Integrand size = 27, antiderivative size = 123 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {2 \sqrt {x+x^2+x^3}}{3 \left (1+x+x^2\right )}+\frac {1}{3} \arctan \left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{3 \sqrt {3}} \]
2*(x^3+x^2+x)^(1/2)/(3*x^2+3*x+3)+1/3*arctan((x^3+x^2+x)^(1/2)/(x^2+x+1))+ 1/3*2^(1/2)*arctanh(2^(1/2)*(x^3+x^2+x)^(1/2)/(x^2+x+1))+1/9*arctanh(3^(1/ 2)*(x^3+x^2+x)^(1/2)/(x^2+x+1))*3^(1/2)
Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {\sqrt {x} \left (6 \sqrt {x}+3 \sqrt {1+x+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )+3 \sqrt {2} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {1+x+x^2}}\right )+\sqrt {3} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )\right )}{9 \sqrt {x \left (1+x+x^2\right )}} \]
(Sqrt[x]*(6*Sqrt[x] + 3*Sqrt[1 + x + x^2]*ArcTan[Sqrt[x]/Sqrt[1 + x + x^2] ] + 3*Sqrt[2]*Sqrt[1 + x + x^2]*ArcTanh[(Sqrt[2]*Sqrt[x])/Sqrt[1 + x + x^2 ]] + Sqrt[3]*Sqrt[1 + x + x^2]*ArcTanh[(Sqrt[3]*Sqrt[x])/Sqrt[1 + x + x^2] ]))/(9*Sqrt[x*(1 + x + x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.95 (sec) , antiderivative size = 887, normalized size of antiderivative = 7.21, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.
| \(\displaystyle \int \frac {x^6+1}{\sqrt {x^3+x^2+x} \left (1-x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2+x+1} \int \frac {x^6+1}{\sqrt {x} \sqrt {x^2+x+1} \left (1-x^6\right )}dx}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2+x+1} \int \frac {x^6+1}{\sqrt {x^2+x+1} \left (1-x^6\right )}d\sqrt {x}}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2+x+1} \int \left (\frac {2}{\sqrt {x^2+x+1} \left (1-x^6\right )}-\frac {1}{\sqrt {x^2+x+1}}\right )d\sqrt {x}}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2+x+1} \left (\frac {\sqrt [3]{-1} \sqrt {x} \left (\sqrt [3]{-1} x+1\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1}}+\frac {1}{6} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )-\frac {i \arctan \left (\frac {-2 \sqrt [3]{-1} \left (2+\sqrt [3]{-1}\right ) x-3 i \sqrt {3}+1}{4 \sqrt {2} \sqrt {x^2+x+1}}\right )}{12 \sqrt {2}}+\frac {i \arctan \left (\frac {2 (-1)^{2/3} \left (2-(-1)^{2/3}\right ) x+3 i \sqrt {3}+1}{4 \sqrt {2} \sqrt {x^2+x+1}}\right )}{12 \sqrt {2}}+\frac {\sqrt [6]{-1} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{2 \sqrt {6} \left (1+\sqrt [3]{-1}\right )}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{6 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{6 \sqrt {3}}+\frac {\sqrt [6]{-1} \text {arctanh}\left (\frac {\left (1+2 \sqrt [3]{-1}\right ) x+\sqrt [3]{-1}+2}{2 \sqrt {1+i \sqrt {3}} \sqrt {x^2+x+1}}\right )}{12 \sqrt {1+i \sqrt {3}}}-\frac {(-1)^{5/6} \text {arctanh}\left (\frac {\left (1-2 (-1)^{2/3}\right ) x-(-1)^{2/3}+2}{2 \sqrt {1-i \sqrt {3}} \sqrt {x^2+x+1}}\right )}{12 \sqrt {1-i \sqrt {3}}}+\frac {\sqrt [6]{-1} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{6 \left (3 i+\sqrt {3}\right ) \sqrt {x^2+x+1}}+\frac {\sqrt [3]{-1} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{6 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1}}+\frac {(x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1}}-\frac {(x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{3 \sqrt {x^2+x+1}}-\frac {(-1)^{5/6} \left (1-(-1)^{2/3} x\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt [6]{-1} \sqrt {x}\right ),1+\sqrt [3]{-1}\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {\frac {1-(-1)^{2/3} x}{\sqrt [3]{-1} x+1}} \sqrt {x^2+x+1}}+\frac {\sqrt [6]{-1} \left (1-(-1)^{2/3} x\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt [6]{-1} \sqrt {x}\right ),1+\sqrt [3]{-1}\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {\frac {1-(-1)^{2/3} x}{\sqrt [3]{-1} x+1}} \sqrt {x^2+x+1}}+\frac {\sqrt {x} \left (1-(-1)^{2/3} x\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1}}\right )}{\sqrt {x^3+x^2+x}}\) |
(2*Sqrt[x]*Sqrt[1 + x + x^2]*(((-1)^(1/3)*Sqrt[x]*(1 + (-1)^(1/3)*x))/(3*( 1 + (-1)^(1/3))*Sqrt[1 + x + x^2]) + (Sqrt[x]*(1 - (-1)^(2/3)*x))/(3*(1 + (-1)^(1/3))*Sqrt[1 + x + x^2]) + ArcTan[Sqrt[x]/Sqrt[1 + x + x^2]]/6 - ((I /12)*ArcTan[(1 - (3*I)*Sqrt[3] - 2*(-1)^(1/3)*(2 + (-1)^(1/3))*x)/(4*Sqrt[ 2]*Sqrt[1 + x + x^2])])/Sqrt[2] + ((I/12)*ArcTan[(1 + (3*I)*Sqrt[3] + 2*(- 1)^(2/3)*(2 - (-1)^(2/3))*x)/(4*Sqrt[2]*Sqrt[1 + x + x^2])])/Sqrt[2] + Arc Tanh[(Sqrt[2]*Sqrt[x])/Sqrt[1 + x + x^2]]/(6*Sqrt[2]) + ((-1)^(1/6)*ArcTan h[(Sqrt[2]*Sqrt[x])/Sqrt[1 + x + x^2]])/(2*Sqrt[6]*(1 + (-1)^(1/3))) + Arc Tanh[(Sqrt[3]*Sqrt[x])/Sqrt[1 + x + x^2]]/(6*Sqrt[3]) + ((-1)^(1/6)*ArcTan h[(2 + (-1)^(1/3) + (1 + 2*(-1)^(1/3))*x)/(2*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 + x + x^2])])/(12*Sqrt[1 + I*Sqrt[3]]) - ((-1)^(5/6)*ArcTanh[(2 - (-1)^(2/3) + (1 - 2*(-1)^(2/3))*x)/(2*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + x + x^2])])/(12*S qrt[1 - I*Sqrt[3]]) - ((1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*A rcTan[Sqrt[x]], 1/4])/(3*Sqrt[1 + x + x^2]) + ((1 + x)*Sqrt[(1 + x + x^2)/ (1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/(12*(1 + (-1)^(1/3))*Sqrt[1 + x + x^2]) + ((-1)^(1/3)*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[ 2*ArcTan[Sqrt[x]], 1/4])/(6*(1 + (-1)^(1/3))*Sqrt[1 + x + x^2]) + ((-1)^(1 /6)*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4 ])/(6*(3*I + Sqrt[3])*Sqrt[1 + x + x^2]) + ((-1)^(1/6)*(1 - (-1)^(2/3)*x)* EllipticF[ArcTan[(-1)^(1/6)*Sqrt[x]], 1 + (-1)^(1/3)])/(3*(1 + (-1)^(1/...
