Integrand size = 22, antiderivative size = 118 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^5}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^5}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^5}+\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
-1/4*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^5+x)^(1/3)))*2^(2/3)+1/4*ln(-2 *x+2^(2/3)*(x^5+x)^(1/3))*2^(2/3)-1/8*ln(2*x^2+2^(2/3)*x*(x^5+x)^(1/3)+2^( 1/3)*(x^5+x)^(2/3))*2^(2/3)
Time = 15.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^5}}\right )-2 \log \left (-2 x+2^{2/3} \sqrt [3]{x+x^5}\right )+\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^5}+\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
-1/4*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^5)^(1/3))] - 2*Log[ -2*x + 2^(2/3)*(x + x^5)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(x + x^5)^(1/3) + 2^(1/3)*(x + x^5)^(2/3)])/2^(1/3)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.11 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2467, 25, 2035, 7266, 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.
| \(\displaystyle \int \frac {x^2+1}{\left (x^2-1\right ) \sqrt [3]{x^5+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \int -\frac {x^2+1}{\sqrt [3]{x} \left (1-x^2\right ) \sqrt [3]{x^4+1}}dx}{\sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {x^2+1}{\sqrt [3]{x} \left (1-x^2\right ) \sqrt [3]{x^4+1}}dx}{\sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {\sqrt [3]{x} \left (x^2+1\right )}{\left (1-x^2\right ) \sqrt [3]{x^4+1}}d\sqrt [3]{x}}{\sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {x+1}{(1-x) \sqrt [3]{x^2+1}}dx^{2/3}}{2 \sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \left (\frac {2}{(1-x) \sqrt [3]{x^2+1}}-\frac {1}{\sqrt [3]{x^2+1}}\right )dx^{2/3}}{2 \sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \left (2 x^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )+\frac {\log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\frac {2^{2/3} \left (x^{2/3}+1\right )^2}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{3 \sqrt [3]{2}}-\frac {\log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{4 \sqrt [3]{2}}\right )}{2 \sqrt [3]{x^5+x}}\) |
(-3*x^(1/3)*(1 + x^4)^(1/3)*(2*x^(2/3)*AppellF1[1/6, 1, 1/3, 7/6, x^2, -x^ 2] + ArcTan[(1 - (2*2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3]]/(2^(1 /3)*Sqrt[3]) + ArcTan[(1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3 ]]/(2*2^(1/3)*Sqrt[3]) - x^(2/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -x^2] + Log[(1 - x^(2/3))^2*(1 + x^(2/3))]/(12*2^(1/3)) + Log[1 + (2^(2/3)*(1 + x^ (2/3))^2)/(1 + x^2)^(2/3) - (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)]/(6*2^ (1/3)) - Log[1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)]/(3*2^(1/3)) - Lo g[1 + x^(2/3) - 2^(2/3)*(1 + x^2)^(1/3)]/(4*2^(1/3))))/(2*(x + x^5)^(1/3))
3.18.53.3.1 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_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, 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 = 14.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}+x \right )}{3 x}\right )+2 \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}} x +{\left (\left (x^{4}+1\right ) x \right )}^{\frac {2}{3}}}{x^{2}}\right )\right )}{8}\) | \(97\) |
| trager | \(\text {Expression too large to display}\) | \(951\) |
1/8*2^(2/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*((x^4+1)*x)^(1/3)+x)/x) +2*ln((-2^(1/3)*x+((x^4+1)*x)^(1/3))/x)-ln((2^(2/3)*x^2+2^(1/3)*((x^4+1)*x )^(1/3)*x+((x^4+1)*x)^(2/3))/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (86) = 172\).
Time = 2.39 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.49 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=-\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{12} - 24 \, x^{10} - 57 \, x^{8} - 56 \, x^{6} - 57 \, x^{4} - 24 \, x^{2} + 1\right )} + 24 \, \sqrt {2} {\left (x^{9} - x^{7} - x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} - 12 \cdot 2^{\frac {1}{6}} {\left (x^{8} + 14 \, x^{6} + 6 \, x^{4} + 14 \, x^{2} + 1\right )} {\left (x^{5} + x\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{12} + 48 \, x^{10} + 15 \, x^{8} + 88 \, x^{6} + 15 \, x^{4} + 48 \, x^{2} + 1\right )}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{8} + 14 \, x^{6} + 6 \, x^{4} + 14 \, x^{2} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{5} + x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} + 6 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} {\left (x^{4} + 4 \, x^{2} + 1\right )}}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + x\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{2} + 1\right )} - 6 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x}{x^{4} - 2 \, x^{2} + 1}\right ) \]
-1/12*sqrt(3)*2^(2/3)*arctan(-1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^12 - 24*x^10 - 57*x^8 - 56*x^6 - 57*x^4 - 24*x^2 + 1) + 24*sqrt(2)*(x^9 - x^7 - x^3 + x)*(x^5 + x)^(1/3) - 12*2^(1/6)*(x^8 + 14*x^6 + 6*x^4 + 14*x^2 + 1)*(x^5 + x)^(2/3))/(x^12 + 48*x^10 + 15*x^8 + 88*x^6 + 15*x^4 + 48*x^2 + 1)) - 1/2 4*2^(2/3)*log((2^(2/3)*(x^8 + 14*x^6 + 6*x^4 + 14*x^2 + 1) + 12*2^(1/3)*(x ^5 + x^3 + x)*(x^5 + x)^(1/3) + 6*(x^5 + x)^(2/3)*(x^4 + 4*x^2 + 1))/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)) + 1/12*2^(2/3)*log((3*2^(2/3)*(x^5 + x)^(2/3 ) - 2^(1/3)*(x^4 - 2*x^2 + 1) - 6*(x^5 + x)^(1/3)*x)/(x^4 - 2*x^2 + 1))
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int \frac {x^{2} + 1}{\sqrt [3]{x \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{5} + x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}} \,d x } \]
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{5} + x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx=\int \frac {x^2+1}{\left (x^2-1\right )\,{\left (x^5+x\right )}^{1/3}} \,d x \]