Integrand size = 44, antiderivative size = 669 \[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{1-a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1-a}}-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{1+a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1+a}}+\frac {\log \left (1+\left (-\sqrt [3]{1-a}+\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1-a}}+\frac {\log \left (-1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1+a}}-\frac {\log \left (1+\left (\sqrt [3]{1-a}-\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1-a}}-\frac {\log \left (1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1+a}} \]
1/2*3^(1/2)*arctan(-1/3*3^(1/2)+1/3*(1-a)^(1/3)*(2*x^2-2)*(x/(a*x^7-3*a*x^ 5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)*3^(1/2))/(1-a)^(1/3)-1/2*3^(1/2)*a rctan(1/3*3^(1/2)+1/3*(1+a)^(1/3)*(2*x^2-2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3- 3*x^4-a*x+3*x^2-1))^(1/3)*3^(1/2))/(1+a)^(1/3)+1/2*ln(1+(-(1-a)^(1/3)+(1-a )^(1/3)*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3))/(1-a )^(1/3)+1/2*ln(-1+(-(1+a)^(1/3)+(1+a)^(1/3)*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a *x^3-3*x^4-a*x+3*x^2-1))^(1/3))/(1+a)^(1/3)-1/4*ln(1+((1-a)^(1/3)-(1-a)^(1 /3)*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)+((1-a)^(2 /3)-2*(1-a)^(2/3)*x^2+(1-a)^(2/3)*x^4)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4 -a*x+3*x^2-1))^(2/3))/(1-a)^(1/3)-1/4*ln(1+(-(1+a)^(1/3)+(1+a)^(1/3)*x^2)* (x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)+((1+a)^(2/3)-2*(1+ a)^(2/3)*x^2+(1+a)^(2/3)*x^4)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^ 2-1))^(2/3))/(1+a)^(1/3)
Result contains complex when optimal does not.
Time = 3.29 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.69 \[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=\frac {\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right ) \left (-2 \sqrt {-6+6 i \sqrt {3}} \sqrt [3]{1+a} \arctan \left (\frac {3 \sqrt [3]{-1+a} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+a} \sqrt [3]{x}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{1+a x}}\right )+2 \sqrt {-6+6 i \sqrt {3}} \sqrt [3]{-1+a} \arctan \left (\frac {3 \sqrt [3]{1+a} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+a} \sqrt [3]{x}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{1+a x}}\right )+\left (1+i \sqrt {3}\right ) \left (2 \sqrt [3]{1+a} \log \left (2 \sqrt [3]{-1+a} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{1+a x}\right )-2 \sqrt [3]{-1+a} \log \left (2 \sqrt [3]{1+a} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{1+a x}\right )-\sqrt [3]{1+a} \log \left (\left (-\sqrt [3]{-1+a} \sqrt [3]{x}+\sqrt [3]{1+a x}\right ) \left (2 i \sqrt [3]{-1+a} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{1+a x}\right )\right )+\sqrt [3]{-1+a} \log \left (\left (-\sqrt [3]{1+a} \sqrt [3]{x}+\sqrt [3]{1+a x}\right ) \left (2 i \sqrt [3]{1+a} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{1+a x}\right )\right )\right )\right )}{8 \sqrt [3]{-1+a} \sqrt [3]{1+a} \sqrt [3]{x}} \]
((1 + a*x)^(1/3)*(x/((1 + a*x)*(-1 + x^2)^3))^(1/3)*(-1 + x^2)*(-2*Sqrt[-6 + (6*I)*Sqrt[3]]*(1 + a)^(1/3)*ArcTan[(3*(-1 + a)^(1/3)*x^(1/3))/(Sqrt[3] *(-1 + a)^(1/3)*x^(1/3) - (3*I + Sqrt[3])*(1 + a*x)^(1/3))] + 2*Sqrt[-6 + (6*I)*Sqrt[3]]*(-1 + a)^(1/3)*ArcTan[(3*(1 + a)^(1/3)*x^(1/3))/(Sqrt[3]*(1 + a)^(1/3)*x^(1/3) - (3*I + Sqrt[3])*(1 + a*x)^(1/3))] + (1 + I*Sqrt[3])* (2*(1 + a)^(1/3)*Log[2*(-1 + a)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(1 + a*x)^ (1/3)] - 2*(-1 + a)^(1/3)*Log[2*(1 + a)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(1 + a*x)^(1/3)] - (1 + a)^(1/3)*Log[(-((-1 + a)^(1/3)*x^(1/3)) + (1 + a*x)^ (1/3))*((2*I)*(-1 + a)^(1/3)*x^(1/3) + (I + Sqrt[3])*(1 + a*x)^(1/3))] + ( -1 + a)^(1/3)*Log[(-((1 + a)^(1/3)*x^(1/3)) + (1 + a*x)^(1/3))*((2*I)*(1 + a)^(1/3)*x^(1/3) + (I + Sqrt[3])*(1 + a*x)^(1/3))])))/(8*(-1 + a)^(1/3)*( 1 + a)^(1/3)*x^(1/3))
Time = 0.48 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.38, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {7239, 7269, 25, 615, 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.
