Integrand size = 25, antiderivative size = 140 \[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=-\frac {3 \left (-1+2 x^2\right ) \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{x+x^3}}\right )}{4 \sqrt [3]{2}}-\frac {3 \log \left (-x+\sqrt [3]{2} \sqrt [3]{x+x^3}\right )}{4 \sqrt [3]{2}}+\frac {3 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{x+x^3}+2^{2/3} \left (x+x^3\right )^{2/3}\right )}{8 \sqrt [3]{2}} \]
-3/4*(2*x^2-1)*(x^3+x)^(1/3)/x^3-3/8*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(1/3) *(x^3+x)^(1/3)))*2^(2/3)-3/8*ln(-x+2^(1/3)*(x^3+x)^(1/3))*2^(2/3)+3/16*ln( x^2+2^(1/3)*x*(x^3+x)^(1/3)+2^(2/3)*(x^3+x)^(2/3))*2^(2/3)
Time = 1.69 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=\frac {3 \sqrt [3]{x+x^3} \left (4 \sqrt [3]{1+x^2}-8 x^2 \sqrt [3]{1+x^2}-2\ 2^{2/3} \sqrt {3} x^{8/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{2} \sqrt [3]{1+x^2}}\right )-2\ 2^{2/3} x^{8/3} \log \left (-x^{2/3}+\sqrt [3]{2} \sqrt [3]{1+x^2}\right )+2^{2/3} x^{8/3} \log \left (x^{4/3}+\sqrt [3]{2} x^{2/3} \sqrt [3]{1+x^2}+2^{2/3} \left (1+x^2\right )^{2/3}\right )\right )}{16 x^3 \sqrt [3]{1+x^2}} \]
(3*(x + x^3)^(1/3)*(4*(1 + x^2)^(1/3) - 8*x^2*(1 + x^2)^(1/3) - 2*2^(2/3)* Sqrt[3]*x^(8/3)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*2^(1/3)*(1 + x^2)^(1 /3))] - 2*2^(2/3)*x^(8/3)*Log[-x^(2/3) + 2^(1/3)*(1 + x^2)^(1/3)] + 2^(2/3 )*x^(8/3)*Log[x^(4/3) + 2^(1/3)*x^(2/3)*(1 + x^2)^(1/3) + 2^(2/3)*(1 + x^2 )^(2/3)]))/(16*x^3*(1 + x^2)^(1/3))
Time = 0.42 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2467, 25, 442, 27, 445, 27, 368, 965, 992}
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.
| \(\displaystyle \int \frac {\left (x^2-4\right ) \sqrt [3]{x^3+x}}{x^4 \left (x^2+2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^3+x} \int -\frac {\left (4-x^2\right ) \sqrt [3]{x^2+1}}{x^{11/3} \left (x^2+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \int \frac {\left (4-x^2\right ) \sqrt [3]{x^2+1}}{x^{11/3} \left (x^2+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (\frac {3}{16} \int -\frac {8 \left (5 x^2+4\right )}{3 x^{5/3} \left (x^2+1\right )^{2/3} \left (x^2+2\right )}dx-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (-\frac {1}{2} \int \frac {5 x^2+4}{x^{5/3} \left (x^2+1\right )^{2/3} \left (x^2+2\right )}dx-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (\frac {1}{2} \left (\frac {3}{4} \int -\frac {4 \sqrt [3]{x}}{\left (x^2+1\right )^{2/3} \left (x^2+2\right )}dx+\frac {3 \sqrt [3]{x^2+1}}{x^{2/3}}\right )-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (\frac {1}{2} \left (\frac {3 \sqrt [3]{x^2+1}}{x^{2/3}}-3 \int \frac {\sqrt [3]{x}}{\left (x^2+1\right )^{2/3} \left (x^2+2\right )}dx\right )-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (\frac {1}{2} \left (\frac {3 \sqrt [3]{x^2+1}}{x^{2/3}}-9 \int \frac {x}{\left (x^2+1\right )^{2/3} \left (x^2+2\right )}d\sqrt [3]{x}\right )-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (\frac {1}{2} \left (\frac {3 \sqrt [3]{x^2+1}}{x^{2/3}}-\frac {9}{2} \int \frac {x^{2/3}}{(x+1)^{2/3} (x+2)}dx^{2/3}\right )-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
| \(\Big \downarrow \) 992 |
| \(\displaystyle -\frac {\sqrt [3]{x^3+x} \left (\frac {1}{2} \left (\frac {3 \sqrt [3]{x^2+1}}{x^{2/3}}-\frac {9}{2} \left (-\frac {\arctan \left (\frac {\frac {2^{2/3} x^{2/3}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (\frac {x^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{x+1}\right )}{2 \sqrt [3]{2}}+\frac {\log (x+2)}{6 \sqrt [3]{2}}\right )\right )-\frac {3 \sqrt [3]{x^2+1}}{4 x^{8/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2+1}}\) |
-(((x + x^3)^(1/3)*((-3*(1 + x^2)^(1/3))/(4*x^(8/3)) + ((3*(1 + x^2)^(1/3) )/x^(2/3) - (9*(-(ArcTan[(1 + (2^(2/3)*x^(2/3))/(1 + x)^(1/3))/Sqrt[3]]/(2 ^(1/3)*Sqrt[3])) + Log[2 + x]/(6*2^(1/3)) - Log[x^(2/3)/2^(1/3) - (1 + x)^ (1/3)]/(2*2^(1/3))))/2)/2))/(x^(1/3)*(1 + x^2)^(1/3)))
3.20.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 ))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*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 = 15.34 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {\frac {3 \,2^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{3}}{8}+\frac {3 \,2^{\frac {2}{3}} x^{3} \ln \left (2\right )}{16}-\frac {3 \,2^{\frac {2}{3}} x^{3} \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{8}+\frac {3 \,2^{\frac {2}{3}} x^{3} \ln \left (\frac {2^{\frac {2}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} x^{2}+2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{16}-\frac {3 x^{2} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{2}+\frac {3 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{4}}{x^{3}}\) | \(153\) |
| trager | \(\text {Expression too large to display}\) | \(748\) |
| risch | \(\text {Expression too large to display}\) | \(1587\) |
3/16/x^3*(2*2^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2*2^(1/3)*(x*(x^2+1))^(1/3 )+x)/x)*x^3+2^(2/3)*x^3*ln(2)-2*2^(2/3)*x^3*ln((-2^(2/3)*x+2*(x*(x^2+1))^( 1/3))/x)+2^(2/3)*x^3*ln((2^(2/3)*(x*(x^2+1))^(1/3)*x+2^(1/3)*x^2+2*(x*(x^2 +1))^(2/3))/x^2)-8*x^2*(x*(x^2+1))^(1/3)+4*(x*(x^2+1))^(1/3))
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (104) = 208\).
