Integrand size = 24, antiderivative size = 162 \[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [8]{2}}+\frac {\arctan \left (\frac {2^{5/8} x \sqrt [4]{x^2+x^6}}{\sqrt [4]{2} x^2-\sqrt {x^2+x^6}}\right )}{2\ 2^{5/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [8]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^2+x^6}}{2^{5/8}}}{x \sqrt [4]{x^2+x^6}}\right )}{2\ 2^{5/8}} \] Output:
-1/4*arctan(2^(1/8)*x/(x^6+x^2)^(1/4))*2^(7/8)+1/4*arctan(2^(5/8)*x*(x^6+x ^2)^(1/4)/(x^2*2^(1/4)-(x^6+x^2)^(1/2)))*2^(3/8)-1/4*arctanh(2^(1/8)*x/(x^ 6+x^2)^(1/4))*2^(7/8)-1/4*arctanh((1/2*x^2*2^(5/8)+1/2*(x^6+x^2)^(1/2)*2^( 3/8))/x/(x^6+x^2)^(1/4))*2^(3/8)
Time = 0.92 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=-\frac {\sqrt {x} \sqrt [4]{1+x^4} \left (\sqrt {2} \arctan \left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\arctan \left (\frac {2^{5/8} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt [4]{2} x-\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/8} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+2^{3/4} \sqrt {1+x^4}}\right )\right )}{2\ 2^{5/8} \sqrt [4]{x^2+x^6}} \] Input:
Integrate[(-1 + x^8)/((x^2 + x^6)^(1/4)*(1 + x^8)),x]
Output:
-1/2*(Sqrt[x]*(1 + x^4)^(1/4)*(Sqrt[2]*ArcTan[(2^(1/8)*Sqrt[x])/(1 + x^4)^ (1/4)] - ArcTan[(2^(5/8)*Sqrt[x]*(1 + x^4)^(1/4))/(2^(1/4)*x - Sqrt[1 + x^ 4])] + Sqrt[2]*ArcTanh[(2^(1/8)*Sqrt[x])/(1 + x^4)^(1/4)] + ArcTanh[(2*2^( 3/8)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + 2^(3/4)*Sqrt[1 + x^4])]))/(2^(5/8)*(x ^2 + x^6)^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.62, 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 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4+1} \int -\frac {1-x^8}{\sqrt {x} \sqrt [4]{x^4+1} \left (x^8+1\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {1-x^8}{\sqrt {x} \sqrt [4]{x^4+1} \left (x^8+1\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {\left (1-x^4\right ) \left (x^4+1\right )^{3/4}}{\sqrt {x} \left (x^8+1\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \frac {\left (1-x^4\right ) \left (x^4+1\right )^{3/4}}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x^4+1\right )^{3/4}}{i-x^4}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x^4+1\right )^{3/4}}{x^4+i}\right )d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},-i x^4,-x^4\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},i x^4,-x^4\right )\right )}{\sqrt [4]{x^6+x^2}}\) |
Input:
Int[(-1 + x^8)/((x^2 + x^6)^(1/4)*(1 + x^8)),x]
Output:
(-2*Sqrt[x]*(1 + x^4)^(1/4)*((1/2 - I/2)*Sqrt[x]*AppellF1[1/8, 1, -3/4, 9/ 8, (-I)*x^4, -x^4] + (1/2 + I/2)*Sqrt[x]*AppellF1[1/8, 1, -3/4, 9/8, I*x^4 , -x^4]))/(x^2 + x^6)^(1/4)
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]))
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 98.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\) | \(37\) |
trager | \(\text {Expression too large to display}\) | \(2800\) |
Input:
int((x^8-1)/(x^6+x^2)^(1/4)/(x^8+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(ln((-_R*x+(x^2*(x^4+1))^(1/4))/x)/_R,_R=RootOf(_Z^8-2))
Result contains complex when optimal does not.
