Integrand size = 28, antiderivative size = 390 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\frac {4 \left (-x^3+x^5\right )^{3/4}}{3 x^2 \left (-1+x^2\right )}+\frac {\sqrt [4]{2} \arctan \left (\frac {3^{7/8} \sqrt {2-\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \arctan \left (\frac {3^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2+\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right ) \]
4/3*(x^5-x^3)^(3/4)/x^2/(x^2-1)+1/3*arctan(3^(7/8)*(2-2^(1/2))^(1/2)*x*(x^ 5-x^3)^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^3)^(1/2)))*2^(1/4)/(51+36*2^(1/2))^(1/ 8)+1/3*(17/3+4*2^(1/2))^(1/8)*arctan(3^(7/8)*(2+2^(1/2))^(1/2)*x*(x^5-x^3) ^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^3)^(1/2)))*2^(1/4)+1/3*arctanh((3^(1/8)*x^2/ (2-2^(1/2))^(1/2)+1/3*(x^5-x^3)^(1/2)*3^(7/8)/(2-2^(1/2))^(1/2))/x/(x^5-x^ 3)^(1/4))*2^(1/4)/(51+36*2^(1/2))^(1/8)+1/3*(17/3+4*2^(1/2))^(1/8)*arctanh ((3^(1/8)*x^2/(2+2^(1/2))^(1/2)+1/3*(x^5-x^3)^(1/2)*3^(7/8)/(2+2^(1/2))^(1 /2))/x/(x^5-x^3)^(1/4))*2^(1/4)
Time = 3.28 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.05 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\frac {x^{3/4} \left (12 \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x}+\sqrt [4]{2} 3^{7/8} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+3^{7/8} \sqrt [4]{34+24 \sqrt {2}} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )+\sqrt [4]{2} 3^{7/8} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+3^{7/8} \sqrt [4]{34+24 \sqrt {2}} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )\right )}{9 \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x^3 \left (-1+x^2\right )}} \]
(x^(3/4)*(12*(17 + 12*Sqrt[2])^(1/8)*x^(1/4) + 2^(1/4)*3^(7/8)*(-1 + x^2)^ (1/4)*ArcTan[(2^(1/4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1/4))/((17 + 12*Sqrt[2]) ^(1/8)*(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + 3^(7/8)*(34 + 24*Sqrt[2]) ^(1/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x^ (1/4)*(-1 + x^2)^(1/4))/(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2])] + 2^(1/4)*3 ^(7/8)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1/4)) /((17 + 12*Sqrt[2])^(1/8)*(3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + 3^(7/8) *(34 + 24*Sqrt[2])^(1/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*3^(7/8)*(17 + 1 2*Sqrt[2])^(1/8)*x^(1/4)*(-1 + x^2)^(1/4))/(3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2])]))/(9*(17 + 12*Sqrt[2])^(1/8)*(x^3*(-1 + x^2))^(1/4))
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.
| \(\displaystyle \int \frac {x^6+1}{\sqrt [4]{x^5-x^3} \left (1-x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2-1} \int \frac {x^6+1}{x^{3/4} \sqrt [4]{x^2-1} \left (1-x^6\right )}dx}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \frac {x^6+1}{\sqrt [4]{x^2-1} \left (1-x^6\right )}d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (\frac {2}{\sqrt [4]{x^2-1} \left (1-x^6\right )}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (-\frac {2}{\left (x^6-1\right ) \sqrt [4]{x^2-1}}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (-\frac {2}{\left (x^6-1\right ) \sqrt [4]{x^2-1}}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
3.30.87.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 68.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17
| method | result | size |
| pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) \left (x^{5}-x^{3}\right )^{\frac {1}{4}}+4 x}{3 \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\) | \(65\) |
| risch | \(\text {Expression too large to display}\) | \(2154\) |
| trager | \(\text {Expression too large to display}\) | \(2163\) |
1/3*(-sum(ln((-_R*x+(x^5-x^3)^(1/4))/x)/_R,_R=RootOf(_Z^8+3))*(x^5-x^3)^(1 /4)+4*x)/(x^5-x^3)^(1/4)
Result contains complex when optimal does not.
Time = 143.79 (sec) , antiderivative size = 2118, normalized size of antiderivative = 5.43 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\text {Too large to display} \]
1/36*(3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(I + 1)*x^4 + (I + 1)*x^2)*log((3^(7/8) *sqrt(2)*(-1)^(1/8)*((109*I + 109)*x^6 - (264*I + 264)*x^5 - (545*I + 545) *x^4 + (264*I + 264)*x^3 + (109*I + 109)*x^2) - 6*3^(3/8)*sqrt(2)*(-1)^(5/ 8)*(-(22*I + 22)*x^6 - (109*I + 109)*x^5 + (110*I + 110)*x^4 + (109*I + 10 9)*x^3 - (22*I + 22)*x^2) - 12*(x^5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(44*I* x^2 + 109*I*x - 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*sqrt(2)* (-1)^(7/8)*(-(109*I - 109)*x^3 + (132*I - 132)*x^2 + (109*I - 109)*x) + 3^ (5/8)*sqrt(2)*(-1)^(3/8)*((44*I - 44)*x^3 + (109*I - 109)*x^2 - (44*I - 44 )*x)) - 12*(x^5 - x^3)^(1/4)*(3^(1/4)*(-1)^(3/4)*(109*I*x^4 - 132*I*x^3 - 109*I*x^2) + 3^(3/4)*(-1)^(1/4)*(-44*I*x^4 - 109*I*x^3 + 44*I*x^2)))/(x^6 + x^4 + x^2)) + 3^(7/8)*sqrt(2)*(-1)^(1/8)*((I - 1)*x^4 - (I - 1)*x^2)*log ((3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(109*I - 109)*x^6 + (264*I - 264)*x^5 + (54 5*I - 545)*x^4 - (264*I - 264)*x^3 - (109*I - 109)*x^2) - 6*3^(3/8)*sqrt(2 )*(-1)^(5/8)*((22*I - 22)*x^6 + (109*I - 109)*x^5 - (110*I - 110)*x^4 - (1 09*I - 109)*x^3 + (22*I - 22)*x^2) - 12*(x^5 - x^3)^(3/4)*(109*x^2 + sqrt( 3)*(44*I*x^2 + 109*I*x - 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8) *sqrt(2)*(-1)^(7/8)*((109*I + 109)*x^3 - (132*I + 132)*x^2 - (109*I + 109) *x) + 3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(44*I + 44)*x^3 - (109*I + 109)*x^2 + ( 44*I + 44)*x)) - 12*(x^5 - x^3)^(1/4)*(3^(3/4)*(-1)^(1/4)*(44*I*x^4 + 109* I*x^3 - 44*I*x^2) + 3^(1/4)*(-1)^(3/4)*(-109*I*x^4 + 132*I*x^3 + 109*I*...
\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx \]
-Integral(x**6/(x**6*(x**5 - x**3)**(1/4) - (x**5 - x**3)**(1/4)), x) - In tegral(1/(x**6*(x**5 - x**3)**(1/4) - (x**5 - x**3)**(1/4)), x)
\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int -\frac {x^6+1}{\left (x^6-1\right )\,{\left (x^5-x^3\right )}^{1/4}} \,d x \]