Integrand size = 32, antiderivative size = 44 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {f} \left (e-2 (d-f) x^3\right )}{\sqrt {d} e}\right )}{6 \sqrt {d} e \sqrt {f}} \]
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {f} \left (e-2 d x^3+2 f x^3\right )}{\sqrt {d} e}\right )}{6 \sqrt {d} e \sqrt {f}} \]
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6, 1690, 1083, 219}
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.
\(\displaystyle \int \frac {x^2}{-4 d f x^6+e^2+4 e f x^3+4 f^2 x^6} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^2}{x^6 \left (4 f^2-4 d f\right )+e^2+4 e f x^3}dx\) |
\(\Big \downarrow \) 1690 |
\(\displaystyle \frac {1}{3} \int \frac {1}{-4 (d-f) f x^6+4 e f x^3+e^2}dx^3\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {2}{3} \int \frac {1}{16 d e^2 f-x^6}d\left (4 e f-8 (d-f) f x^3\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {4 e f-8 f x^3 (d-f)}{4 \sqrt {d} e \sqrt {f}}\right )}{6 \sqrt {d} e \sqrt {f}}\) |
3.6.39.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\frac {2 \left (4 d f -4 f^{2}\right ) x^{3}-4 e f}{4 \sqrt {d f}\, e}\right )}{6 \sqrt {d f}\, e}\) | \(42\) |
risch | \(\frac {\ln \left (\left (-2 \sqrt {d f}-2 f \right ) x^{3}-e \right )}{12 \sqrt {d f}\, e}-\frac {\ln \left (\left (-2 \sqrt {d f}+2 f \right ) x^{3}+e \right )}{12 \sqrt {d f}\, e}\) | \(60\) |
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 3.82 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=\left [\frac {\sqrt {d f} \log \left (-\frac {4 \, {\left (d^{2} f - 2 \, d f^{2} + f^{3}\right )} x^{6} - 4 \, {\left (d e f - e f^{2}\right )} x^{3} + d e^{2} + e^{2} f + 2 \, {\left (2 \, {\left (d e - e f\right )} x^{3} - e^{2}\right )} \sqrt {d f}}{4 \, {\left (d f - f^{2}\right )} x^{6} - 4 \, e f x^{3} - e^{2}}\right )}{12 \, d e f}, \frac {\sqrt {-d f} \arctan \left (-\frac {{\left (2 \, {\left (d - f\right )} x^{3} - e\right )} \sqrt {-d f}}{d e}\right )}{6 \, d e f}\right ] \]
[1/12*sqrt(d*f)*log(-(4*(d^2*f - 2*d*f^2 + f^3)*x^6 - 4*(d*e*f - e*f^2)*x^ 3 + d*e^2 + e^2*f + 2*(2*(d*e - e*f)*x^3 - e^2)*sqrt(d*f))/(4*(d*f - f^2)* x^6 - 4*e*f*x^3 - e^2))/(d*e*f), 1/6*sqrt(-d*f)*arctan(-(2*(d - f)*x^3 - e )*sqrt(-d*f)/(d*e))/(d*e*f)]
