Integrand size = 22, antiderivative size = 187 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\frac {\left (x^2+x^6\right )^{3/4}}{2 x \left (1+x^4\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}+\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{8\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{8\ 2^{3/4}} \]
1/2*(x^6+x^2)^(3/4)/x/(x^4+1)-1/16*arctan(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/ 4)+1/16*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)-1/16*arctanh(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)-1/16*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)
Time = 0.84 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.17 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\frac {\sqrt {x} \left (8 \sqrt {x}-2^{3/4} \sqrt [4]{1+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt [4]{2} \sqrt [4]{1+x^4} \arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )-2^{3/4} \sqrt [4]{1+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \sqrt [4]{1+x^4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{16 \sqrt [4]{x^2+x^6}} \]
(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])]))/(16*(x^2 + x^6 )^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.25, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2467, 25, 1388, 966, 1012}
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.
| \(\displaystyle \int \frac {x^4}{\sqrt [4]{x^6+x^2} \left (x^8-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4+1} \int -\frac {x^{7/2}}{\sqrt [4]{x^4+1} \left (1-x^8\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^{7/2}}{\sqrt [4]{x^4+1} \left (1-x^8\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^{7/2}}{\left (1-x^4\right ) \left (x^4+1\right )^{5/4}}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^4}{\left (1-x^4\right ) \left (x^4+1\right )^{5/4}}d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle -\frac {2 x^5 \sqrt [4]{x^4+1} \operatorname {AppellF1}\left (\frac {9}{8},1,\frac {5}{4},\frac {17}{8},x^4,-x^4\right )}{9 \sqrt [4]{x^6+x^2}}\) |
3.24.60.3.1 Defintions of rubi rules used
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
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_.)*(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 = 53.65 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.50
| method | result | size |
| pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+\ln \left (\frac {-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+16 x}{32 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\) | \(281\) |
| risch | \(\text {Expression too large to display}\) | \(646\) |
| trager | \(\text {Expression too large to display}\) | \(655\) |
1/32*(2*arctan(1/2*(x^2*(x^4+1))^(1/4)/x*2^(3/4))*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)
Result contains complex when optimal does not.
Time = 3.71 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.84 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + 2^{\frac {3}{4}} {\left (-i \, x^{5} - i \, x\right )} \log \left (\frac {4 i \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2 i \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + 2^{\frac {3}{4}} {\left (i \, x^{5} + i \, x\right )} \log \left (\frac {-4 i \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 i \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x\right )} \log \left (\frac {\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (-i \, x^{5} + 2 i \, x^{3} - i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x - \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (-\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x\right )} \log \left (\frac {-\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (i \, x^{5} - 2 i \, x^{3} + i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x\right )} \log \left (\frac {\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (i \, x^{5} - 2 i \, x^{3} + i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x - \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (-\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x\right )} \log \left (\frac {-\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (-i \, x^{5} + 2 i \, x^{3} - i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - 32 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{64 \, {\left (x^{5} + x\right )}} \]
-1/64*(2^(3/4)*(x^5 + x)*log((4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)* (x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) - 2^(3/4)*(x^5 + x)*log(-(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) - 4*sqrt(x^6 + x^2 )*x)/(x^5 - 2*x^3 + x)) + 2^(3/4)*(-I*x^5 - I*x)*log((4*I*2^(1/4)*(x^6 + x ^2)^(1/4)*x^2 - 2*I*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) + 2^(3/4)*(I*x^5 + I*x)*log((-4* I*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*I*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)* (x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) - 2^(1/4)*((I - 1)*x^5 + (I - 1)*x)*log(((2*I + 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt( 2)*(-I*x^5 + 2*I*x^3 - I*x) + 4*sqrt(x^6 + x^2)*x - (2*I - 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*(-(I + 1)*x^5 - (I + 1)*x)*log ((-(2*I - 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(I*x^5 - 2*I*x^3 + I* x) + 4*sqrt(x^6 + x^2)*x + (2*I + 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x ^3 + x)) - 2^(1/4)*((I + 1)*x^5 + (I + 1)*x)*log(((2*I - 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(I*x^5 - 2*I*x^3 + I*x) + 4*sqrt(x^6 + x^2)*x - (2*I + 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*(-(I - 1 )*x^5 - (I - 1)*x)*log((-(2*I + 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2) *(-I*x^5 + 2*I*x^3 - I*x) + 4*sqrt(x^6 + x^2)*x + (2*I - 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 32*(x^6 + x^2)^(3/4))/(x^5 + x)
\[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
\[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx=\int \frac {x^4}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^8-1\right )} \,d x \]