\(\int \frac {-1+x^6}{\sqrt {1+x^4} (1+x^6)} \, dx\) [553]

Optimal result

Integrand size = 22, antiderivative size = 43 \[ \int \frac {-1+x^6}{\sqrt {1+x^4} \left (1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt {1+x^4}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^6}{\sqrt {1+x^4} \left (1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt {1+x^4}}\right ) \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.67 (sec) , antiderivative size = 498, normalized size of antiderivative = 11.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{3 \sqrt {2}}-\frac {1}{6} i \arctan \left (\frac {\sqrt [3]{-1} \left (x^2+\sqrt [3]{-1}\right )}{\sqrt {x^4+1}}\right )-\frac {1}{6} i \arctan \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{-1} x^2+1\right )}{\sqrt {x^4+1}}\right )-\frac {\sqrt [6]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (\sqrt {3}+3 i\right ) \sqrt {x^4+1}}-\frac {\sqrt [3]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \sqrt {x^4+1}}-\frac {(-1)^{2/3} \text {arctanh}\left (\frac {x}{\sqrt {x^4+1}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}-\frac {\sqrt [6]{-1} \text {arctanh}\left (\frac {x}{\sqrt {x^4+1}}\right )}{2 \sqrt {3} \left (1+\sqrt [3]{-1}\right )}-\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {x^4+1}}\right )+\frac {(-1)^{5/6} \text {arctanh}\left (\frac {\sqrt {1-\sqrt [3]{-1}} \left (x^2+\sqrt [3]{-1}\right )}{\sqrt {x^4+1}}\right )}{6 \sqrt {1-\sqrt [3]{-1}}}-\frac {\sqrt [6]{-1} \text {arctanh}\left (\frac {\sqrt [3]{-1} x^2+1}{\sqrt {1+(-1)^{2/3}} \sqrt {x^4+1}}\right )}{6 \sqrt {1+(-1)^{2/3}}}\)

Input:

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

Output:

-1/3*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - (I/6)*ArcTan[((-1)^(1/3)* 
((-1)^(1/3) + x^2))/Sqrt[1 + x^4]] - (I/6)*ArcTan[((-1)^(1/3)*(1 + (-1)^(1 
/3)*x^2))/Sqrt[1 + x^4]] - ArcTanh[x/Sqrt[1 + x^4]]/3 - ((-1)^(1/6)*ArcTan 
h[x/Sqrt[1 + x^4]])/(2*Sqrt[3]*(1 + (-1)^(1/3))) - ((-1)^(2/3)*ArcTanh[x/S 
qrt[1 + x^4]])/(Sqrt[3]*(3*I - Sqrt[3])) + ((-1)^(5/6)*ArcTanh[(Sqrt[1 - ( 
-1)^(1/3)]*((-1)^(1/3) + x^2))/Sqrt[1 + x^4]])/(6*Sqrt[1 - (-1)^(1/3)]) - 
((-1)^(1/6)*ArcTanh[(1 + (-1)^(1/3)*x^2)/(Sqrt[1 + (-1)^(2/3)]*Sqrt[1 + x^ 
4])])/(6*Sqrt[1 + (-1)^(2/3)]) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*El 
lipticF[2*ArcTan[x], 1/2])/(3*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/( 
1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(6*(1 + (-1)^(1/3))*Sqrt[1 + x^4] 
) - ((-1)^(1/3)*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x 
], 1/2])/(3*(1 + (-1)^(1/3))*Sqrt[1 + x^4]) - ((-1)^(1/6)*(1 + x^2)*Sqrt[( 
1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(3*(3*I + Sqrt[3])*Sqrt 
[1 + x^4])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98

method	result	size
elliptic	\(\frac {\left (\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{3}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}\right ) \sqrt {2}}{2}\)	\(42\)
default	\(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{6}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {3}\, x^{2}+\sqrt {3}-2 x}{\sqrt {x^{4}+1}}\right )}{3}+\frac {\operatorname {arctanh}\left (\frac {2 x +\sqrt {3}\, x^{2}+\sqrt {3}}{\sqrt {x^{4}+1}}\right )}{3}\)	\(70\)
pseudoelliptic	\(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{6}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {3}\, x^{2}+\sqrt {3}-2 x}{\sqrt {x^{4}+1}}\right )}{3}+\frac {\operatorname {arctanh}\left (\frac {2 x +\sqrt {3}\, x^{2}+\sqrt {3}}{\sqrt {x^{4}+1}}\right )}{3}\)	\(70\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -\sqrt {x^{4}+1}}{x^{2}+1}\right )}{6}+\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+1}\, x -x^{2}-1}{x^{4}-x^{2}+1}\right )}{3}\)	\(77\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23 \[ \int \frac {-1+x^6}{\sqrt {1+x^4} \left (1+x^6\right )} \, dx=-\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + \frac {1}{3} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 1} x + 1}{x^{4} - x^{2} + 1}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {-1+x^6}{\sqrt {1+x^4} \left (1+x^6\right )} \, dx=-\left (\int \frac {\sqrt {x^{4}+1}}{x^{10}+x^{6}+x^{4}+1}d x \right )+\int \frac {\sqrt {x^{4}+1}\, x^{6}}{x^{10}+x^{6}+x^{4}+1}d x \] Input:

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

Output:

 - int(sqrt(x**4 + 1)/(x**10 + x**6 + x**4 + 1),x) + int((sqrt(x**4 + 1)*x 
**6)/(x**10 + x**6 + x**4 + 1),x)