Integrand size = 30, antiderivative size = 38 \[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=-\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{\sqrt {3}} \] Output:
-2/3*2^(2/3)*arctanh(3^(1/2)*(1-2^(1/3)*x)/(x^3-1)^(1/2))*3^(1/2)
Time = 2.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {-1+x^3}}{\sqrt {3} \left (-1+\sqrt [3]{2} x\right )}\right )}{\sqrt {3}} \] Input:
Integrate[(2^(2/3) + 2*x)/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]
Output:
(2*2^(2/3)*ArcTanh[Sqrt[-1 + x^3]/(Sqrt[3]*(-1 + 2^(1/3)*x))])/Sqrt[3]
Time = 0.41 (sec) , antiderivative size = 38, 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 = {2562, 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.
\(\displaystyle \int \frac {2 x+2^{2/3}}{\left (2^{2/3}-x\right ) \sqrt {x^3-1}} \, dx\) |
\(\Big \downarrow \) 2562 |
\(\displaystyle -2 2^{2/3} \int \frac {1}{1-\frac {3 \left (1-\sqrt [3]{2} x\right )^2}{x^3-1}}d\frac {1-\sqrt [3]{2} x}{\sqrt {x^3-1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {x^3-1}}\right )}{\sqrt {3}}\) |
Input:
Int[(2^(2/3) + 2*x)/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]
Output:
(-2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/Sqrt[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] :> Simp[2*(e/d) Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*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*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.49 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.82
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) \ln \left (\frac {12 \sqrt {x^{3}-1}\, x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) x^{3}-6 \sqrt {x^{3}-1}\, 2^{\frac {2}{3}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right )}{\left (2^{\frac {1}{3}} x -2\right )^{3}}\right )}{3}\) | \(107\) |
default | \(-\frac {4 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {6 \,2^{\frac {2}{3}} \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\) | \(262\) |
elliptic | \(-\frac {4 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {6 \,2^{\frac {2}{3}} \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\) | \(262\) |
Input:
int((2*x+2^(2/3))/(2^(2/3)-x)/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*RootOf(_Z^2-6*2^(1/3))*ln((12*(x^3-1)^(1/2)*x+3*RootOf(_Z^2-6*2^(1/3)) *2^(2/3)*x^2+RootOf(_Z^2-6*2^(1/3))*x^3-6*(x^3-1)^(1/2)*2^(2/3)-6*RootOf(_ Z^2-6*2^(1/3))*2^(1/3)*x+2*RootOf(_Z^2-6*2^(1/3)))/(2^(1/3)*x-2)^3)
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 6.26 \[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \log \left (\frac {x^{18} + 1440 \, x^{15} + 17400 \, x^{12} - 21056 \, x^{9} - 10368 \, x^{6} + 15360 \, x^{3} + 2 \, \sqrt {6} 2^{\frac {1}{6}} {\left (126 \, x^{14} + 2664 \, x^{11} - 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac {2}{3}} {\left (x^{16} + 310 \, x^{13} + 2332 \, x^{10} - 2656 \, x^{7} - 256 \, x^{4} + 512 \, x\right )} + 2^{\frac {1}{3}} {\left (17 \, x^{15} + 1058 \, x^{12} + 2528 \, x^{9} - 5408 \, x^{6} + 2560 \, x^{3} - 512\right )}\right )} \sqrt {x^{3} - 1} + 24 \cdot 2^{\frac {2}{3}} {\left (x^{17} + 121 \, x^{14} + 478 \, x^{11} - 1144 \, x^{8} + 608 \, x^{5} - 64 \, x^{2}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} + 176 \, x^{13} + 83 \, x^{10} - 680 \, x^{7} + 544 \, x^{4} - 128 \, x\right )} - 2048}{x^{18} - 24 \, x^{15} + 240 \, x^{12} - 1280 \, x^{9} + 3840 \, x^{6} - 6144 \, x^{3} + 4096}\right ) \] Input:
integrate((2^(2/3)+2*x)/(2^(2/3)-x)/(x^3-1)^(1/2),x, algorithm="fricas")
Output:
1/6*sqrt(6)*2^(1/6)*log((x^18 + 1440*x^15 + 17400*x^12 - 21056*x^9 - 10368 *x^6 + 15360*x^3 + 2*sqrt(6)*2^(1/6)*(126*x^14 + 2664*x^11 - 4608*x^5 + 23 04*x^2 + 2^(2/3)*(x^16 + 310*x^13 + 2332*x^10 - 2656*x^7 - 256*x^4 + 512*x ) + 2^(1/3)*(17*x^15 + 1058*x^12 + 2528*x^9 - 5408*x^6 + 2560*x^3 - 512))* sqrt(x^3 - 1) + 24*2^(2/3)*(x^17 + 121*x^14 + 478*x^11 - 1144*x^8 + 608*x^ 5 - 64*x^2) + 48*2^(1/3)*(5*x^16 + 176*x^13 + 83*x^10 - 680*x^7 + 544*x^4 - 128*x) - 2048)/(x^18 - 24*x^15 + 240*x^12 - 1280*x^9 + 3840*x^6 - 6144*x ^3 + 4096))
\[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=- \int \frac {2^{\frac {2}{3}}}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx - \int \frac {2 x}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx \] Input:
integrate((2**(2/3)+2*x)/(2**(2/3)-x)/(x**3-1)**(1/2),x)
Output:
-Integral(2**(2/3)/(x*sqrt(x**3 - 1) - 2**(2/3)*sqrt(x**3 - 1)), x) - Inte gral(2*x/(x*sqrt(x**3 - 1) - 2**(2/3)*sqrt(x**3 - 1)), x)
\[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\int { -\frac {2 \, x + 2^{\frac {2}{3}}}{\sqrt {x^{3} - 1} {\left (x - 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate((2^(2/3)+2*x)/(2^(2/3)-x)/(x^3-1)^(1/2),x, algorithm="maxima")
Output:
-integrate((2*x + 2^(2/3))/(sqrt(x^3 - 1)*(x - 2^(2/3))), x)
Exception generated. \[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2^(2/3)+2*x)/(2^(2/3)-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,[2]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Bad Argument
Time = 23.77 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.63 \[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {{\left (\sqrt {x^3-1}-\sqrt {3}+2^{1/3}\,\sqrt {3}\,x\right )}^3\,\left (\sqrt {3}+\sqrt {x^3-1}-2^{1/3}\,\sqrt {3}\,x\right )}{{\left (x-2^{2/3}\right )}^6}\right )}{3} \] Input:
int(-(2*x + 2^(2/3))/((x^3 - 1)^(1/2)*(x - 2^(2/3))),x)
Output:
(2^(2/3)*3^(1/2)*log((((x^3 - 1)^(1/2) - 3^(1/2) + 2^(1/3)*3^(1/2)*x)^3*(3 ^(1/2) + (x^3 - 1)^(1/2) - 2^(1/3)*3^(1/2)*x))/(x - 2^(2/3))^6))/3
\[ \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=2^{\frac {2}{3}} \left (\int \frac {1}{\sqrt {x^{3}-1}\, 2^{\frac {2}{3}}-\sqrt {x^{3}-1}\, x}d x \right )+2 \left (\int \frac {x}{\sqrt {x^{3}-1}\, 2^{\frac {2}{3}}-\sqrt {x^{3}-1}\, x}d x \right ) \] Input:
int((2^(2/3)+2*x)/(2^(2/3)-x)/(x^3-1)^(1/2),x)
Output:
2**(2/3)*int(1/(sqrt(x**3 - 1)*2**(2/3) - sqrt(x**3 - 1)*x),x) + 2*int(x/( sqrt(x**3 - 1)*2**(2/3) - sqrt(x**3 - 1)*x),x)