2/(-3/2-1/2*I*3^(1/2))-1/2*I/(-3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*
(1/(-3/2+1/2*I*3^(1/2))*x-1/2/(-3/2+1/2*I*3^(1/2))+1/2*I/(-3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3+1)^(1/2)*Ell
ipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),-1/3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*
3^(1/2)))^(1/2))+1/2*(1/(3/2-1/2*I*3^(1/2))+1/(3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/2/(-3/2
-1/2*I*3^(1/2))-1/2*I/(-3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/2/(-3/2+1/2*I*3^(1/2))+1
/2*I/(-3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3+1)^(1/2)*3^(1/2)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),1/
3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))-1/2*I*(1/(3/2-1/2*I*3^(1/2))
+1/(3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/2/(-3/2-1/2*I*3^(1/2))-1/2*I/(-3/2-1/2*I*3^(1/2))*
3^(1/2))^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/2/(-3/2+1/2*I*3^(1/2))+1/2*I/(-3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x
^3+1)^(1/2)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),1/3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2)
)/(-3/2-1/2*I*3^(1/2)))^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (79) = 158\).

Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.72 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{24} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} - \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} - \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} + \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {-2 \, \sqrt {3} + 3} - 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} + \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {-2 \, \sqrt {3} + 3} - 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \]

[In]

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

[Out]

-1/24*sqrt(2*sqrt(3) + 3)*log((x^4 - 2*x^3 + 6*x^2 + 2*sqrt(x^3 + 1)*(2*x^2 - sqrt(3)*(x^2 - 2*x) - 2*x + 2)*s
qrt(2*sqrt(3) + 3) + 4*sqrt(3)*(x^3 + 1) + 4*x + 4)/(x^4 + 4*x^3 - 8*x + 4)) + 1/24*sqrt(2*sqrt(3) + 3)*log((x
^4 - 2*x^3 + 6*x^2 - 2*sqrt(x^3 + 1)*(2*x^2 - sqrt(3)*(x^2 - 2*x) - 2*x + 2)*sqrt(2*sqrt(3) + 3) + 4*sqrt(3)*(
x^3 + 1) + 4*x + 4)/(x^4 + 4*x^3 - 8*x + 4)) - 1/24*sqrt(-2*sqrt(3) + 3)*log((x^4 - 2*x^3 + 6*x^2 + 2*sqrt(x^3
+ 1)*(2*x^2 + sqrt(3)*(x^2 - 2*x) - 2*x + 2)*sqrt(-2*sqrt(3) + 3) - 4*sqrt(3)*(x^3 + 1) + 4*x + 4)/(x^4 + 4*x
^3 - 8*x + 4)) + 1/24*sqrt(-2*sqrt(3) + 3)*log((x^4 - 2*x^3 + 6*x^2 - 2*sqrt(x^3 + 1)*(2*x^2 + sqrt(3)*(x^2 -
2*x) - 2*x + 2)*sqrt(-2*sqrt(3) + 3) - 4*sqrt(3)*(x^3 + 1) + 4*x + 4)/(x^4 + 4*x^3 - 8*x + 4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x + 1}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x + 1}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.14 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\left (\Pi \left (\sqrt {3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right );\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (-\sqrt {3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right );\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

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

[Out]

-((ellipticPi(3^(1/2)*((3^(1/2)*1i)/6 + 1/2), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 +
3/2)/((3^(1/2)*1i)/2 - 3/2)) - ellipticPi(-3^(1/2)*((3^(1/2)*1i)/6 + 1/2), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/
2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2
))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(3^(1/2) + 1i)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3
/2))^(1/2))/(2*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)
*1i)/2 + 1/2))^(1/2))