Integrand size = 30, antiderivative size = 42 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )}{\sqrt {-3+2 \sqrt {3}}} \] Output:
-2*arctanh((-3+2*3^(1/2))^(1/2)*(1+x)/(x^3+1)^(1/2))/(-3+2*3^(1/2))^(1/2)
Time = 1.89 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-2 \sqrt {1+\frac {2}{\sqrt {3}}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \] Input:
Integrate[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]
Output:
-2*Sqrt[1 + 2/Sqrt[3]]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)]
Time = 0.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
| \(\displaystyle \int \frac {x+\sqrt {3}+1}{\left (x-\sqrt {3}+1\right ) \sqrt {x^3+1}} \, dx\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle -2 \int \frac {1}{\frac {\left (3-2 \sqrt {3}\right ) (x+1)^2}{x^3+1}+1}d\frac {x+1}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{\sqrt {2 \sqrt {3}-3}}\) |
Input:
Int[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]
Output:
(-2*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/Sqrt[-3 + 2*Sqr t[3]]
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])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.84 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.10
| method | result | size |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) \sqrt {3}\, x^{2}-4 \sqrt {3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) x +48 \sqrt {x^{3}+1}\, \sqrt {3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) \sqrt {3}+72 \sqrt {x^{3}+1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right )}{\left (\sqrt {3}\, x +x -2\right )^{2}}\right )}{6}\) | \(130\) |
| default | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {4 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(245\) |
| elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {4 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(245\) |
Input:
int((1+3^(1/2)+x)/(1-3^(1/2)+x)/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/6*RootOf(_Z^2-24*3^(1/2)-36)*ln(-(6*RootOf(_Z^2-24*3^(1/2)-36)*x^2+4*Ro otOf(_Z^2-24*3^(1/2)-36)*3^(1/2)*x^2-4*3^(1/2)*RootOf(_Z^2-24*3^(1/2)-36)* x+48*(x^3+1)^(1/2)*3^(1/2)+4*RootOf(_Z^2-24*3^(1/2)-36)*3^(1/2)+72*(x^3+1) ^(1/2)+12*RootOf(_Z^2-24*3^(1/2)-36))/(3^(1/2)*x+x-2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (32) = 64\).
Time = 0.18 (sec) , antiderivative size = 202, normalized size of antiderivative = 4.81 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {1}{2} \, \sqrt {\frac {2}{3} \, \sqrt {3} + 1} \log \left (\frac {x^{8} - 16 \, x^{7} + 112 \, x^{6} - 16 \, x^{5} + 112 \, x^{4} + 224 \, x^{3} + 64 \, x^{2} + 4 \, {\left (3 \, x^{6} - 36 \, x^{5} + 54 \, x^{4} - 48 \, x^{3} - 36 \, x^{2} - 2 \, \sqrt {3} {\left (x^{6} - 9 \, x^{5} + 21 \, x^{4} - 4 \, x^{3} + 12 \, x + 4\right )} - 24\right )} \sqrt {x^{3} + 1} \sqrt {\frac {2}{3} \, \sqrt {3} + 1} + 16 \, \sqrt {3} {\left (x^{7} - 2 \, x^{6} + 6 \, x^{5} + 5 \, x^{4} + 2 \, x^{3} + 6 \, x^{2} + 4 \, x + 4\right )} + 128 \, x + 112}{x^{8} + 8 \, x^{7} + 16 \, x^{6} - 16 \, x^{5} - 56 \, x^{4} + 32 \, x^{3} + 64 \, x^{2} - 64 \, x + 16}\right ) \] Input:
integrate((1+3^(1/2)+x)/(1-3^(1/2)+x)/(x^3+1)^(1/2),x, algorithm="fricas")
Output:
1/2*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))
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\int \frac {x + 1 + \sqrt {3}}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - \sqrt {3} + 1\right )}\, dx \] Input:
integrate((1+3**(1/2)+x)/(1-3**(1/2)+x)/(x**3+1)**(1/2),x)
Output:
Integral((x + 1 + sqrt(3))/(sqrt((x + 1)*(x**2 - x + 1))*(x - sqrt(3) + 1) ), x)
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {x^{3} + 1} {\left (x - \sqrt {3} + 1\right )}} \,d x } \] Input:
integrate((1+3^(1/2)+x)/(1-3^(1/2)+x)/(x^3+1)^(1/2),x, algorithm="maxima")
Output:
integrate((x + sqrt(3) + 1)/(sqrt(x^3 + 1)*(x - sqrt(3) + 1)), x)
Exception generated. \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1+3^(1/2)+x)/(1-3^(1/2)+x)/(x^3+1)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[-1,-1]:[1,0,-3]%%},[2]%%%} / %%%{%%{[-2,4]:[1,0,-3]%%} ,[2]%%%}
Timed out. \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Hanged} \] Input:
int((x + 3^(1/2) + 1)/((x^3 + 1)^(1/2)*(x - 3^(1/2) + 1)),x)
Output:
\text{Hanged}
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {4 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )+\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x^{2}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )+2 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )+6 \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )+6 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )}{\sqrt {3}} \] Input:
int((1+3^(1/2)+x)/(1-3^(1/2)+x)/(x^3+1)^(1/2),x)
Output:
(4*sqrt(3)*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x) + sqrt(3)*int((sqrt(x**3 + 1)*x**2)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x) + 2*sqrt(3)*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x) + 6*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x) + 6*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x))/sqrt(3)