Integrand size = 26, antiderivative size = 452 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=-\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \arctan \left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+\sqrt {2} \sqrt {-x^2+x^6}}\right )-\frac {1}{4} \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-\sqrt {2} \sqrt {-x^2+x^6}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} x \sqrt [4]{-x^2+x^6}+\sqrt {4-2 \sqrt {2}} \sqrt {-x^2+x^6}\right ) \]
-1/4*(2^(1/2)-1)^(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)))-1/4*(2^(1/2)-1)^(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)))-1/4*(1+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)))-1/4*(2^(1/2)-1)^(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))+1/8*(1+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))-1/8*(1+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 = 0.70 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.36 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=-\frac {\sqrt {x} \sqrt [4]{-1+x^4} \left (\sqrt {-1+i} \arctan \left (\frac {\sqrt {-1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {-1-i} \arctan \left (\frac {\sqrt {-1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {1+i} \arctan \left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {1-i} \arctan \left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )\right )}{2 \sqrt {2} \sqrt [4]{x^2 \left (-1+x^4\right )}} \]
-1/2*(Sqrt[x]*(-1 + x^4)^(1/4)*(Sqrt[-1 + I]*ArcTan[(Sqrt[-1 - I]*Sqrt[x]) /(-1 + x^4)^(1/4)] + Sqrt[-1 - I]*ArcTan[(Sqrt[-1 + I]*Sqrt[x])/(-1 + x^4) ^(1/4)] + Sqrt[1 + I]*ArcTan[(Sqrt[1 - I]*Sqrt[x])/(-1 + x^4)^(1/4)] + Sqr t[1 - I]*ArcTan[(Sqrt[1 + I]*Sqrt[x])/(-1 + x^4)^(1/4)]))/(Sqrt[2]*(x^2*(- 1 + x^4))^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2467, 966, 937, 936}
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.
| \(\displaystyle \int \frac {x^4-1}{\left (x^4+1\right ) \sqrt [4]{x^6-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4-1} \int \frac {\left (x^4-1\right )^{3/4}}{\sqrt {x} \left (x^4+1\right )}dx}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^4-1} \int \frac {\left (x^4-1\right )^{3/4}}{x^4+1}d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {2 \sqrt {x} \left (x^4-1\right ) \int \frac {\left (1-x^4\right )^{3/4}}{x^4+1}d\sqrt {x}}{\left (1-x^4\right )^{3/4} \sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {2 x \left (x^4-1\right ) \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},x^4,-x^4\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{x^6-x^2}}\) |
(2*x*(-1 + x^4)*AppellF1[1/8, -3/4, 1, 9/8, x^4, -x^4])/((1 - x^4)^(3/4)*( -x^2 + x^6)^(1/4))
3.31.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
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]
Time = 52.36 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {\left (2+\sqrt {2}\right ) \left (\ln \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}+2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )-\ln \left (\frac {-\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}-2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )\right ) \sqrt {-2+2 \sqrt {2}}}{16}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (-2+\sqrt {2}\right ) \left (2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )+\ln \left (\frac {\sqrt {2}\, x^{2}-x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )\right )}{16}\) | \(390\) |
| trager | \(\text {Expression too large to display}\) | \(2859\) |
-1/16*(2+2^(1/2))*(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)-1/16*(2+2*2^(1/2))^(1/2)*( -2+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)-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((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))
Result contains complex when optimal does not.
Time = 66.62 (sec) , antiderivative size = 1141, normalized size of antiderivative = 2.52 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\text {Too large to display} \]
-1/16*sqrt(2)*sqrt(-I + 1)*log(-(2*sqrt(2)*sqrt(-I + 1)*sqrt(x^6 - x^2)*(( 35880521239*I + 106079915533)*x^5 + (212159831066*I - 71761042478)*x^3 - ( 35880521239*I + 106079915533)*x) - sqrt(2)*sqrt(-I + 1)*((35099697147*I - 70980218386)*x^9 - (283920873544*I + 140398788588)*x^7 - (210598182882*I - 425881310316)*x^5 + (283920873544*I + 140398788588)*x^3 + (35099697147*I - 70980218386)*x) + 4*(x^6 - x^2)^(3/4)*(-(35099697147*I - 70980218386)*x^ 4 + (141960436772*I + 70199394294)*x^2 + 35099697147*I - 70980218386) + 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)) + 1/16*sqrt(2)*sqrt(-I - 1)*log(-(2*sqrt(2)*sqrt(-I - 1)*sqrt(x^6 - x^2)* (-(35880521239*I - 106079915533)*x^5 - (212159831066*I + 71761042478)*x^3 + (35880521239*I - 106079915533)*x) - sqrt(2)*sqrt(-I - 1)*((35099697147*I + 70980218386)*x^9 - (283920873544*I - 140398788588)*x^7 - (210598182882* I + 425881310316)*x^5 + (283920873544*I - 140398788588)*x^3 + (35099697147 *I + 70980218386)*x) + 4*(x^6 - x^2)^(3/4)*((35099697147*I + 70980218386)* x^4 - (141960436772*I - 70199394294)*x^2 - 35099697147*I - 70980218386) + 4*(x^6 - x^2)^(1/4)*((35880521239*I - 106079915533)*x^6 + (212159831066*I + 71761042478)*x^4 - (35880521239*I - 106079915533)*x^2))/(x^9 + 2*x^5 + x )) - 1/16*sqrt(2)*sqrt(-I - 1)*log(-(2*sqrt(2)*sqrt(-I - 1)*sqrt(x^6 - x^2 )*((35880521239*I - 106079915533)*x^5 + (212159831066*I + 71761042478)*...
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}\, dx \]
Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**2*(x - 1)*(x + 1)*(x**2 + 1))**(1 /4)*(x**4 + 1)), x)
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {x^4-1}{\left (x^4+1\right )\,{\left (x^6-x^2\right )}^{1/4}} \,d x \]