\(\int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx\) [2269]

Optimal result

Integrand size = 17, antiderivative size = 172 \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left (2 \sqrt [3]{2}-2 \sqrt [3]{2} x\right ) \sqrt [3]{1-x^3}+4 \left (1-x^3\right )^{2/3}\right )}{4 \sqrt [3]{2}} \] Output:

1/4*3^(1/2)*arctan(3^(1/2)*(-x^3+1)^(1/3)/(2^(1/3)-2^(1/3)*x+(-x^3+1)^(1/3 
)))*2^(2/3)+1/4*ln(-2^(1/3)+2^(1/3)*x+2*(-x^3+1)^(1/3))*2^(2/3)-1/8*ln(2^( 
2/3)-2*2^(2/3)*x+2^(2/3)*x^2+(2*2^(1/3)-2*2^(1/3)*x)*(-x^3+1)^(1/3)+4*(-x^ 
3+1)^(2/3))*2^(2/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )+2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )-\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2-2 (-1+x) \sqrt [3]{2-2 x^3}+4 \left (1-x^3\right )^{2/3}\right )}{4 \sqrt [3]{2}} \] Input:

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

Output:

(2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^3)^(1/3))/(2^(1/3) - 2^(1/3)*x + (1 - x^ 
3)^(1/3))] + 2*Log[-2^(1/3) + 2^(1/3)*x + 2*(1 - x^3)^(1/3)] - Log[2^(2/3) 
 - 2*2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x)*(2 - 2*x^3)^(1/3) + 4*(1 - x^3)^ 
(2/3)])/(4*2^(1/3))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.56, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2574}

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.

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

\(\Big \downarrow \) 2574

\(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}}\)

Input:

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

Output:

-1/2*(Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/2^( 
1/3) - Log[(1 - x)*(1 + x)^2]/(4*2^(1/3)) + (3*Log[-1 + x + 2^(2/3)*(1 - x 
^3)^(1/3)])/(4*2^(1/3))
 

Defintions of rubi rules used

rule 2574

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ 
Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq 
rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 
)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ 
(1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + 
a*d^3, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 6.18 (sec) , antiderivative size = 1143, normalized size of antiderivative = 6.65

method	result	size
trager	\(\text {Expression too large to display}\)	\(1143\)

Input:

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

Output:

1/4*RootOf(_Z^3-4)*ln(-(8*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z 
^2)^2*RootOf(_Z^3-4)^2*x-6*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_ 
Z^2)*RootOf(_Z^3-4)^3*x+8*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+2*_Z*Ro 
otOf(_Z^3-4)+4*_Z^2)*(-x^3+1)^(2/3)+26*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4 
)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*(-x^3+1)^(1/3)*x+9*RootOf(_Z^3-4)^2*(-x^3+ 
1)^(1/3)*x-26*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4 
*_Z^2)*(-x^3+1)^(1/3)+28*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^ 
2)*x^2-9*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)-21*RootOf(_Z^3-4)*x^2+24*RootOf(R 
ootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x-18*RootOf(_Z^3-4)*x+52*(-x^3 
+1)^(2/3)+28*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)-21*RootOf 
(_Z^3-4))/(1+x)^2)-1/4*ln((20*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+ 
4*_Z^2)^2*RootOf(_Z^3-4)^2*x+6*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4) 
+4*_Z^2)*RootOf(_Z^3-4)^3*x+8*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+2*_ 
Z*RootOf(_Z^3-4)+4*_Z^2)*(-x^3+1)^(2/3)-18*RootOf(_Z^3-4)*RootOf(RootOf(_Z 
^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*(-x^3+1)^(1/3)*x-13*RootOf(_Z^3-4)^2*( 
-x^3+1)^(1/3)*x+18*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3 
-4)+4*_Z^2)*(-x^3+1)^(1/3)-70*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+ 
4*_Z^2)*x^2+13*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)-21*RootOf(_Z^3-4)*x^2-20*Ro 
otOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x-6*RootOf(_Z^3-4)*x-36* 
(-x^3+1)^(2/3)-70*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)-2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (132) = 264\).

Time = 1.77 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.73 \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\frac {1}{6} \cdot 2^{\frac {1}{6}} \sqrt {\frac {3}{2}} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {3}{2}} {\left (8 \cdot 2^{\frac {2}{3}} {\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )} - 4 \, {\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {4 \cdot 2^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 2^{\frac {1}{3}} {\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} - 2 \, {\left (3 \, x^{3} - x^{2} + x - 3\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} - 2 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 4 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) \] Input:

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

Output:

1/6*2^(1/6)*sqrt(3/2)*arctan(1/3*2^(1/6)*sqrt(3/2)*(8*2^(2/3)*(x^4 + 2*x^3 
 + 2*x^2 + 2*x + 1)*(-x^3 + 1)^(2/3) + 2^(1/3)*(13*x^6 + 2*x^5 + 19*x^4 - 
4*x^3 + 19*x^2 + 2*x + 13) - 4*(5*x^5 - 5*x^4 + 6*x^3 - 6*x^2 + 5*x - 5)*( 
-x^3 + 1)^(1/3))/(3*x^6 - 18*x^5 - 3*x^4 - 28*x^3 - 3*x^2 - 18*x + 3)) - 1 
/24*2^(2/3)*log((4*2^(2/3)*(-x^3 + 1)^(2/3)*(x^2 + 1) + 2^(1/3)*(5*x^4 + 6 
*x^2 + 5) - 2*(3*x^3 - x^2 + x - 3)*(-x^3 + 1)^(1/3))/(x^4 + 4*x^3 + 6*x^2 
 + 4*x + 1)) + 1/12*2^(2/3)*log((2^(2/3)*(x^2 + 2*x + 1) - 2*2^(1/3)*(-x^3 
 + 1)^(1/3)*(x - 1) - 4*(-x^3 + 1)^(2/3))/(x^2 + 2*x + 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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