\(\int \frac {1+x}{(1-\sqrt {3}+x) \sqrt {1+x^3}} \, dx\) [172]

Optimal result

Integrand size = 25, antiderivative size = 145 \[ \int \frac {1+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )}{\sqrt {-3+2 \sqrt {3}}}+\frac {\sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \] Output:

-arctanh((-3+2*3^(1/2))^(1/2)*(1+x)/(x^3+1)^(1/2))/(-3+2*3^(1/2))^(1/2)+1/ 
3*(1/2*6^(1/2)+1/2*2^(1/2))*(1+x)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*Ellipt 
icF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*3^(3/4)/((1+x)/(1+x+3^(1/2) 
)^2)^(1/2)/(x^3+1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.40 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.84 \[ \int \frac {1+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \sqrt {6} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (\sqrt {-i+\sqrt {3}+2 i x} \left ((1+2 i)-i \sqrt {3}+\left ((-2-i)+\sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 i \sqrt {i+\sqrt {3}-2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 i \sqrt {3}}{-3+(2+i) \sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (-3+(2+i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1+x^3}} \] Input:

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

Output:

(-2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x 
]*((1 + 2*I) - I*Sqrt[3] + ((-2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I 
 + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + ( 
2*I)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[ 
3])/(-3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^ 
(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((-3 + (2 + I)*Sqrt[3])*Sqrt[I + S 
qrt[3] - (2*I)*x]*Sqrt[1 + x^3])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2566, 27, 759, 2565, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2566

\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {x^3+1}}dx+\frac {1}{12} \int \frac {6 \left (x+\sqrt {3}+1\right )}{\left (x-\sqrt {3}+1\right ) \sqrt {x^3+1}}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 759

\(\displaystyle \frac {1}{2} \int \frac {x+\sqrt {3}+1}{\left (x-\sqrt {3}+1\right ) \sqrt {x^3+1}}dx+\frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\)

\(\Big \downarrow \) 2565

\(\displaystyle \frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\int \frac {1}{\frac {\left (3-2 \sqrt {3}\right ) (x+1)^2}{x^3+1}+1}d\frac {x+1}{\sqrt {x^3+1}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{\sqrt {2 \sqrt {3}-3}}\)

Input:

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

Output:

-(ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]/Sqrt[-3 + 2*Sqrt[3 
]]) + (Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*E 
llipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^ 
(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 759

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]*Sqrt[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 2565

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d)   S 
ubst[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 2566

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> Simp[-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d 
^3))   Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/(c*d*(b*c^3 - 28*a* 
d^3))   Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), 
 x], 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] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3) 
, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (119 ) = 238\).

Time = 1.65 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\)	\(245\)
elliptic	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\)	\(245\)

Input:

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

Output:

2*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1 
/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2 
)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2 
+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))-2*(3/2-1/2*I*3^(1/2))*((x+1)/ 
(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1 
/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*Ellip 
ticPi(((x+1)/(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 [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.43 \[ \int \frac {1+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {1}{4} \, \sqrt {\frac {2}{3} \, \sqrt {3} + 1} \log \left (\frac {x^{8} - 16 \, x^{7} + 112 \, x^{6} - 16 \, x^{5} + 112 \, x^{4} + 224 \, x^{3} + 64 \, x^{2} + 4 \, {\left (3 \, x^{6} - 36 \, x^{5} + 54 \, x^{4} - 48 \, x^{3} - 36 \, x^{2} - 2 \, \sqrt {3} {\left (x^{6} - 9 \, x^{5} + 21 \, x^{4} - 4 \, x^{3} + 12 \, x + 4\right )} - 24\right )} \sqrt {x^{3} + 1} \sqrt {\frac {2}{3} \, \sqrt {3} + 1} + 16 \, \sqrt {3} {\left (x^{7} - 2 \, x^{6} + 6 \, x^{5} + 5 \, x^{4} + 2 \, x^{3} + 6 \, x^{2} + 4 \, x + 4\right )} + 128 \, x + 112}{x^{8} + 8 \, x^{7} + 16 \, x^{6} - 16 \, x^{5} - 56 \, x^{4} + 32 \, x^{3} + 64 \, x^{2} - 64 \, x + 16}\right ) + {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:

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

Output:

1/4*sqrt(2/3*sqrt(3) + 1)*log((x^8 - 16*x^7 + 112*x^6 - 16*x^5 + 112*x^4 + 
 224*x^3 + 64*x^2 + 4*(3*x^6 - 36*x^5 + 54*x^4 - 48*x^3 - 36*x^2 - 2*sqrt( 
3)*(x^6 - 9*x^5 + 21*x^4 - 4*x^3 + 12*x + 4) - 24)*sqrt(x^3 + 1)*sqrt(2/3* 
sqrt(3) + 1) + 16*sqrt(3)*(x^7 - 2*x^6 + 6*x^5 + 5*x^4 + 2*x^3 + 6*x^2 + 4 
*x + 4) + 128*x + 112)/(x^8 + 8*x^7 + 16*x^6 - 16*x^5 - 56*x^4 + 32*x^3 + 
64*x^2 - 64*x + 16)) + weierstrassPInverse(0, -4, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}}{x^{4}+x^{3}-3 x^{2}+4 x -2}d x \right )+\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{4}+x^{3}-3 x^{2}+4 x -2}d x \right )+3 \left (\int \frac {\sqrt {x^{3}+1}}{x^{4}+x^{3}-3 x^{2}+4 x -2}d x \right )}{\sqrt {3}} \] Input:

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

Output:

(sqrt(3)*int(sqrt(x**3 + 1)/(x**4 + x**3 - 3*x**2 + 4*x - 2),x) + sqrt(3)* 
int((sqrt(x**3 + 1)*x)/(x**4 + x**3 - 3*x**2 + 4*x - 2),x) + 3*int(sqrt(x* 
*3 + 1)/(x**4 + x**3 - 3*x**2 + 4*x - 2),x))/sqrt(3)