\(\int \frac {1+x^8}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\) [2343]

Optimal result

Integrand size = 24, antiderivative size = 185 \[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=-\frac {\left (x^2+x^6\right )^{3/4}}{x \left (1+x^4\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}+\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{4\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{4\ 2^{3/4}} \] Output:

-(x^6+x^2)^(3/4)/x/(x^4+1)-1/8*arctan(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)+1 
/8*arctan(2^(3/4)*x*(x^6+x^2)^(1/4)/(2^(1/2)*x^2-(x^6+x^2)^(1/2)))*2^(1/4) 
-1/8*arctanh(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)-1/8*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^(1/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.17 \[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=-\frac {\sqrt {x} \left (8 \sqrt {x}+2^{3/4} \sqrt [4]{1+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \sqrt [4]{1+x^4} \arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )+2^{3/4} \sqrt [4]{1+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt [4]{2} \sqrt [4]{1+x^4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{8 \sqrt [4]{x^2+x^6}} \] Input:

Integrate[(1 + x^8)/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]
 

Output:

-1/8*(Sqrt[x]*(8*Sqrt[x] + 2^(3/4)*(1 + x^4)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x] 
)/(1 + x^4)^(1/4)] - 2^(1/4)*(1 + x^4)^(1/4)*ArcTan[(2^(3/4)*Sqrt[x]*(1 + 
x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] + 2^(3/4)*(1 + x^4)^(1/4)*ArcTanh 
[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] + 2^(1/4)*(1 + x^4)^(1/4)*ArcTanh[(2*2 
^(1/4)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + Sqrt[2]*Sqrt[1 + x^4])]))/(x^2 + x^ 
6)^(1/4)
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.78 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.54, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2467, 25, 1388, 2035, 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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4+1} \int -\frac {x^8+1}{\sqrt {x} \sqrt [4]{x^4+1} \left (1-x^8\right )}dx}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^8+1}{\sqrt {x} \sqrt [4]{x^4+1} \left (1-x^8\right )}dx}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 1388

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^8+1}{\sqrt {x} \left (1-x^4\right ) \left (x^4+1\right )^{5/4}}dx}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^8+1}{\left (1-x^4\right ) \left (x^4+1\right )^{5/4}}d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \left (2 \sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {5}{4},\frac {9}{8},x^4,-x^4\right )-\sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {5}{4},\frac {9}{8},-x^4\right )-\frac {1}{9} x^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},\frac {5}{4},\frac {17}{8},-x^4\right )\right )}{\sqrt [4]{x^6+x^2}}\)

Input:

Int[(1 + x^8)/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]
 

Output:

(-2*Sqrt[x]*(1 + x^4)^(1/4)*(2*Sqrt[x]*AppellF1[1/8, 1, 5/4, 9/8, x^4, -x^ 
4] - Sqrt[x]*Hypergeometric2F1[1/8, 5/4, 9/8, -x^4] - (x^(9/2)*Hypergeomet 
ric2F1[9/8, 5/4, 17/8, -x^4])/9))/(x^2 + x^6)^(1/4)
 

Defintions of rubi rules used

rule 25

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

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 2009

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

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 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 = 32.68 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.50

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-\ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+\ln \left (\frac {-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-16 x}{16 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\)	\(277\)
risch	\(\text {Expression too large to display}\)	\(647\)
trager	\(\text {Expression too large to display}\)	\(656\)

Input:

int((x^8+1)/(x^6+x^2)^(1/4)/(x^8-1),x,method=_RETURNVERBOSE)
 

Output:

1/16*(2*arctan(1/2*2^(3/4)*(x^2*(x^4+1))^(1/4)/x)*2^(3/4)*(x^2*(x^4+1))^(1 
/4)-ln((2^(1/4)*x+(x^2*(x^4+1))^(1/4))/(-2^(1/4)*x+(x^2*(x^4+1))^(1/4)))*2 
^(3/4)*(x^2*(x^4+1))^(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^(1/4)*(x^2*(x^4+1))^(1/4)+2*arctan((2^(1/4)*(x^2*(x^4+1))^(1 
/4)+x)/x)*2^(1/4)*(x^2*(x^4+1))^(1/4)+2*arctan((2^(1/4)*(x^2*(x^4+1))^(1/4 
)-x)/x)*2^(1/4)*(x^2*(x^4+1))^(1/4)-16*x)/(x^2*(x^4+1))^(1/4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (144) = 288\).

