\(\int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx\) [795]

Optimal result

Integrand size = 15, antiderivative size = 61 \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\frac {\arctan \left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )}{2 \sqrt [4]{2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\frac {\arctan \left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )-\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{x+x^3}}{1+x}\right )}{2 \sqrt [4]{2}} \] Input:

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

Output:

(ArcTan[(2^(3/4)*(x + x^3)^(1/4))/(1 + x)] - ArcTanh[(2^(3/4)*(x + x^3)^(1 
/4))/(1 + x)])/(2*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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 616

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

\(\Big \downarrow \) 1888

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

Input:

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

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 7.43 (sec) , antiderivative size = 512, normalized size of antiderivative = 8.39

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} \sqrt {x^{3}+x}\, x^{2}+4 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{3}+x \right )^{\frac {1}{4}} x^{3}+2 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{3}+x \right )^{\frac {1}{4}} x^{2}+x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )-6 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}-16 \left (x^{3}+x \right )^{\frac {3}{4}} x -2 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{2}-16 \left (x^{3}+x \right )^{\frac {3}{4}}+12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\left (-1+x \right )^{4}}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}+4 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{3}+x \right )^{\frac {1}{4}} x^{3}+2 \sqrt {x^{3}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{3}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+6 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -16 \left (x^{3}+x \right )^{\frac {3}{4}} x -12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+2 \left (x^{3}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}-16 \left (x^{3}+x \right )^{\frac {3}{4}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (-1+x \right )^{4}}\right )}{8}\)	\(512\)

Input:

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

Output:

1/8*RootOf(_Z^4-8)*ln(-(2*RootOf(_Z^4-8)^3*(x^3+x)^(1/2)*x^2+4*(x^3+x)^(1/ 
2)*RootOf(_Z^4-8)^3*x-2*RootOf(_Z^4-8)^2*(x^3+x)^(1/4)*x^3+2*(x^3+x)^(1/2) 
*RootOf(_Z^4-8)^3-6*RootOf(_Z^4-8)^2*(x^3+x)^(1/4)*x^2+x^4*RootOf(_Z^4-8)- 
6*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x+12*RootOf(_Z^4-8)*x^3-16*(x^3+x)^(3/4)* 
x-2*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2+6*RootOf(_Z^4-8)*x^2-16*(x^3+x)^(3/4)+1 
2*RootOf(_Z^4-8)*x+RootOf(_Z^4-8))/(-1+x)^4)-1/8*RootOf(_Z^2+RootOf(_Z^4-8 
)^2)*ln(-(2*RootOf(_Z^4-8)^2*(x^3+x)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x 
^2+4*(x^3+x)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x+2*Root 
Of(_Z^4-8)^2*(x^3+x)^(1/4)*x^3+2*(x^3+x)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-8)^ 
2)*RootOf(_Z^4-8)^2+6*RootOf(_Z^4-8)^2*(x^3+x)^(1/4)*x^2-RootOf(_Z^2+RootO 
f(_Z^4-8)^2)*x^4+6*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2*x-16*(x^3+x)^(3/4)*x-12* 
RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^3+2*(x^3+x)^(1/4)*RootOf(_Z^4-8)^2-16*(x^3 
+x)^(3/4)-6*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^2-12*RootOf(_Z^2+RootOf(_Z^4-8 
)^2)*x-RootOf(_Z^2+RootOf(_Z^4-8)^2))/(-1+x)^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (45) = 90\).

Time = 3.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 4.66 \[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx=\frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2 \, {\left (2^{\frac {3}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{4} + 12 \, x^{3} + 6 \, x^{2} + 12 \, x + 1\right )} + 4 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} + 8 \cdot 2^{\frac {1}{4}} \sqrt {x^{3} + x} {\left (x^{2} + 2 \, x + 1\right )} + 16 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{4} + 12 \, x^{3} + 6 \, x^{2} + 12 \, x + 1\right )} - 4 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{4}} + 8 \cdot 2^{\frac {1}{4}} \sqrt {x^{3} + x} {\left (x^{2} + 2 \, x + 1\right )} - 16 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x + 1\right )}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) \] Input:

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

Output:

1/8*2^(3/4)*arctan(2*(2^(3/4)*(x^3 + 3*x^2 + 3*x + 1)*(x^3 + x)^(1/4) + 4* 
2^(1/4)*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 1/16*2 
^(3/4)*log((2^(3/4)*(x^4 + 12*x^3 + 6*x^2 + 12*x + 1) + 4*sqrt(2)*(x^3 + 3 
*x^2 + 3*x + 1)*(x^3 + x)^(1/4) + 8*2^(1/4)*sqrt(x^3 + x)*(x^2 + 2*x + 1) 
+ 16*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) + 1/16*2^(3 
/4)*log(-(2^(3/4)*(x^4 + 12*x^3 + 6*x^2 + 12*x + 1) - 4*sqrt(2)*(x^3 + 3*x 
^2 + 3*x + 1)*(x^3 + x)^(1/4) + 8*2^(1/4)*sqrt(x^3 + x)*(x^2 + 2*x + 1) - 
16*(x^3 + x)^(3/4)*(x + 1))/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))
 

Sympy [F]

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

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

Output:

Integral(1/((x*(x**2 + 1))**(1/4)*(x - 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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