Integrand size = 32, antiderivative size = 501 \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\frac {2 \left (-1+x^4\right ) \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+\sqrt {2} \sqrt {-x^2+x^6}}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-\sqrt {2} \sqrt {-x^2+x^6}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} x \sqrt [4]{-x^2+x^6}+\sqrt {4-2 \sqrt {2}} \sqrt {-x^2+x^6}\right ) \]
2/5*(x^4-1)*(x^6-x^2)^(1/4)/x^3+1/8*(2+2*2^(1/2))^(1/2)*arctan((2-2^(1/2)) ^(1/2)*x/(-(2+2^(1/2))^(1/2)*x+2^(3/4)*(x^6-x^2)^(1/4)))+1/8*(2+2*2^(1/2)) ^(1/2)*arctan((2-2^(1/2))^(1/2)*x/((2+2^(1/2))^(1/2)*x+2^(3/4)*(x^6-x^2)^( 1/4)))-1/8*(-2+2*2^(1/2))^(1/2)*arctan(2^(3/4)*(2+2^(1/2))^(1/2)*x*(x^6-x^ 2)^(1/4)/(-2*x^2+2^(1/2)*(x^6-x^2)^(1/2)))-1/8*(2+2*2^(1/2))^(1/2)*arctanh ((2^(1/4)*x^2/(2-2^(1/2))^(1/2)+1/2*(x^6-x^2)^(1/2)*2^(3/4)/(2-2^(1/2))^(1 /2))/x/(x^6-x^2)^(1/4))-1/16*(-2+2*2^(1/2))^(1/2)*ln(-2*x^2+2^(3/4)*(2+2^( 1/2))^(1/2)*x*(x^6-x^2)^(1/4)-2^(1/2)*(x^6-x^2)^(1/2))+1/16*(-2+2*2^(1/2)) ^(1/2)*ln(2*(2-2^(1/2))^(1/2)*x^2+2*2^(1/4)*x*(x^6-x^2)^(1/4)+(4-2*2^(1/2) )^(1/2)*(x^6-x^2)^(1/2))
Result contains complex when optimal does not.
Time = 0.00 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\frac {\sqrt [4]{x^2 \left (-1+x^4\right )} \left (-8 \sqrt [4]{-1+x^4}+8 x^4 \sqrt [4]{-1+x^4}+5 \sqrt {1+i} x^{5/2} \arctan \left (\frac {\sqrt {-1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+5 \sqrt {1-i} x^{5/2} \arctan \left (\frac {\sqrt {-1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-5 \sqrt {-1+i} x^{5/2} \arctan \left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-5 \sqrt {-1-i} x^{5/2} \arctan \left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )\right )}{20 x^3 \sqrt [4]{-1+x^4}} \]
((x^2*(-1 + x^4))^(1/4)*(-8*(-1 + x^4)^(1/4) + 8*x^4*(-1 + x^4)^(1/4) + 5* Sqrt[1 + I]*x^(5/2)*ArcTan[(Sqrt[-1 - I]*Sqrt[x])/(-1 + x^4)^(1/4)] + 5*Sq rt[1 - I]*x^(5/2)*ArcTan[(Sqrt[-1 + I]*Sqrt[x])/(-1 + x^4)^(1/4)] - 5*Sqrt [-1 + I]*x^(5/2)*ArcTan[(Sqrt[1 - I]*Sqrt[x])/(-1 + x^4)^(1/4)] - 5*Sqrt[- 1 - I]*x^(5/2)*ArcTan[(Sqrt[1 + I]*Sqrt[x])/(-1 + x^4)^(1/4)]))/(20*x^3*(- 1 + x^4)^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.66 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2467, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^6-x^2} \left (x^8+x^4+1\right )}{x^4 \left (x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^6-x^2} \int \frac {\sqrt [4]{x^4-1} \left (x^8+x^4+1\right )}{x^{7/2} \left (x^4+1\right )}dx}{\sqrt {x} \sqrt [4]{x^4-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt [4]{x^6-x^2} \int \frac {\sqrt [4]{x^4-1} \left (x^8+x^4+1\right )}{x^3 \left (x^4+1\right )}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4-1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt [4]{x^6-x^2} \int \left (\sqrt [4]{x^4-1} x-\frac {\sqrt [4]{x^4-1} x}{x^4+1}+\frac {\sqrt [4]{x^4-1}}{x^3}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt [4]{x^6-x^2} \left (-\frac {\sqrt [4]{x^4-1} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},x^4,-x^4\right )}{3 \sqrt [4]{1-x^4}}+\frac {\sqrt [4]{x^4-1} x^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{8},\frac {11}{8},x^4\right )}{3 \sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1} \operatorname {Hypergeometric2F1}\left (-\frac {5}{8},-\frac {1}{4},\frac {3}{8},x^4\right )}{5 \sqrt [4]{1-x^4} x^{5/2}}\right )}{\sqrt {x} \sqrt [4]{x^4-1}}\) |
(2*(-x^2 + x^6)^(1/4)*(-1/3*(x^(3/2)*(-1 + x^4)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, x^4, -x^4])/(1 - x^4)^(1/4) - ((-1 + x^4)^(1/4)*Hypergeometric2F1 [-5/8, -1/4, 3/8, x^4])/(5*x^(5/2)*(1 - x^4)^(1/4)) + (x^(3/2)*(-1 + x^4)^ (1/4)*Hypergeometric2F1[-1/4, 3/8, 11/8, x^4])/(3*(1 - x^4)^(1/4))))/(Sqrt [x]*(-1 + x^4)^(1/4))
3.31.77.3.1 Defintions of rubi rules used
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]
Time = 132.22 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.81