3.19.16.3.1 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 = 2.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(\frac {2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{9}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right )}{3}-\frac {\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )}{3}\) | \(80\) |
| default | \(-\frac {-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}}{3}+\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}\) | \(111\) |
| pseudoelliptic | \(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-3 \arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right ) \sqrt {x \left (x^{2}+x +1\right )}+6 x}{9 \sqrt {x \left (x^{2}+x +1\right )}}\) | \(111\) |
| trager | \(\frac {2 \sqrt {x^{3}+x^{2}+x}}{3 \left (x^{2}+x +1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}+x^{2}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}-x +1}\right )}{6}\) | \(180\) |
| elliptic | \(\text {Expression too large to display}\) | \(1330\) |
2/3*x/(x*(x^2+x+1))^(1/2)+1/9*3^(1/2)*arctanh(1/3*(x*(x^2+x+1))^(1/2)/x*3^ (1/2))+1/3*2^(1/2)*arctanh(1/2*(x*(x^2+x+1))^(1/2)/x*2^(1/2))-1/3*arctan(( x*(x^2+x+1))^(1/2)/x)
Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.59 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {3 \, \sqrt {2} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 14 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 3 \, x + 1\right )} + 19 \, x^{2} + 14 \, x + 1}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \sqrt {3} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 20 \, x^{3} + 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - 6 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + 24 \, \sqrt {x^{3} + x^{2} + x}}{36 \, {\left (x^{2} + x + 1\right )}} \]
1/36*(3*sqrt(2)*(x^2 + x + 1)*log((x^4 + 14*x^3 + 4*sqrt(2)*sqrt(x^3 + x^2 + x)*(x^2 + 3*x + 1) + 19*x^2 + 14*x + 1)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1) ) + sqrt(3)*(x^2 + x + 1)*log((x^4 + 20*x^3 + 4*sqrt(3)*sqrt(x^3 + x^2 + x )*(x^2 + 4*x + 1) + 30*x^2 + 20*x + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 6*(x^2 + x + 1)*arctan(1/2*(x^2 + 1)/sqrt(x^3 + x^2 + x)) + 24*sqrt(x^3 + x^2 + x))/(x^2 + x + 1)
\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx - \int \frac {1}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx \]
-Integral(x**6/(x**6*sqrt(x**3 + x**2 + x) - sqrt(x**3 + x**2 + x)), x) - Integral(1/(x**6*sqrt(x**3 + x**2 + x) - sqrt(x**3 + x**2 + x)), x)
\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{3} + x^{2} + x}} \,d x } \]
\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{3} + x^{2} + x}} \,d x } \]
Time = 5.50 (sec) , antiderivative size = 1195, normalized size of antiderivative = 9.72 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\text {Too large to display} \]
(2*((3^(1/2)*1i)/6 - 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2) *1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/(( 3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(-1, asin((x/((3^(1/2)*1i)/2 - 1/2)) ^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*( (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) - (2*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1 /2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2) )^(1/2)*ellipticF(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3 ^(1/2)*1i)/2 + 1/2))^(1/2) - (2*((3^(1/2)*1i)/6 - 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((ellipticE(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/ 2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) - ((x/((3^(1/2)*1i)/2 - 1/2))^(1/2 )*(x/((3^(1/2)*1i)/2 + 1/2) + 1)^(1/2))/(1 - x/((3^(1/2)*1i)/2 - 1/2))^(1/ 2))/(((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2) + 1) - ellipticF(asin(( x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2))/(x^2 + x^3 - x*((3^(1 /2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) + (2*((3^(1/2)*1i)/2 - 1/2) *(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1 i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^...