Integrand size = 29, antiderivative size = 291 \[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\frac {1}{3} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{6\ 2^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )}{\sqrt {2}}+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{6\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right )}{\sqrt {2}} \]
1/3*arctan(x/(x^5+x^3)^(1/4))+1/4*arctan(2^(1/4)*x/(x^5+x^3)^(1/4))*2^(3/4 )-1/12*arctan(2^(3/4)*x*(x^5+x^3)^(1/4)/(2^(1/2)*x^2-(x^5+x^3)^(1/2)))*2^( 1/4)+1/2*2^(1/2)*arctan(2^(1/2)*x*(x^5+x^3)^(1/4)/(-x^2+(x^5+x^3)^(1/2)))+ 1/3*arctanh(x/(x^5+x^3)^(1/4))+1/4*arctanh(2^(1/4)*x/(x^5+x^3)^(1/4))*2^(3 /4)+1/12*arctanh((1/2*x^2*2^(3/4)+1/2*(x^5+x^3)^(1/2)*2^(1/4))/x/(x^5+x^3) ^(1/4))*2^(1/4)+1/2*2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2*(x^5+x^3)^(1/2)*2 ^(1/2))/x/(x^5+x^3)^(1/4))
\[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx \]
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.
| \(\displaystyle \int \frac {x^6+x^3+1}{\sqrt [4]{x^5+x^3} \left (1-x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2+1} \int \frac {x^6+x^3+1}{x^{3/4} \sqrt [4]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [4]{x^5+x^3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {x^6+x^3+1}{\sqrt [4]{x^2+1} \left (1-x^6\right )}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {x^3+2}{\sqrt [4]{x^2+1} \left (1-x^6\right )}-\frac {1}{\sqrt [4]{x^2+1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {-x^3-2}{\sqrt [4]{x^2+1} \left (x^6-1\right )}-\frac {1}{\sqrt [4]{x^2+1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {-x^3-2}{\sqrt [4]{x^2+1} \left (x^6-1\right )}-\frac {1}{\sqrt [4]{x^2+1}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\) |
3.29.47.3.1 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 = 23.41 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.37
| method | result | size |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{3}-\frac {\ln \left (\frac {-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) 2^{\frac {1}{4}}}{24}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}}}{12}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}}}{12}+\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right )}{6}-\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{6}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}}}{4}+\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{8}-\frac {\ln \left (\frac {-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) \sqrt {2}}{4}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{2}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{2}\) | \(399\) |
| trager | \(\text {Expression too large to display}\) | \(1424\) |
-1/3*arctan((x^3*(x^2+1))^(1/4)/x)-1/24*ln((-(x^3*(x^2+1))^(1/4)*2^(3/4)*x +2^(1/2)*x^2+(x^3*(x^2+1))^(1/2))/((x^3*(x^2+1))^(1/4)*2^(3/4)*x+2^(1/2)*x ^2+(x^3*(x^2+1))^(1/2)))*2^(1/4)-1/12*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)+ x)/x)*2^(1/4)-1/12*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)-x)/x)*2^(1/4)+1/6*l n(((x^3*(x^2+1))^(1/4)+x)/x)-1/6*ln(((x^3*(x^2+1))^(1/4)-x)/x)-1/4*arctan( 1/2*(x^3*(x^2+1))^(1/4)/x*2^(3/4))*2^(3/4)+1/8*ln((-2^(1/4)*x-(x^3*(x^2+1) )^(1/4))/(2^(1/4)*x-(x^3*(x^2+1))^(1/4)))*2^(3/4)-1/4*ln((-(x^3*(x^2+1))^( 1/4)*2^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2))/((x^3*(x^2+1))^(1/4)*2^(1/2)*x+x^2 +(x^3*(x^2+1))^(1/2)))*2^(1/2)-1/2*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)+x)/ x)*2^(1/2)-1/2*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)-x)/x)*2^(1/2)
Result contains complex when optimal does not.
Time = 37.15 (sec) , antiderivative size = 1113, normalized size of antiderivative = 3.82 \[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\text {Too large to display} \]
1/16*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(x^4 + 2*x^3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/16*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(x^4 + 2*x^3 + x^2) - 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) + 1/16*I*2^(3/4)*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^( 3/4)*(I*x^4 + 2*I*x^3 + I*x^2) + 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/16*I*2^(3/4)*log(-(4*sqrt(2)*(x^5 + x ^3)^(1/4)*x^2 - 2^(3/4)*(-I*x^4 - 2*I*x^3 - I*x^2) - 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - (1/8*I - 1/8)*sqrt( 2)*log((4*I*(x^5 + x^3)^(1/4)*x^2 + (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*(-(I - 1)*x^4 + (I - 1)*x^3 - (I - 1)*x^2) + 4*(x^5 + x^3)^(3/4))/ (x^4 + x^3 + x^2)) + (1/8*I - 1/8)*sqrt(2)*log((4*I*(x^5 + x^3)^(1/4)*x^2 - (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*((I - 1)*x^4 - (I - 1)*x^3 + (I - 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) + (1/8*I + 1/8)* sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4)*x^2 - (2*I - 2)*sqrt(2)*sqrt(x^5 + x^3 )*x + sqrt(2)*((I + 1)*x^4 - (I + 1)*x^3 + (I + 1)*x^2) + 4*(x^5 + x^3)^(3 /4))/(x^4 + x^3 + x^2)) - (1/8*I + 1/8)*sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4 )*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*(-(I + 1)*x^4 + (I + 1)*x^3 - (I + 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - (1/48*I - 1/48)*2^(1/4)*log(-2*(4*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + (2*I + 2)*...
\[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=- \int \frac {x^{3}}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx \]
-Integral(x**3/(x**6*(x**5 + x**3)**(1/4) - (x**5 + x**3)**(1/4)), x) - In tegral(x**6/(x**6*(x**5 + x**3)**(1/4) - (x**5 + x**3)**(1/4)), x) - Integ ral(1/(x**6*(x**5 + x**3)**(1/4) - (x**5 + x**3)**(1/4)), x)
\[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + x^{3} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + x^{3} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int -\frac {x^6+x^3+1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^6-1\right )} \,d x \]