Time = 68.39 (sec) , antiderivative size = 1624, normalized size of antiderivative = 10.02 \[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=\text {Too large to display} \] Input:
integrate((x^8-1)/(x^6+x^2)^(1/4)/(x^8+1),x, algorithm="fricas")
Output:
-1/16*2^(7/8)*log(-(2^(7/8)*(x^9 - 2*x^7 + 4*x^5 - 2*x^3 + x) + 2*(x^6 + x ^2)^(3/4)*(2*x^4 - 2*x^2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x ^2)*(2^(5/8)*(x^5 - 2*x^3 + x) - 2*2^(1/8)*(x^5 - x^3 + x)) - 2^(3/8)*(x^9 - 4*x^7 + 4*x^5 - 4*x^3 + x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) + 1/16*2^(7/8)*log((2^(7 /8)*(x^9 - 2*x^7 + 4*x^5 - 2*x^3 + x) - 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2*x^2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(x^5 - 2*x^ 3 + x) - 2*2^(1/8)*(x^5 - x^3 + x)) - 2^(3/8)*(x^9 - 4*x^7 + 4*x^5 - 4*x^3 + x) + 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) + 1/16*I*2^(7/8)*log((2^(7/8)*(I*x^9 - 2*I*x^7 + 4*I*x^5 - 2*I*x^3 + I*x) - 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2*x^2 - sqrt(2)*( x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(-I*x^5 + 2*I*x^3 - I*x ) + 2*2^(1/8)*(I*x^5 - I*x^3 + I*x)) + 2^(3/8)*(-I*x^9 + 4*I*x^7 - 4*I*x^5 + 4*I*x^3 - I*x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2 ^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) - 1/16*I*2^(7/8)*log((2^(7/8)*(-I*x^ 9 + 2*I*x^7 - 4*I*x^5 + 2*I*x^3 - I*x) - 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2*x^ 2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(I*x^5 - 2 *I*x^3 + I*x) + 2*2^(1/8)*(-I*x^5 + I*x^3 - I*x)) + 2^(3/8)*(I*x^9 - 4*I*x ^7 + 4*I*x^5 - 4*I*x^3 + I*x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) + (1/16*I + 1/16)*2^(...
\[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{8} + 1\right )}\, dx \] Input:
integrate((x**8-1)/(x**6+x**2)**(1/4)/(x**8+1),x)
Output:
Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)/((x**2*(x**4 + 1))**(1/4)*( x**8 + 1)), x)
\[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \] 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)
\[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \] 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)
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)
\[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=\frac {\frac {4 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {5}{4}}}{3}+\frac {2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}} x^{4}}{3}+\frac {2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}}}{3}-2 \sqrt {x^{4}+1}\, \left (\int \frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{16}+2 \sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{8}+2 \sqrt {x}\, x^{4}+\sqrt {x}}d x \right ) x^{4}-2 \sqrt {x^{4}+1}\, \left (\int \frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{16}+2 \sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{8}+2 \sqrt {x}\, x^{4}+\sqrt {x}}d x \right )+\frac {2 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{11}}{x^{16}+2 x^{12}+2 x^{8}+2 x^{4}+1}d x \right ) x^{4}}{3}+\frac {2 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{11}}{x^{16}+2 x^{12}+2 x^{8}+2 x^{4}+1}d x \right )}{3}-\frac {4 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{3}}{x^{16}+2 x^{12}+2 x^{8}+2 x^{4}+1}d x \right ) x^{4}}{3}-\frac {4 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{3}}{x^{16}+2 x^{12}+2 x^{8}+2 x^{4}+1}d x \right )}{3}}{\sqrt {x^{4}+1}\, \left (x^{4}+1\right )} \] Input:
int((x^8-1)/(x^6+x^2)^(1/4)/(x^8+1),x)
Output:
(2*(2*sqrt(x)*(x**4 + 1)**(5/4) + sqrt(x)*(x**4 + 1)**(1/4)*x**4 + sqrt(x) *(x**4 + 1)**(1/4) - 3*sqrt(x**4 + 1)*int((x**4 + 1)**(3/4)/(sqrt(x)*x**16 + 2*sqrt(x)*x**12 + 2*sqrt(x)*x**8 + 2*sqrt(x)*x**4 + sqrt(x)),x)*x**4 - 3*sqrt(x**4 + 1)*int((x**4 + 1)**(3/4)/(sqrt(x)*x**16 + 2*sqrt(x)*x**12 + 2*sqrt(x)*x**8 + 2*sqrt(x)*x**4 + sqrt(x)),x) + sqrt(x**4 + 1)*int((sqrt(x )*(x**4 + 1)**(3/4)*x**11)/(x**16 + 2*x**12 + 2*x**8 + 2*x**4 + 1),x)*x**4 + sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4)*x**11)/(x**16 + 2*x**12 + 2*x**8 + 2*x**4 + 1),x) - 2*sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4) *x**3)/(x**16 + 2*x**12 + 2*x**8 + 2*x**4 + 1),x)*x**4 - 2*sqrt(x**4 + 1)* int((sqrt(x)*(x**4 + 1)**(3/4)*x**3)/(x**16 + 2*x**12 + 2*x**8 + 2*x**4 + 1),x)))/(3*sqrt(x**4 + 1)*(x**4 + 1))