| \(\displaystyle \int \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \sqrt [3]{\frac {x}{\left (x^2-1\right )^3 (a x+1)}}dx\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle -\frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \int -\frac {\sqrt [3]{x}}{\sqrt [3]{a x+1} \left (1-x^2\right )}dx}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \int \frac {\sqrt [3]{x}}{\sqrt [3]{a x+1} \left (1-x^2\right )}dx}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \int \left (\frac {\sqrt [3]{x}}{2 (1-x) \sqrt [3]{a x+1}}+\frac {\sqrt [3]{x}}{2 (x+1) \sqrt [3]{a x+1}}\right )dx}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a-1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a-1}}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a+1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a+1}}+\frac {\log (1-x)}{4 \sqrt [3]{a+1}}-\frac {\log (x+1)}{4 \sqrt [3]{a-1}}+\frac {3 \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a-1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a-1}}-\frac {3 \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a+1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a+1}}\right )}{\sqrt [3]{x}}\) |
((1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*((Sqrt[3]* ArcTan[1/Sqrt[3] + (2*(1 + a*x)^(1/3))/(Sqrt[3]*(-1 + a)^(1/3)*x^(1/3))])/ (2*(-1 + a)^(1/3)) - (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + a*x)^(1/3))/(Sqrt [3]*(1 + a)^(1/3)*x^(1/3))])/(2*(1 + a)^(1/3)) + Log[1 - x]/(4*(1 + a)^(1/ 3)) - Log[1 + x]/(4*(-1 + a)^(1/3)) + (3*Log[-x^(1/3) + (1 + a*x)^(1/3)/(- 1 + a)^(1/3)])/(4*(-1 + a)^(1/3)) - (3*Log[-x^(1/3) + (1 + a*x)^(1/3)/(1 + a)^(1/3)])/(4*(1 + a)^(1/3))))/x^(1/3)
3.32.16.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Simp[ a^IntPart[p]*((a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[ p])*z^(q*FracPart[p]))) Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a , m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !F reeQ[z, x]
\[\int \left (\frac {x}{a \,x^{7}-3 a \,x^{5}+x^{6}+3 a \,x^{3}-3 x^{4}-a x +3 x^{2}-1}\right )^{\frac {1}{3}}d x\]
Time = 0.35 (sec) , antiderivative size = 3267, normalized size of antiderivative = 4.88 \[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=\text {Too large to display} \]
[1/4*(sqrt(3)*(a^2 - 1)*sqrt((-a + 1)^(1/3)/(a - 1))*log(-((3*a - 2)*x - s qrt(3)*((a*x^3 - a*x + x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 - a)*x^5 + (a - 1)*x^ 4 - 2*(a^2 - a)*x^3 - 2*(a - 1)*x^2 + (a^2 - a)*x + a - 1)*(x/(a*x^7 - 3*a *x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) - (a*x + 1)*(-a + 1 )^(1/3))*sqrt((-a + 1)^(1/3)/(a - 1)) + 3*(a*x^3 - a*x + x^2 - 1)*(-a + 1) ^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/ 3) + 1)/(x + 1)) + sqrt(3)*(a^2 - 1)*sqrt(-1/(a + 1)^(2/3))*log(((3*a + 2) *x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 + a)*x^5 + (a + 1)*x^4 - 2*(a^2 + a)*x^3 - 2*(a + 1)*x^2 + (a^2 + a)*x + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x + 1)*(a + 1)^(1/3))*sqrt(-1/(a + 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a + 1)^(1 /3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x - 1)) - (a + 1)^(2/3)*(a - 1)*log((x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^ 7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x ^4 - 2*(a + 1)*x^2 + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) + (a + 1)*(-a + 1)^(2/3)*log((x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3 *x^2 - 1))^(1/3) + ((a - 1)*x^4 - 2*(a - 1)*x^2 + a - 1)*(x/(a*x^7 - 3*...
\[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=\int \sqrt [3]{\frac {x}{a x^{7} - 3 a x^{5} + 3 a x^{3} - a x + x^{6} - 3 x^{4} + 3 x^{2} - 1}}\, dx \]
\[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=\int { \left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}} \,d x } \]
\[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=\int { \left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}} \,d x } \]
Timed out. \[ \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx=\int {\left (-\frac {x}{-a\,x^7-x^6+3\,a\,x^5+3\,x^4-3\,a\,x^3-3\,x^2+a\,x+1}\right )}^{1/3} \,d x \]