\(\int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx\) [572]

Optimal result

Integrand size = 15, antiderivative size = 78 \[ \int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx=\sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log (-1+x)+\log \left (1-\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(78)=156\).

Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx=\frac {\sqrt [3]{\frac {-1+x}{1+x}} \left (3 \sqrt [3]{-1+x}+3 \sqrt [3]{-1+x} x+2 \sqrt {3} \sqrt [3]{1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )+2 \sqrt [3]{1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\sqrt [3]{1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{3 \sqrt [3]{-1+x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {60, 71}

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.

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 71

\(\displaystyle \sqrt [3]{x-1} (x+1)^{2/3}-\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {1}{2} \log (x-1)-\frac {3}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 71

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> 
 With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( 
Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + 
 b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.39 (sec) , antiderivative size = 573, normalized size of antiderivative = 7.35

method	result	size
risch	\(\left (-1+x \right )^{\frac {1}{3}} \left (1+x \right )^{\frac {2}{3}}+\frac {\left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{-1+x}\right )}{3}-\frac {2 \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x +1}{-1+x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {2 \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x +1}{-1+x}\right )}{3}\right ) \left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{\left (-1+x \right )^{\frac {2}{3}} \left (1+x \right )^{\frac {1}{3}}}\)	\(573\)

Input:

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

Output:

(-1+x)^(1/3)*(1+x)^(2/3)+(2/3*RootOf(_Z^2+_Z+1)*ln(-(2*RootOf(_Z^2+_Z+1)^2 
*x^2+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3)+3*RootOf(_Z^2+_Z+1)*(x^3-x^2- 
x+1)^(1/3)*x-2*RootOf(_Z^2+_Z+1)^2*x+5*RootOf(_Z^2+_Z+1)*x^2-3*RootOf(_Z^2 
+_Z+1)*(x^3-x^2-x+1)^(1/3)-4*RootOf(_Z^2+_Z+1)*x+2*x^2-RootOf(_Z^2+_Z+1)-2 
)/(-1+x))-2/3*ln((-2*RootOf(_Z^2+_Z+1)^2*x^2+3*RootOf(_Z^2+_Z+1)*(x^3-x^2- 
x+1)^(2/3)+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)*x+2*RootOf(_Z^2+_Z+1)^2 
*x+RootOf(_Z^2+_Z+1)*x^2+3*(x^3-x^2-x+1)^(2/3)-3*RootOf(_Z^2+_Z+1)*(x^3-x^ 
2-x+1)^(1/3)+3*(x^3-x^2-x+1)^(1/3)*x+x^2-3*(x^3-x^2-x+1)^(1/3)-RootOf(_Z^2 
+_Z+1)-2*x+1)/(-1+x))*RootOf(_Z^2+_Z+1)-2/3*ln((-2*RootOf(_Z^2+_Z+1)^2*x^2 
+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3)+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1) 
^(1/3)*x+2*RootOf(_Z^2+_Z+1)^2*x+RootOf(_Z^2+_Z+1)*x^2+3*(x^3-x^2-x+1)^(2/ 
3)-3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)+3*(x^3-x^2-x+1)^(1/3)*x+x^2-3*( 
x^3-x^2-x+1)^(1/3)-RootOf(_Z^2+_Z+1)-2*x+1)/(-1+x)))/(-1+x)^(2/3)*((-1+x)^ 
2*(1+x))^(1/3)/(1+x)^(1/3)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx=-\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x + 1\right )}}\right ) + {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \log \left (\frac {{\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} + {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {2}{3}} + x + 1}{x + 1}\right ) + \frac {2}{3} \, \log \left (\frac {{\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} - x - 1}{x + 1}\right ) \] Input:

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

Output:

-2/3*sqrt(3)*arctan(1/3*(sqrt(3)*(x + 1) + 2*sqrt(3)*(x + 1)^(2/3)*(x - 1) 
^(1/3))/(x + 1)) + (x + 1)^(2/3)*(x - 1)^(1/3) - 1/3*log(((x + 1)^(2/3)*(x 
 - 1)^(1/3) + (x + 1)^(1/3)*(x - 1)^(2/3) + x + 1)/(x + 1)) + 2/3*log(((x 
+ 1)^(2/3)*(x - 1)^(1/3) - x - 1)/(x + 1))
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx=\frac {2^{\frac {2}{3}} \left (x - 1\right )^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{2 \Gamma \left (\frac {7}{3}\right )} \] Input:

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

Output:

2**(2/3)*(x - 1)**(4/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), (x - 1)*exp_p 
olar(I*pi)/2)/(2*gamma(7/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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