Integrand size = 24, antiderivative size = 162 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 2^{3/8}}+\frac {\arctan \left (\frac {2^{5/8} x \sqrt [4]{x^2+x^6}}{\sqrt [4]{2} x^2-\sqrt {x^2+x^6}}\right )}{2\ 2^{7/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 2^{3/8}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^2+x^6}}{2^{5/8}}}{x \sqrt [4]{x^2+x^6}}\right )}{2\ 2^{7/8}} \] Output:
1/4*arctan(2^(1/8)*x/(x^6+x^2)^(1/4))*2^(5/8)+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^(1/8)-1/4*arctanh(2^(1/8)*x/(x^6 +x^2)^(1/4))*2^(5/8)+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^(1/8)
Time = 0.83 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (\sqrt {2} \arctan \left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\arctan \left (\frac {2^{5/8} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt [4]{2} x-\sqrt {1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/8} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+2^{3/4} \sqrt {1+x^4}}\right )\right )}{2\ 2^{7/8} \sqrt {x} \sqrt [4]{1+x^4}} \] Input:
Integrate[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^8),x]
Output:
((x^2 + x^6)^(1/4)*(Sqrt[2]*ArcTan[(2^(1/8)*Sqrt[x])/(1 + x^4)^(1/4)] + Ar cTan[(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*2^(7/8)*Sqrt[x]*(1 + x^4)^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2467, 25, 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.
\(\displaystyle \int \frac {\left (x^4-1\right ) \sqrt [4]{x^6+x^2}}{x^8+1} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x^6+x^2} \int -\frac {\sqrt {x} \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+1}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x^6+x^2} \int \frac {\sqrt {x} \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+1}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {x \sqrt [4]{x^4+1}}{x^8+1}-\frac {x^5 \sqrt [4]{x^4+1}}{x^8+1}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \left (\frac {1}{6} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},-i x^4,-x^4\right )+\frac {1}{6} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},i x^4,-x^4\right )+\left (\frac {1}{6}-\frac {i}{6}\right ) x^{3/2} \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {3}{4},\frac {11}{8},-i x^4,-x^4\right )+\left (\frac {1}{6}+\frac {i}{6}\right ) x^{3/2} \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {3}{4},\frac {11}{8},i x^4,-x^4\right )-\frac {1}{3} x^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {3}{4},\frac {11}{8},-x^4\right )\right )}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
Input:
Int[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^8),x]
Output:
(-2*(x^2 + x^6)^(1/4)*((x^(3/2)*AppellF1[3/8, 1, -1/4, 11/8, (-I)*x^4, -x^ 4])/6 + (x^(3/2)*AppellF1[3/8, 1, -1/4, 11/8, I*x^4, -x^4])/6 + (1/6 - I/6 )*x^(3/2)*AppellF1[3/8, 1, 3/4, 11/8, (-I)*x^4, -x^4] + (1/6 + I/6)*x^(3/2 )*AppellF1[3/8, 1, 3/4, 11/8, I*x^4, -x^4] - (x^(3/2)*Hypergeometric2F1[3/ 8, 3/4, 11/8, -x^4])/3))/(Sqrt[x]*(1 + x^4)^(1/4))
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 = 85.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3}}\right )}{4}\) | \(37\) |
trager | \(\text {Expression too large to display}\) | \(2867\) |
Input:
int((x^4-1)*(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^3,_R=RootOf(_Z^8-2))
Result contains complex when optimal does not.