Time = 29.26 (sec) , antiderivative size = 669, normalized size of antiderivative = 3.62 \[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx =\text {Too large to display} \] Input:

integrate((x^8+1)/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="fricas")
 

Output:

-1/128*(2*8^(3/4)*(x^5 + x)*arctan((x^9 + 4*x^7 + 6*x^5 + 4*x^3 + 8^(3/4)* 
(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 4*sqrt(2)*sqrt(x^6 + x^2)*(x^5 
 + 2*x^3 + x) + 2*8^(1/4)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + x)/(x^9 - 
28*x^7 + 6*x^5 - 28*x^3 + x)) + 2*8^(3/4)*(x^5 + x)*arctan(-(x^9 + 4*x^7 + 
 6*x^5 + 4*x^3 - 8^(3/4)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 4*sqr 
t(2)*sqrt(x^6 + x^2)*(x^5 + 2*x^3 + x) - 2*8^(1/4)*(x^6 + x^2)^(3/4)*(x^4 
- 6*x^2 + 1) + x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) + 8*2^(3/4)*(x^5 + 
x)*arctan(-1/4*(2^(3/4)*(x^6 + x^2)^(1/4)*(x^4 + 1) - 2*2^(1/4)*(x^6 + x^2 
)^(3/4))/(x^5 + x)) + 4*2^(3/4)*(x^5 + 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)) - 4*2^(3/4)*(x^5 + 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)) + 8^(3/4)*(x^5 + x)*log(2*(4*8 
^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 8^(3/4)*(x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 
2*x^3 + x) + 8*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) - 8^(3/4)*(x^5 + x)*l 
og(-2*(4*8^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 8^(3/4)*(x^6 + x^2)^(3/4) - sqrt( 
2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) + 128*(x^6 
+ x^2)^(3/4))/(x^5 + x)
 

Sympy [F]

\[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int \frac {x^{8} + 1}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \] Input:

integrate((x**8+1)/(x**6+x**2)**(1/4)/(x**8-1),x)
 

Output:

Integral((x**8 + 1)/((x**2*(x**4 + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)*( 
x**4 + 1)), x)
 

Maxima [F]

\[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int { \frac {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^8+1)/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="maxima")
 

Output:

integrate((x^8 + 1)/((x^8 - 1)*(x^6 + x^2)^(1/4)), x)
 

Giac [F]

\[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int { \frac {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^8+1)/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="giac")
 

Output:

integrate((x^8 + 1)/((x^8 - 1)*(x^6 + x^2)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int \frac {x^8+1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^8-1\right )} \,d x \] Input:

int((x^8 + 1)/((x^2 + x^6)^(1/4)*(x^8 - 1)),x)
 

Output:

int((x^8 + 1)/((x^2 + x^6)^(1/4)*(x^8 - 1)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\frac {2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {5}{4}}+2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}} x^{4}+2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}}+5 \sqrt {x^{4}+1}\, \left (\int \frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+\sqrt {x}\, x^{8}-\sqrt {x}\, x^{4}-\sqrt {x}}d x \right ) x^{4}+5 \sqrt {x^{4}+1}\, \left (\int \frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+\sqrt {x}\, x^{8}-\sqrt {x}\, x^{4}-\sqrt {x}}d x \right )+\sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{7}}{x^{12}+x^{8}-x^{4}-1}d x \right ) x^{4}+\sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{7}}{x^{12}+x^{8}-x^{4}-1}d x \right )}{3 \sqrt {x^{4}+1}\, \left (x^{4}+1\right )} \] Input:

int((x^8+1)/(x^6+x^2)^(1/4)/(x^8-1),x)
 

Output:

(2*sqrt(x)*(x**4 + 1)**(5/4) + 2*sqrt(x)*(x**4 + 1)**(1/4)*x**4 + 2*sqrt(x 
)*(x**4 + 1)**(1/4) + 5*sqrt(x**4 + 1)*int((x**4 + 1)**(3/4)/(sqrt(x)*x**1 
2 + sqrt(x)*x**8 - sqrt(x)*x**4 - sqrt(x)),x)*x**4 + 5*sqrt(x**4 + 1)*int( 
(x**4 + 1)**(3/4)/(sqrt(x)*x**12 + sqrt(x)*x**8 - sqrt(x)*x**4 - sqrt(x)), 
x) + sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4)*x**7)/(x**12 + x**8 - x 
**4 - 1),x)*x**4 + sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4)*x**7)/(x* 
*12 + x**8 - x**4 - 1),x))/(3*sqrt(x**4 + 1)*(x**4 + 1))