Time = 1.49 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.06 \[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (24 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{4} + 5 \, x^{2} + 2\right )} {\left (x^{3} + x\right )}^{\frac {2}{3}} - 12 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{5} + 22 \, x^{3} + 4 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} - 2^{\frac {5}{6}} {\left (91 \, x^{6} + 168 \, x^{4} + 84 \, x^{2} + 8\right )}\right )}}{6 \, {\left (53 \, x^{6} + 48 \, x^{4} - 12 \, x^{2} - 8\right )}}\right ) - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + x\right )}^{\frac {2}{3}} {\left (2 \, x^{2} + 1\right )} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{4} + 22 \, x^{2} + 4\right )} + 6 \, {\left (5 \, x^{3} + 4 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{2} + 4}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} + 2\right )} + 6 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{x^{2} + 2}\right ) - 12 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{2} - 1\right )}}{16 \, x^{3}} \]
1/16*(2*sqrt(3)*2^(2/3)*(-1)^(1/3)*x^3*arctan(1/6*sqrt(3)*2^(1/6)*(24*sqrt (2)*(-1)^(1/3)*(2*x^4 + 5*x^2 + 2)*(x^3 + x)^(2/3) - 12*2^(1/6)*(-1)^(2/3) *(19*x^5 + 22*x^3 + 4*x)*(x^3 + x)^(1/3) - 2^(5/6)*(91*x^6 + 168*x^4 + 84* x^2 + 8))/(53*x^6 + 48*x^4 - 12*x^2 - 8)) - 2^(2/3)*(-1)^(1/3)*x^3*log((12 *2^(1/3)*(-1)^(2/3)*(x^3 + x)^(2/3)*(2*x^2 + 1) - 2^(2/3)*(-1)^(1/3)*(19*x ^4 + 22*x^2 + 4) + 6*(5*x^3 + 4*x)*(x^3 + x)^(1/3))/(x^4 + 4*x^2 + 4)) + 2 *2^(2/3)*(-1)^(1/3)*x^3*log((3*2^(2/3)*(-1)^(1/3)*(x^3 + x)^(1/3)*x - 2^(1 /3)*(-1)^(2/3)*(x^2 + 2) + 6*(x^3 + x)^(2/3))/(x^2 + 2)) - 12*(x^3 + x)^(1 /3)*(2*x^2 - 1))/x^3
\[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} + 1\right )} \left (x - 2\right ) \left (x + 2\right )}{x^{4} \left (x^{2} + 2\right )}\, dx \]
\[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=\int { \frac {{\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{2} - 4\right )}}{{\left (x^{2} + 2\right )} x^{4}} \,d x } \]
-3/56*(18*x^5 + 7*(x^3 + x)*x^2 + 8*x^3 - 10*x)*(x^2 + 1)^(1/3)/(x^(17/3) + 2*x^(11/3)) + integrate(9/7*(3*x^4 - x^2 - 4)*(x^2 + 1)^(1/3)/(x^(23/3) + 4*x^(17/3) + 4*x^(11/3)), x)
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69 \[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=\frac {3}{4} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + \frac {3}{16} \cdot 4^{\frac {1}{3}} \log \left (\left (\frac {1}{2}\right )^{\frac {2}{3}} + \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right ) - \frac {3}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {1}{2}\right )^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {9}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \]
3/4*sqrt(3)*(1/2)^(1/3)*arctan(2/3*sqrt(3)*(1/2)^(2/3)*((1/2)^(1/3) + 2*(1 /x^2 + 1)^(1/3))) + 3/4*(1/x^2 + 1)^(4/3) + 3/16*4^(1/3)*log((1/2)^(2/3) + (1/2)^(1/3)*(1/x^2 + 1)^(1/3) + (1/x^2 + 1)^(2/3)) - 3/4*(1/2)^(1/3)*log( abs(-(1/2)^(1/3) + (1/x^2 + 1)^(1/3))) - 9/4*(1/x^2 + 1)^(1/3)
Timed out. \[ \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx=\int \frac {\left (x^2-4\right )\,{\left (x^3+x\right )}^{1/3}}{x^4\,\left (x^2+2\right )} \,d x \]