Time = 8.11 (sec) , antiderivative size = 1135, normalized size of antiderivative = 7.01 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\text {Too large to display} \] Input:
integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8+1),x, algorithm="fricas")
Output:
(1/128*I - 1/128)*8^(7/8)*sqrt(2)*log(-(2*8^(3/4)*(I*x^7 + I*x^3) - (x^6 + x^2)^(3/4)*((4*I + 4)*8^(1/8)*sqrt(2)*x^2 + 8^(5/8)*sqrt(2)*(-(I + 1)*x^4 - I - 1)) - 8*sqrt(x^6 + x^2)*(x^5 - sqrt(2)*x^3 + x) + 2*8^(1/4)*(-I*x^9 - 4*I*x^5 - I*x) - (x^6 + x^2)^(1/4)*(-(I - 1)*8^(7/8)*sqrt(2)*x^4 - 2*8^ (3/8)*sqrt(2)*(-(I - 1)*x^6 - (I - 1)*x^2)))/(x^9 + x)) - (1/128*I - 1/128 )*8^(7/8)*sqrt(2)*log(-(2*8^(3/4)*(I*x^7 + I*x^3) - (x^6 + x^2)^(3/4)*(-(4 *I + 4)*8^(1/8)*sqrt(2)*x^2 + 8^(5/8)*sqrt(2)*((I + 1)*x^4 + I + 1)) - 8*s qrt(x^6 + x^2)*(x^5 - sqrt(2)*x^3 + x) + 2*8^(1/4)*(-I*x^9 - 4*I*x^5 - I*x ) - (x^6 + x^2)^(1/4)*((I - 1)*8^(7/8)*sqrt(2)*x^4 - 2*8^(3/8)*sqrt(2)*((I - 1)*x^6 + (I - 1)*x^2)))/(x^9 + x)) - (1/128*I + 1/128)*8^(7/8)*sqrt(2)* log(-(2*8^(3/4)*(-I*x^7 - I*x^3) - (x^6 + x^2)^(3/4)*(-(4*I - 4)*8^(1/8)*s qrt(2)*x^2 + 8^(5/8)*sqrt(2)*((I - 1)*x^4 + I - 1)) - 8*sqrt(x^6 + x^2)*(x ^5 - sqrt(2)*x^3 + x) + 2*8^(1/4)*(I*x^9 + 4*I*x^5 + I*x) - (x^6 + x^2)^(1 /4)*((I + 1)*8^(7/8)*sqrt(2)*x^4 - 2*8^(3/8)*sqrt(2)*((I + 1)*x^6 + (I + 1 )*x^2)))/(x^9 + x)) + (1/128*I + 1/128)*8^(7/8)*sqrt(2)*log(-(2*8^(3/4)*(- I*x^7 - I*x^3) - (x^6 + x^2)^(3/4)*((4*I - 4)*8^(1/8)*sqrt(2)*x^2 + 8^(5/8 )*sqrt(2)*(-(I - 1)*x^4 - I + 1)) - 8*sqrt(x^6 + x^2)*(x^5 - sqrt(2)*x^3 + x) + 2*8^(1/4)*(I*x^9 + 4*I*x^5 + I*x) - (x^6 + x^2)^(1/4)*(-(I + 1)*8^(7 /8)*sqrt(2)*x^4 - 2*8^(3/8)*sqrt(2)*(-(I + 1)*x^6 - (I + 1)*x^2)))/(x^9 + x)) - 1/64*8^(7/8)*log((8^(3/4)*(x^7 + x^3) + (x^6 + x^2)^(3/4)*(4*8^(1...
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{8} + 1}\, dx \] Input:
integrate((x**4-1)*(x**6+x**2)**(1/4)/(x**8+1),x)
Output:
Integral((x**2*(x**4 + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)/(x**8 + 1), x )
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + 1} \,d x } \] Input:
integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8+1),x, algorithm="maxima")
Output:
integrate((x^6 + x^2)^(1/4)*(x^4 - 1)/(x^8 + 1), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + 1} \,d x } \] Input:
integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8+1),x, algorithm="giac")
Output:
integrate((x^6 + x^2)^(1/4)*(x^4 - 1)/(x^8 + 1), x)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8+1} \,d x \] Input:
int(((x^2 + x^6)^(1/4)*(x^4 - 1))/(x^8 + 1),x)
Output:
int(((x^2 + x^6)^(1/4)*(x^4 - 1))/(x^8 + 1), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx=\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}} x^{4}}{x^{8}+1}d x -\left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{8}+1}d x \right ) \] Input:
int((x^4-1)*(x^6+x^2)^(1/4)/(x^8+1),x)
Output:
int((sqrt(x)*(x**4 + 1)**(1/4)*x**4)/(x**8 + 1),x) - int((sqrt(x)*(x**4 + 1)**(1/4))/(x**8 + 1),x)