Time = 0.49 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.70 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=- \frac {\frac {\sqrt {\frac {1}{d f}} \log {\left (x^{3} + \frac {- d e \sqrt {\frac {1}{d f}} - e}{2 d - 2 f} \right )}}{12} - \frac {\sqrt {\frac {1}{d f}} \log {\left (x^{3} + \frac {d e \sqrt {\frac {1}{d f}} - e}{2 d - 2 f} \right )}}{12}}{e} \]
-(sqrt(1/(d*f))*log(x**3 + (-d*e*sqrt(1/(d*f)) - e)/(2*d - 2*f))/12 - sqrt (1/(d*f))*log(x**3 + (d*e*sqrt(1/(d*f)) - e)/(2*d - 2*f))/12)/e
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.52 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=\frac {\log \left (\frac {2 \, {\left (d f - f^{2}\right )} x^{3} - e f + \sqrt {d f} e}{2 \, {\left (d f - f^{2}\right )} x^{3} - e f - \sqrt {d f} e}\right )}{12 \, \sqrt {d f} e} \]
1/12*log((2*(d*f - f^2)*x^3 - e*f + sqrt(d*f)*e)/(2*(d*f - f^2)*x^3 - e*f - sqrt(d*f)*e))/(sqrt(d*f)*e)
Time = 0.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=-\frac {\arctan \left (\frac {2 \, d f x^{3} - 2 \, f^{2} x^{3} - e f}{\sqrt {-d f} e}\right )}{6 \, \sqrt {-d f} e} \]
Time = 17.36 (sec) , antiderivative size = 923, normalized size of antiderivative = 20.98 \[ \int \frac {x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx=\frac {\mathrm {atan}\left (\frac {\frac {\left (x^3\,\left (32\,d^3\,f^3-96\,d^2\,f^4+96\,d\,f^5-32\,f^6\right )+\frac {x^3\,\left (-64\,e\,d^3\,f^4+192\,e\,d^2\,f^5-192\,e\,d\,f^6+64\,e\,f^7\right )+16\,e^2\,f^6-48\,d\,e^2\,f^5+48\,d^2\,e^2\,f^4-16\,d^3\,e^2\,f^3-\frac {\frac {x^3\,\left (-384\,d^3\,e^2\,f^5+1152\,d^2\,e^2\,f^6-1152\,d\,e^2\,f^7+384\,e^2\,f^8\right )}{12}+16\,e^3\,f^7-48\,d\,e^3\,f^6+48\,d^2\,e^3\,f^5-16\,d^3\,e^3\,f^4}{\sqrt {d}\,e\,\sqrt {f}}}{\sqrt {d}\,e\,\sqrt {f}}\right )\,1{}\mathrm {i}}{\sqrt {d}\,e\,\sqrt {f}}+\frac {\left (x^3\,\left (32\,d^3\,f^3-96\,d^2\,f^4+96\,d\,f^5-32\,f^6\right )-\frac {x^3\,\left (-64\,e\,d^3\,f^4+192\,e\,d^2\,f^5-192\,e\,d\,f^6+64\,e\,f^7\right )+16\,e^2\,f^6-48\,d\,e^2\,f^5+48\,d^2\,e^2\,f^4-16\,d^3\,e^2\,f^3+\frac {\frac {x^3\,\left (-384\,d^3\,e^2\,f^5+1152\,d^2\,e^2\,f^6-1152\,d\,e^2\,f^7+384\,e^2\,f^8\right )}{12}+16\,e^3\,f^7-48\,d\,e^3\,f^6+48\,d^2\,e^3\,f^5-16\,d^3\,e^3\,f^4}{\sqrt {d}\,e\,\sqrt {f}}}{\sqrt {d}\,e\,\sqrt {f}}\right )\,1{}\mathrm {i}}{\sqrt {d}\,e\,\sqrt {f}}}{\frac {x^3\,\left (32\,d^3\,f^3-96\,d^2\,f^4+96\,d\,f^5-32\,f^6\right )+\frac {x^3\,\left (-64\,e\,d^3\,f^4+192\,e\,d^2\,f^5-192\,e\,d\,f^6+64\,e\,f^7\right )+16\,e^2\,f^6-48\,d\,e^2\,f^5+48\,d^2\,e^2\,f^4-16\,d^3\,e^2\,f^3-\frac {\frac {x^3\,\left (-384\,d^3\,e^2\,f^5+1152\,d^2\,e^2\,f^6-1152\,d\,e^2\,f^7+384\,e^2\,f^8\right )}{12}+16\,e^3\,f^7-48\,d\,e^3\,f^6+48\,d^2\,e^3\,f^5-16\,d^3\,e^3\,f^4}{\sqrt {d}\,e\,\sqrt {f}}}{\sqrt {d}\,e\,\sqrt {f}}}{\sqrt {d}\,e\,\sqrt {f}}-\frac {x^3\,\left (32\,d^3\,f^3-96\,d^2\,f^4+96\,d\,f^5-32\,f^6\right )-\frac {x^3\,\left (-64\,e\,d^3\,f^4+192\,e\,d^2\,f^5-192\,e\,d\,f^6+64\,e\,f^7\right )+16\,e^2\,f^6-48\,d\,e^2\,f^5+48\,d^2\,e^2\,f^4-16\,d^3\,e^2\,f^3+\frac {\frac {x^3\,\left (-384\,d^3\,e^2\,f^5+1152\,d^2\,e^2\,f^6-1152\,d\,e^2\,f^7+384\,e^2\,f^8\right )}{12}+16\,e^3\,f^7-48\,d\,e^3\,f^6+48\,d^2\,e^3\,f^5-16\,d^3\,e^3\,f^4}{\sqrt {d}\,e\,\sqrt {f}}}{\sqrt {d}\,e\,\sqrt {f}}}{\sqrt {d}\,e\,\sqrt {f}}}\right )\,1{}\mathrm {i}}{6\,\sqrt {d}\,e\,\sqrt {f}} \]
(atan((((x^3*(96*d*f^5 - 32*f^6 - 96*d^2*f^4 + 32*d^3*f^3) + (x^3*(64*e*f^ 7 + 192*d^2*e*f^5 - 64*d^3*e*f^4 - 192*d*e*f^6) + 16*e^2*f^6 - 48*d*e^2*f^ 5 + 48*d^2*e^2*f^4 - 16*d^3*e^2*f^3 - ((x^3*(384*e^2*f^8 - 1152*d*e^2*f^7 + 1152*d^2*e^2*f^6 - 384*d^3*e^2*f^5))/12 + 16*e^3*f^7 - 48*d*e^3*f^6 + 48 *d^2*e^3*f^5 - 16*d^3*e^3*f^4)/(d^(1/2)*e*f^(1/2)))/(d^(1/2)*e*f^(1/2)))*1 i)/(d^(1/2)*e*f^(1/2)) + ((x^3*(96*d*f^5 - 32*f^6 - 96*d^2*f^4 + 32*d^3*f^ 3) - (x^3*(64*e*f^7 + 192*d^2*e*f^5 - 64*d^3*e*f^4 - 192*d*e*f^6) + 16*e^2 *f^6 - 48*d*e^2*f^5 + 48*d^2*e^2*f^4 - 16*d^3*e^2*f^3 + ((x^3*(384*e^2*f^8 - 1152*d*e^2*f^7 + 1152*d^2*e^2*f^6 - 384*d^3*e^2*f^5))/12 + 16*e^3*f^7 - 48*d*e^3*f^6 + 48*d^2*e^3*f^5 - 16*d^3*e^3*f^4)/(d^(1/2)*e*f^(1/2)))/(d^( 1/2)*e*f^(1/2)))*1i)/(d^(1/2)*e*f^(1/2)))/((x^3*(96*d*f^5 - 32*f^6 - 96*d^ 2*f^4 + 32*d^3*f^3) + (x^3*(64*e*f^7 + 192*d^2*e*f^5 - 64*d^3*e*f^4 - 192* d*e*f^6) + 16*e^2*f^6 - 48*d*e^2*f^5 + 48*d^2*e^2*f^4 - 16*d^3*e^2*f^3 - ( (x^3*(384*e^2*f^8 - 1152*d*e^2*f^7 + 1152*d^2*e^2*f^6 - 384*d^3*e^2*f^5))/ 12 + 16*e^3*f^7 - 48*d*e^3*f^6 + 48*d^2*e^3*f^5 - 16*d^3*e^3*f^4)/(d^(1/2) *e*f^(1/2)))/(d^(1/2)*e*f^(1/2)))/(d^(1/2)*e*f^(1/2)) - (x^3*(96*d*f^5 - 3 2*f^6 - 96*d^2*f^4 + 32*d^3*f^3) - (x^3*(64*e*f^7 + 192*d^2*e*f^5 - 64*d^3 *e*f^4 - 192*d*e*f^6) + 16*e^2*f^6 - 48*d*e^2*f^5 + 48*d^2*e^2*f^4 - 16*d^ 3*e^2*f^3 + ((x^3*(384*e^2*f^8 - 1152*d*e^2*f^7 + 1152*d^2*e^2*f^6 - 384*d ^3*e^2*f^5))/12 + 16*e^3*f^7 - 48*d*e^3*f^6 + 48*d^2*e^3*f^5 - 16*d^3*e...