3.10.2 \(\int \frac {(1+x^2) \sqrt [4]{x^2+x^6}}{x^2 (-1+x^2)} \, dx\) [902]

3.10.2.1 Optimal result

Integrand size = 27, antiderivative size = 68 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (-1+x^2\right )} \, dx=\frac {2 \sqrt [4]{x^2+x^6}}{x}+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right ) \]

output

2*(x^6+x^2)^(1/4)/x+2^(1/4)*arctan(2^(1/4)*x/(x^6+x^2)^(1/4))-2^(1/4)*arct 
anh(2^(1/4)*x/(x^6+x^2)^(1/4))
 

3.10.2.2 Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.49 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (-1+x^2\right )} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (2 \sqrt [4]{1+x^4}+\sqrt [4]{2} \sqrt {x} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )\right )}{x \sqrt [4]{1+x^4}} \]

input

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

output

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

3.10.2.3 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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

input

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

output

$Aborted
 

3.10.2.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

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]
 

3.10.2.4 Maple [A] (verified)

Time = 11.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.40

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}} x -\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x +4 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2 x}\)	\(95\)

input

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

output

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

3.10.2.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.15 (sec) , antiderivative size = 339, normalized size of antiderivative = 4.99 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (-1+x^2\right )} \, dx=-\frac {2^{\frac {1}{4}} x \log \left (-\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x + 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} x \log \left (-\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x + 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) + i \cdot 2^{\frac {1}{4}} x \log \left (\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (i \, x^{5} + 2 i \, x^{3} + i \, x\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) - i \cdot 2^{\frac {1}{4}} x \log \left (\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (-i \, x^{5} - 2 i \, x^{3} - i \, x\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) - 8 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}}{4 \, x} \]

input

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

output

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

3.10.2.6 Sympy [F]

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

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

3.10.2.7 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)
 

3.10.2.8 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)
 

3.10.2.9 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)