| method | result | size |
| pseudoelliptic | \(\frac {32 \left (x^{4}-1\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+5 x^{3} \left (\left (-\ln \left (\frac {-\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}-2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )+\ln \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}+2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )\right ) \sqrt {-2+2 \sqrt {2}}-\sqrt {2+2 \sqrt {2}}\, \left (2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {\sqrt {2}\, x^{2}-x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+\ln \left (\frac {x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )\right )\right )}{80 x^{3}}\) | \(408\) |
| trager | \(\text {Expression too large to display}\) | \(2950\) |
| risch | \(\text {Expression too large to display}\) | \(7288\) |
1/80*(32*(x^4-1)*(x^6-x^2)^(1/4)+5*x^3*((-ln((-(2+2*2^(1/2))^(1/2)*x*(x^6- x^2)^(1/4)+2^(1/2)*x^2+(x^6-x^2)^(1/2))/x^2)-2*arctan((x*(-2+2*2^(1/2))^(1 /2)-2*(x^6-x^2)^(1/4))/(2+2*2^(1/2))^(1/2)/x)+ln(((2+2*2^(1/2))^(1/2)*x*(x ^6-x^2)^(1/4)+2^(1/2)*x^2+(x^6-x^2)^(1/2))/x^2)+2*arctan((x*(-2+2*2^(1/2)) ^(1/2)+2*(x^6-x^2)^(1/4))/(2+2*2^(1/2))^(1/2)/x))*(-2+2*2^(1/2))^(1/2)-(2+ 2*2^(1/2))^(1/2)*(2*arctan(((2+2*2^(1/2))^(1/2)*x+2*(x^6-x^2)^(1/4))/x/(-2 +2*2^(1/2))^(1/2))-ln((2^(1/2)*x^2-x*(-2+2*2^(1/2))^(1/2)*(x^6-x^2)^(1/4)+ (x^6-x^2)^(1/2))/x^2)+ln((x*(-2+2*2^(1/2))^(1/2)*(x^6-x^2)^(1/4)+2^(1/2)*x ^2+(x^6-x^2)^(1/2))/x^2)-2*arctan(((2+2*2^(1/2))^(1/2)*x-2*(x^6-x^2)^(1/4) )/x/(-2+2*2^(1/2))^(1/2)))))/x^3
Result contains complex when optimal does not.
Time = 20.76 (sec) , antiderivative size = 1108, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\text {Too large to display} \]
1/80*(5*sqrt(-I - 1)*x^3*log(-(4*sqrt(-I - 1)*sqrt(x^6 - x^2)*((7098021838 6*I + 35099697147)*x^5 + (70199394294*I - 141960436772)*x^3 - (70980218386 *I + 35099697147)*x) + 4*(x^6 - x^2)^(3/4)*(-(35099697147*I - 70980218386) *x^4 + (141960436772*I + 70199394294)*x^2 + 35099697147*I - 70980218386) - sqrt(-I - 1)*(-(35880521239*I + 106079915533)*x^9 - (424319662132*I - 143 522084956)*x^7 + (215283127434*I + 636479493198)*x^5 + (424319662132*I - 1 43522084956)*x^3 - (35880521239*I + 106079915533)*x) + 4*((35880521239*I + 106079915533)*x^6 + (212159831066*I - 71761042478)*x^4 - (35880521239*I + 106079915533)*x^2)*(x^6 - x^2)^(1/4))/(x^9 + 2*x^5 + x)) + 5*sqrt(-I + 1) *x^3*log(-(4*sqrt(-I + 1)*sqrt(x^6 - x^2)*(-(70980218386*I - 35099697147)* x^5 - (70199394294*I + 141960436772)*x^3 + (70980218386*I - 35099697147)*x ) + 4*(x^6 - x^2)^(3/4)*((35099697147*I + 70980218386)*x^4 - (141960436772 *I - 70199394294)*x^2 - 35099697147*I - 70980218386) - sqrt(-I + 1)*(-(358 80521239*I - 106079915533)*x^9 - (424319662132*I + 143522084956)*x^7 + (21 5283127434*I - 636479493198)*x^5 + (424319662132*I + 143522084956)*x^3 - ( 35880521239*I - 106079915533)*x) + 4*(x^6 - x^2)^(1/4)*((35880521239*I - 1 06079915533)*x^6 + (212159831066*I + 71761042478)*x^4 - (35880521239*I - 1 06079915533)*x^2))/(x^9 + 2*x^5 + x)) - 5*sqrt(-I + 1)*x^3*log(-(4*sqrt(-I + 1)*sqrt(x^6 - x^2)*((70980218386*I - 35099697147)*x^5 + (70199394294*I + 141960436772)*x^3 - (70980218386*I - 35099697147)*x) + 4*(x^6 - x^2)^...
\[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{4} \left (x^{4} + 1\right )}\, dx \]
Integral((x**2*(x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1)/(x**4*(x**4 + 1)), x)
\[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\int { \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}}}{{\left (x^{4} + 1\right )} x^{4}} \,d x } \]
\[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\int { \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}}}{{\left (x^{4} + 1\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1+x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\int \frac {{\left (x^6-x^2\right )}^{1/4}\,\left (x^8+x^4+1\right )}{x^4\,\left (x^4+1\right )} \,d x \]