\(\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} (1-x^6)} \, dx\) [2987]

Optimal result

Integrand size = 28, antiderivative size = 390 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\frac {4 \left (-x^3+x^5\right )^{3/4}}{3 x^2 \left (-1+x^2\right )}+\frac {\sqrt [4]{2} \arctan \left (\frac {3^{7/8} \sqrt {2-\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \arctan \left (\frac {3^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2+\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right ) \] Output:

4/3*(x^5-x^3)^(3/4)/x^2/(x^2-1)+1/3*arctan(3^(7/8)*(2-2^(1/2))^(1/2)*x*(x^ 
5-x^3)^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^3)^(1/2)))*2^(1/4)/(51+36*2^(1/2))^(1/ 
8)+1/3*(17/3+4*2^(1/2))^(1/8)*arctan(3^(7/8)*(2+2^(1/2))^(1/2)*x*(x^5-x^3) 
^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^3)^(1/2)))*2^(1/4)+1/3*arctanh((3^(1/8)*x^2/ 
(2-2^(1/2))^(1/2)+1/3*(x^5-x^3)^(1/2)*3^(7/8)/(2-2^(1/2))^(1/2))/x/(x^5-x^ 
3)^(1/4))*2^(1/4)/(51+36*2^(1/2))^(1/8)+1/3*(17/3+4*2^(1/2))^(1/8)*arctanh 
((3^(1/8)*x^2/(2+2^(1/2))^(1/2)+1/3*(x^5-x^3)^(1/2)*3^(7/8)/(2+2^(1/2))^(1 
/2))/x/(x^5-x^3)^(1/4))*2^(1/4)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 3.46 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.05 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\frac {x^{3/4} \left (12 \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x}+\sqrt [4]{2} 3^{7/8} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+3^{7/8} \sqrt [4]{34+24 \sqrt {2}} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )+\sqrt [4]{2} 3^{7/8} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+3^{7/8} \sqrt [4]{34+24 \sqrt {2}} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )\right )}{9 \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x^3 \left (-1+x^2\right )}} \] Input:

Integrate[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
 

Output:

(x^(3/4)*(12*(17 + 12*Sqrt[2])^(1/8)*x^(1/4) + 2^(1/4)*3^(7/8)*(-1 + x^2)^ 
(1/4)*ArcTan[(2^(1/4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1/4))/((17 + 12*Sqrt[2]) 
^(1/8)*(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + 3^(7/8)*(34 + 24*Sqrt[2]) 
^(1/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x^ 
(1/4)*(-1 + x^2)^(1/4))/(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2])] + 2^(1/4)*3 
^(7/8)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1/4)) 
/((17 + 12*Sqrt[2])^(1/8)*(3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + 3^(7/8) 
*(34 + 24*Sqrt[2])^(1/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*3^(7/8)*(17 + 1 
2*Sqrt[2])^(1/8)*x^(1/4)*(-1 + x^2)^(1/4))/(3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + 
x^2])]))/(9*(17 + 12*Sqrt[2])^(1/8)*(x^3*(-1 + x^2))^(1/4))
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^6+1}{\sqrt [4]{x^5-x^3} \left (1-x^6\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2-1} \int \frac {x^6+1}{x^{3/4} \sqrt [4]{x^2-1} \left (1-x^6\right )}dx}{\sqrt [4]{x^5-x^3}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \frac {x^6+1}{\sqrt [4]{x^2-1} \left (1-x^6\right )}d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (\frac {2}{\sqrt [4]{x^2-1} \left (1-x^6\right )}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (-\frac {2}{\left (x^6-1\right ) \sqrt [4]{x^2-1}}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (-\frac {2}{\left (x^6-1\right ) \sqrt [4]{x^2-1}}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\)

Input:

Int[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
 

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 86.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17

method	result	size
pseudoelliptic	\(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) \left (x^{5}-x^{3}\right )^{\frac {1}{4}}+4 x}{3 \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\)	\(65\)
risch	\(\text {Expression too large to display}\)	\(2154\)
trager	\(\text {Expression too large to display}\)	\(2163\)

Input:

int((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x,method=_RETURNVERBOSE)
 

Output:

1/3*(-sum(ln((-_R*x+(x^5-x^3)^(1/4))/x)/_R,_R=RootOf(_Z^8+3))*(x^5-x^3)^(1 
/4)+4*x)/(x^5-x^3)^(1/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 143.26 (sec) , antiderivative size = 1950, normalized size of antiderivative = 5.00 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="fricas")
 

Output:

