Integrand size = 23, antiderivative size = 157 \[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=-\frac {\arctan \left (\frac {1+\sqrt [3]{2} \sqrt [3]{d} x}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\arctan \left (\sqrt {-1+d x^3}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {-1+d x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}} \]
-1/6*arctan((1+2^(1/3)*d^(1/3)*x)/(d*x^3-1)^(1/2))*2^(1/3)/d^(2/3)-1/18*ar ctan((d*x^3-1)^(1/2))*2^(1/3)/d^(2/3)-1/18*arctanh((1-2^(1/3)*d^(1/3)*x)*3 ^(1/2)/(d*x^3-1)^(1/2))*2^(1/3)/d^(2/3)*3^(1/2)-1/18*arctanh(1/3*(d*x^3-1) ^(1/2)*3^(1/2))*2^(1/3)/d^(2/3)*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.34 \[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=\frac {x^2 \sqrt {1-d x^3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},d x^3,\frac {d x^3}{4}\right )}{8 \sqrt {-1+d x^3}} \]
Time = 0.21 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {987}
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.
| \(\displaystyle \int \frac {x}{\left (4-d x^3\right ) \sqrt {d x^3-1}} \, dx\) |
| \(\Big \downarrow \) 987 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{d} x+1}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\arctan \left (\sqrt {d x^3-1}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {d x^3-1}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}\) |
-1/3*ArcTan[(1 + 2^(1/3)*d^(1/3)*x)/Sqrt[-1 + d*x^3]]/(2^(2/3)*d^(2/3)) - ArcTan[Sqrt[-1 + d*x^3]]/(9*2^(2/3)*d^(2/3)) - ArcTanh[(Sqrt[3]*(1 - 2^(1/ 3)*d^(1/3)*x))/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*Sqrt[3]*d^(2/3)) - ArcTanh[Sqr t[-1 + d*x^3]/Sqrt[3]]/(3*2^(2/3)*Sqrt[3]*d^(2/3))
3.1.74.3.1 Defintions of rubi rules used
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi th[{q = Rt[d/c, 3]}, Simp[(-q)*(ArcTan[Sqrt[c + d*x^3]/Rt[-c, 2]]/(9*2^(2/3 )*b*Rt[-c, 2])), x] + (-Simp[q*(ArcTan[Rt[-c, 2]*((1 - 2^(1/3)*q*x)/Sqrt[c + d*x^3])]/(3*2^(2/3)*b*Rt[-c, 2])), x] - Simp[q*(ArcTanh[Sqrt[c + d*x^3]/( Sqrt[3]*Rt[-c, 2])]/(3*2^(2/3)*Sqrt[3]*b*Rt[-c, 2])), x] - Simp[q*(ArcTanh[ Sqrt[3]*Rt[-c, 2]*((1 + 2^(1/3)*q*x)/Sqrt[c + d*x^3])]/(3*2^(2/3)*Sqrt[3]*b *Rt[-c, 2])), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4* b*c - a*d, 0] && NegQ[c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.66 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-4\right )}{\sum }\frac {\sqrt {-\frac {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}{2}}\, \sqrt {\frac {x -\frac {1}{d^{\frac {1}{3}}}}{-\frac {3}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}}}\, \sqrt {2}\, \sqrt {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} d +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}-i \sqrt {3}\, d^{\frac {1}{3}}+\underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}+d^{\frac {1}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {-i \left (x +\frac {1}{2 d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right ) \sqrt {3}\, d^{\frac {1}{3}}}}{3}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d^{\frac {2}{3}}}{3}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {1}{3}}}{6}-\frac {i \sqrt {3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha \,d^{\frac {1}{3}}}{2}-\frac {1}{2}, \sqrt {-\frac {i \sqrt {3}}{d^{\frac {1}{3}} \left (-\frac {3}{2 d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {4}{3}} \sqrt {d \,x^{3}-1}}\right )}{9}\) | \(240\) |
| elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-4\right )}{\sum }\frac {\sqrt {-\frac {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}{2}}\, \sqrt {\frac {x -\frac {1}{d^{\frac {1}{3}}}}{-\frac {3}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}}}\, \sqrt {2}\, \sqrt {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} d +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}-i \sqrt {3}\, d^{\frac {1}{3}}+\underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}+d^{\frac {1}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {-i \left (x +\frac {1}{2 d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right ) \sqrt {3}\, d^{\frac {1}{3}}}}{3}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d^{\frac {2}{3}}}{3}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {1}{3}}}{6}-\frac {i \sqrt {3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha \,d^{\frac {1}{3}}}{2}-\frac {1}{2}, \sqrt {-\frac {i \sqrt {3}}{d^{\frac {1}{3}} \left (-\frac {3}{2 d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {4}{3}} \sqrt {d \,x^{3}-1}}\right )}{9}\) | \(240\) |
-1/9*I*2^(1/2)*sum(1/_alpha/d^(4/3)*(-1/2*I*(2*x+1/d^(1/3)+I*3^(1/2)/d^(1/ 3))*d^(1/3))^(1/2)*((x-1/d^(1/3))/(-3/d^(1/3)-I*3^(1/2)/d^(1/3)))^(1/2)*(1 /2*I*(2*x+1/d^(1/3)-I*3^(1/2)/d^(1/3))*d^(1/3))^(1/2)/(d*x^3-1)^(1/2)*(-2* _alpha^2*d+I*3^(1/2)*_alpha*d^(2/3)-I*3^(1/2)*d^(1/3)+_alpha*d^(2/3)+d^(1/ 3))*EllipticPi(1/3*3^(1/2)*(-I*(x+1/2/d^(1/3)+1/2*I*3^(1/2)/d^(1/3))*3^(1/ 2)*d^(1/3))^(1/2),1/3*I*3^(1/2)*_alpha^2*d^(2/3)-1/6*I*3^(1/2)*_alpha*d^(1 /3)-1/6*I*3^(1/2)+1/2*_alpha*d^(1/3)-1/2,(-I*3^(1/2)/d^(1/3)/(-3/2/d^(1/3) -1/2*I*3^(1/2)/d^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-4))
Leaf count of result is larger than twice the leaf count of optimal. 1669 vs. \(2 (110) = 220\).
Time = 0.39 (sec) , antiderivative size = 1669, normalized size of antiderivative = 10.63 \[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=\text {Too large to display} \]
-1/36*(1/432)^(1/6)*(sqrt(-3) + 1)*(d^(-4))^(1/6)*log((d^3*x^9 + 66*d^2*x^ 6 - 72*d*x^3 - 24*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x + sqrt(-3)*(d^5 *x^7 + d^4*x^4 - 2*d^3*x))*(d^(-4))^(2/3) - 6*(1/2)^(1/3)*(d^4*x^8 + 7*d^3 *x^5 - 8*d^2*x^2 - sqrt(-3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2))*(d^(-4))^(1 /3) + 6*sqrt(d*x^3 - 1)*(648*(1/432)^(5/6)*(sqrt(-3)*d^5*x^5 - d^5*x^5)*(d ^(-4))^(5/6) + sqrt(1/3)*(5*d^4*x^6 + 20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x + sqrt(-3)*(d^3*x^7 + 16*d^2*x ^4 - 8*d*x))*(d^(-4))^(1/6)) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) + 1/36*(1/432)^(1/6)*(sqrt(-3) + 1)*(d^(-4))^(1/6)*log((d^3*x^9 + 66*d^2* x^6 - 72*d*x^3 - 24*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x + sqrt(-3)*(d ^5*x^7 + d^4*x^4 - 2*d^3*x))*(d^(-4))^(2/3) - 6*(1/2)^(1/3)*(d^4*x^8 + 7*d ^3*x^5 - 8*d^2*x^2 - sqrt(-3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2))*(d^(-4))^ (1/3) - 6*sqrt(d*x^3 - 1)*(648*(1/432)^(5/6)*(sqrt(-3)*d^5*x^5 - d^5*x^5)* (d^(-4))^(5/6) + sqrt(1/3)*(5*d^4*x^6 + 20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x + sqrt(-3)*(d^3*x^7 + 16*d^2 *x^4 - 8*d*x))*(d^(-4))^(1/6)) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64 )) - 1/36*(1/432)^(1/6)*(sqrt(-3) - 1)*(d^(-4))^(1/6)*log((d^3*x^9 + 66*d^ 2*x^6 - 72*d*x^3 - 24*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x - sqrt(-3)* (d^5*x^7 + d^4*x^4 - 2*d^3*x))*(d^(-4))^(2/3) - 6*(1/2)^(1/3)*(d^4*x^8 + 7 *d^3*x^5 - 8*d^2*x^2 + sqrt(-3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2))*(d^(...
