\(\int \frac {(-1+x^2) \sqrt [4]{x^2+x^6}}{x^2 (1+x^2)} \, dx\) [1754]

Optimal result

Integrand size = 27, antiderivative size = 118 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\frac {2 \sqrt [4]{x^2+x^6}}{x}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{\sqrt [4]{2}} \] Output:

2*(x^6+x^2)^(1/4)/x-1/2*arctan(2^(3/4)*x*(x^6+x^2)^(1/4)/(2^(1/2)*x^2-(x^6 
+x^2)^(1/2)))*2^(3/4)-1/2*arctanh((1/2*x^2*2^(3/4)+1/2*(x^6+x^2)^(1/2)*2^( 
1/4))/x/(x^6+x^2)^(1/4))*2^(3/4)
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=-\frac {\sqrt [4]{x^2+x^6} \left (-4 \sqrt [4]{1+x^4}+2^{3/4} \sqrt {x} \arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )+2^{3/4} \sqrt {x} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{2 x \sqrt [4]{1+x^4}} \] Input:

Integrate[((-1 + x^2)*(x^2 + x^6)^(1/4))/(x^2*(1 + x^2)),x]
 

Output:

-1/2*((x^2 + x^6)^(1/4)*(-4*(1 + x^4)^(1/4) + 2^(3/4)*Sqrt[x]*ArcTan[(2^(3 
/4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] + 2^(3/4)*Sqrt[x 
]*ArcTanh[(2*2^(1/4)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + Sqrt[2]*Sqrt[1 + x^4] 
)]))/(x*(1 + x^4)^(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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \left (-2 \int \frac {x \sqrt [4]{x^4+1}}{x^2+1}d\sqrt {x}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-\frac {1}{8},\frac {7}{8},-x^4\right )}{\sqrt {x}}\right )}{\sqrt {x} \sqrt [4]{x^4+1}}\)

Input:

Int[((-1 + x^2)*(x^2 + x^6)^(1/4))/(x^2*(1 + x^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.00 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.35

method	result	size
pseudoelliptic	\(\frac {-\ln \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) 2^{\frac {3}{4}} x -2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {3}{4}} x -2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {3}{4}} x +8 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{4 x}\)	\(159\)

Input:

int((x^2-1)*(x^6+x^2)^(1/4)/x^2/(x^2+1),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (97) = 194\).

Time = 4.32 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.84 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\frac {4 \cdot 2^{\frac {3}{4}} x \arctan \left (\frac {2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) - 2^{\frac {3}{4}} x \log \left (\frac {x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}\right ) + 2^{\frac {3}{4}} x \log \left (\frac {x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}\right ) + 16 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}}{8 \, x} \] Input:

integrate((x^2-1)*(x^6+x^2)^(1/4)/x^2/(x^2+1),x, algorithm="fricas")
 

Output:

1/8*(4*2^(3/4)*x*arctan(1/2*(2^(3/4)*(x^6 + x^2)^(3/4) - 2^(1/4)*(x^6 + x^ 
2)^(1/4)*(x^4 + 1))/(x^5 + x)) - 2^(3/4)*x*log((x^5 + 2*x^3 + 4*2^(1/4)*(x 
^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x + 2*2^(3/4)*(x^6 + x^2)^ 
(3/4) + x)/(x^5 + 2*x^3 + x)) + 2^(3/4)*x*log((x^5 + 2*x^3 - 4*2^(1/4)*(x^ 
6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x - 2*2^(3/4)*(x^6 + x^2)^( 
3/4) + x)/(x^5 + 2*x^3 + x)) + 16*(x^6 + x^2)^(1/4))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((x^2-1)*(x^6+x^2)^(1/4)/x^2/(x^2+1),x, algorithm="maxima")
 

Output:

integrate((x^6 + x^2)^(1/4)*(x^2 - 1)/((x^2 + 1)*x^2), x)
 

Giac [F]

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

integrate((x^2-1)*(x^6+x^2)^(1/4)/x^2/(x^2+1),x, algorithm="giac")
 

Output:

integrate((x^6 + x^2)^(1/4)*(x^2 - 1)/((x^2 + 1)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^2-1\right )}{x^2\,\left (x^2+1\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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