1/12*(sqrt(2)*(-1/3)^(1/8)*((I - 1)*x^4 - (I - 1)*x^2)*log(-(6*sqrt(2)*sqr 
t(x^5 - x^3)*((-1/3)^(7/8)*((109*I + 109)*x^3 - (132*I + 132)*x^2 - (109*I 
 + 109)*x) + (-1/3)^(3/8)*(-(44*I + 44)*x^3 - (109*I + 109)*x^2 + (44*I + 
44)*x)) + 4*(x^5 - x^3)^(3/4)*(109*x^2 + 3*sqrt(-1/3)*(44*x^2 + 109*x - 44 
) - 132*x - 109) + sqrt(2)*(6*(-1/3)^(5/8)*((22*I - 22)*x^6 + (109*I - 109 
)*x^5 - (110*I - 110)*x^4 - (109*I - 109)*x^3 + (22*I - 22)*x^2) - (-1/3)^ 
(1/8)*(-(109*I - 109)*x^6 + (264*I - 264)*x^5 + (545*I - 545)*x^4 - (264*I 
 - 264)*x^3 - (109*I - 109)*x^2)) + 12*(x^5 - x^3)^(1/4)*((-1/3)^(3/4)*(-1 
09*I*x^4 + 132*I*x^3 + 109*I*x^2) + (-1/3)^(1/4)*(44*I*x^4 + 109*I*x^3 - 4 
4*I*x^2)))/(x^6 + x^4 + x^2)) + sqrt(2)*(-1/3)^(1/8)*(-(I + 1)*x^4 + (I + 
1)*x^2)*log(-(6*sqrt(2)*sqrt(x^5 - x^3)*((-1/3)^(7/8)*(-(109*I - 109)*x^3 
+ (132*I - 132)*x^2 + (109*I - 109)*x) + (-1/3)^(3/8)*((44*I - 44)*x^3 + ( 
109*I - 109)*x^2 - (44*I - 44)*x)) + 4*(x^5 - x^3)^(3/4)*(109*x^2 + 3*sqrt 
(-1/3)*(44*x^2 + 109*x - 44) - 132*x - 109) + sqrt(2)*(6*(-1/3)^(5/8)*(-(2 
2*I + 22)*x^6 - (109*I + 109)*x^5 + (110*I + 110)*x^4 + (109*I + 109)*x^3 
- (22*I + 22)*x^2) - (-1/3)^(1/8)*((109*I + 109)*x^6 - (264*I + 264)*x^5 - 
 (545*I + 545)*x^4 + (264*I + 264)*x^3 + (109*I + 109)*x^2)) + 12*(x^5 - x 
^3)^(1/4)*((-1/3)^(3/4)*(109*I*x^4 - 132*I*x^3 - 109*I*x^2) + (-1/3)^(1/4) 
*(-44*I*x^4 - 109*I*x^3 + 44*I*x^2)))/(x^6 + x^4 + x^2)) + sqrt(2)*(-1/3)^ 
(1/8)*((I + 1)*x^4 - (I + 1)*x^2)*log(-(6*sqrt(2)*sqrt(x^5 - x^3)*((-1/...
 

Sympy [F]

\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx \] Input:

integrate((x**6+1)/(x**5-x**3)**(1/4)/(-x**6+1),x)
 

Output:

-Integral(x**6/(x**6*(x**5 - x**3)**(1/4) - (x**5 - x**3)**(1/4)), x) - In 
tegral(1/(x**6*(x**5 - x**3)**(1/4) - (x**5 - x**3)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="maxima")
 

Output:

-integrate((x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
 

Giac [F]

\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="giac")
 

Output:

integrate(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int -\frac {x^6+1}{\left (x^6-1\right )\,{\left (x^5-x^3\right )}^{1/4}} \,d x \] Input:

int(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)),x)
 

Output:

int(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=-\left (\int \frac {x^{6}}{x^{\frac {27}{4}} \left (x^{2}-1\right )^{\frac {1}{4}}-x^{\frac {3}{4}} \left (x^{2}-1\right )^{\frac {1}{4}}}d x \right )-\left (\int \frac {1}{x^{\frac {27}{4}} \left (x^{2}-1\right )^{\frac {1}{4}}-x^{\frac {3}{4}} \left (x^{2}-1\right )^{\frac {1}{4}}}d x \right ) \] Input:

int((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x)
 

Output:

 - (int(x**6/(x**(3/4)*(x**2 - 1)**(1/4)*x**6 - x**(3/4)*(x**2 - 1)**(1/4) 
),x) + int(1/(x**(3/4)*(x**2 - 1)**(1/4)*x**6 - x**(3/4)*(x**2 - 1)**(1/4) 
),x))