\[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=- \int \frac {x}{d x^{3} \sqrt {d x^{3} - 1} - 4 \sqrt {d x^{3} - 1}}\, dx \]
\[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} - 1} {\left (d x^{3} - 4\right )}} \,d x } \]
\[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} - 1} {\left (d x^{3} - 4\right )}} \,d x } \]
Time = 15.03 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.11 \[ \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx=\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {\left (54\,\sqrt {d\,x^3-1}+54\,\sqrt {3}-54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x\right )\,{\left (\sqrt {d\,x^3-1}-\sqrt {3}+2^{1/3}\,\sqrt {3}\,d^{1/3}\,x\right )}^3}{{\left (2^{2/3}-d^{1/3}\,x\right )}^6}\right )}{2916\,d^{2/3}}+\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {{\left (2\,\sqrt {d\,x^3-1}+2\,\sqrt {3}+2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,3{}\mathrm {i}\right )}^3\,\left (108\,\sqrt {3}-108\,\sqrt {d\,x^3-1}+54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,162{}\mathrm {i}\right )}{{\left (2^{2/3}+2\,d^{1/3}\,x-2^{2/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^6}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{2916\,d^{2/3}}+\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {{\left (2\,\sqrt {d\,x^3-1}-2\,\sqrt {3}-2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,3{}\mathrm {i}\right )}^3\,\left (108\,\sqrt {d\,x^3-1}+108\,\sqrt {3}+54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x-2^{1/3}\,d^{1/3}\,x\,162{}\mathrm {i}\right )}{{\left (2^{2/3}+2\,d^{1/3}\,x+2^{2/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^6}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{2916\,d^{2/3}} \]
(3^(1/2)*314928^(1/3)*log(((54*(d*x^3 - 1)^(1/2) + 54*3^(1/2) - 54*2^(1/3) *3^(1/2)*d^(1/3)*x)*((d*x^3 - 1)^(1/2) - 3^(1/2) + 2^(1/3)*3^(1/2)*d^(1/3) *x)^3)/(2^(2/3) - d^(1/3)*x)^6))/(2916*d^(2/3)) + (3^(1/2)*314928^(1/3)*lo g(((2*(d*x^3 - 1)^(1/2) + 2*3^(1/2) + 2^(1/3)*d^(1/3)*x*3i + 2^(1/3)*3^(1/ 2)*d^(1/3)*x)^3*(108*3^(1/2) - 108*(d*x^3 - 1)^(1/2) + 2^(1/3)*d^(1/3)*x*1 62i + 54*2^(1/3)*3^(1/2)*d^(1/3)*x))/(2^(2/3) - 2^(2/3)*3^(1/2)*1i + 2*d^( 1/3)*x)^6)*((3^(1/2)*1i)/2 - 1/2)^(1/2))/(2916*d^(2/3)) + (3^(1/2)*314928^ (1/3)*log(((2*(d*x^3 - 1)^(1/2) - 2*3^(1/2) + 2^(1/3)*d^(1/3)*x*3i - 2^(1/ 3)*3^(1/2)*d^(1/3)*x)^3*(108*(d*x^3 - 1)^(1/2) + 108*3^(1/2) - 2^(1/3)*d^( 1/3)*x*162i + 54*2^(1/3)*3^(1/2)*d^(1/3)*x))/(2^(2/3)*3^(1/2)*1i + 2^(2/3) + 2*d^(1/3)*x)^6)*((3^(1/2)*1i)/2 + 1/2)^(1/2)*1i)/(2916*d^(2/3))