Integrand size = 22, antiderivative size = 533 \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx=-\frac {1}{a d x}-\frac {c^{7/6} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{7/6} \left (c d^2+a e^2\right )}+\frac {c^{7/6} d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{7/6} \left (c d^2+a e^2\right )}-\frac {c^{7/6} d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{7/6} \left (c d^2+a e^2\right )}+\frac {e^{7/3} \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{4/3} \left (c d^2+a e^2\right )}+\frac {c^{2/3} e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{c} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \left (c d^2+a e^2\right )}+\frac {c^{7/6} d \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{2 \sqrt {3} a^{7/6} \left (c d^2+a e^2\right )}+\frac {e^{7/3} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{4/3} \left (c d^2+a e^2\right )}-\frac {c^{2/3} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 a^{2/3} \left (c d^2+a e^2\right )}-\frac {e^{7/3} \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{4/3} \left (c d^2+a e^2\right )}+\frac {c^{2/3} e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{c} x^2+c^{2/3} x^4\right )}{12 a^{2/3} \left (c d^2+a e^2\right )} \] Output:
-1/a/d/x-1/3*c^(7/6)*d*arctan(c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)-1/6 *c^(7/6)*d*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)-1/6* c^(7/6)*d*arctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)+1/3*e^ (7/3)*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)*3^(1/2)/d^(1/3))*3^(1/2)/d^(4/3)/(a *e^2+c*d^2)+1/6*c^(2/3)*e*arctan(1/3*(a^(1/3)-2*c^(1/3)*x^2)*3^(1/2)/a^(1/ 3))*3^(1/2)/a^(2/3)/(a*e^2+c*d^2)+1/6*c^(7/6)*d*arctanh(3^(1/2)*a^(1/6)*c^ (1/6)*x/(a^(1/3)+c^(1/3)*x^2))*3^(1/2)/a^(7/6)/(a*e^2+c*d^2)+1/3*e^(7/3)*l n(d^(1/3)+e^(1/3)*x)/d^(4/3)/(a*e^2+c*d^2)-1/6*c^(2/3)*e*ln(a^(1/3)+c^(1/3 )*x^2)/a^(2/3)/(a*e^2+c*d^2)-1/6*e^(7/3)*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2 /3)*x^2)/d^(4/3)/(a*e^2+c*d^2)+1/12*c^(2/3)*e*ln(a^(2/3)-a^(1/3)*c^(1/3)*x ^2+c^(2/3)*x^4)/a^(2/3)/(a*e^2+c*d^2)
Time = 0.50 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx=\frac {-\frac {12 c d}{a}-\frac {12 e^2}{d}-\frac {4 c^{7/6} d x \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{7/6}}+\frac {2 c^{2/3} \left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) x \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{7/6}}-\frac {2 c^{7/6} d x \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{7/6}}+\frac {2 \sqrt {3} c^{2/3} e x \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{2/3}}+\frac {4 \sqrt {3} e^{7/3} x \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{4/3}}+\frac {4 e^{7/3} x \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{4/3}}-\frac {2 c^{2/3} e x \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{a^{2/3}}-\frac {\sqrt {3} c^{7/6} d x \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{7/6}}+\frac {c^{2/3} e x \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{2/3}}+\frac {\sqrt {3} c^{7/6} d x \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{7/6}}+\frac {c^{2/3} e x \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{2/3}}-\frac {2 e^{7/3} x \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{4/3}}}{12 \left (c d^2+a e^2\right ) x} \] Input:
Integrate[1/(x^2*(d + e*x^3)*(a + c*x^6)),x]
Output:
((-12*c*d)/a - (12*e^2)/d - (4*c^(7/6)*d*x*ArcTan[(c^(1/6)*x)/a^(1/6)])/a^ (7/6) + (2*c^(2/3)*(Sqrt[c]*d + Sqrt[3]*Sqrt[a]*e)*x*ArcTan[Sqrt[3] - (2*c ^(1/6)*x)/a^(1/6)])/a^(7/6) - (2*c^(7/6)*d*x*ArcTan[Sqrt[3] + (2*c^(1/6)*x )/a^(1/6)])/a^(7/6) + (2*Sqrt[3]*c^(2/3)*e*x*ArcTan[Sqrt[3] + (2*c^(1/6)*x )/a^(1/6)])/a^(2/3) + (4*Sqrt[3]*e^(7/3)*x*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/ 3))/Sqrt[3]])/d^(4/3) + (4*e^(7/3)*x*Log[d^(1/3) + e^(1/3)*x])/d^(4/3) - ( 2*c^(2/3)*e*x*Log[a^(1/3) + c^(1/3)*x^2])/a^(2/3) - (Sqrt[3]*c^(7/6)*d*x*L og[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/a^(7/6) + (c^(2/3)* e*x*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/a^(2/3) + (Sqr t[3]*c^(7/6)*d*x*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/a ^(7/6) + (c^(2/3)*e*x*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^ 2])/a^(2/3) - (2*e^(7/3)*x*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2]) /d^(4/3))/(12*(c*d^2 + a*e^2)*x)
Time = 0.92 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1837, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+c x^6\right ) \left (d+e x^3\right )} \, dx\) |
| \(\Big \downarrow \) 1837 |
| \(\displaystyle \int \left (-\frac {c x \left (a e+c d x^3\right )}{a \left (a+c x^6\right ) \left (a e^2+c d^2\right )}-\frac {e^3 x}{d \left (d+e x^3\right ) \left (a e^2+c d^2\right )}+\frac {1}{a d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{2 \sqrt {3} (-a)^{7/6} \left (a e^2+c d^2\right )}+\frac {c^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}{2 \sqrt {3} (-a)^{7/6} \left (a e^2+c d^2\right )}+\frac {e^{7/3} \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{4/3} \left (a e^2+c d^2\right )}+\frac {c^{2/3} \left (\sqrt {-a} e+\sqrt {c} d\right ) \log \left (-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{12 (-a)^{7/6} \left (a e^2+c d^2\right )}-\frac {c^{2/3} \left (\sqrt {c} d-\sqrt {-a} e\right ) \log \left (\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{12 (-a)^{7/6} \left (a e^2+c d^2\right )}+\frac {c^{2/3} \left (\sqrt {c} d-\sqrt {-a} e\right ) \log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{6 (-a)^{7/6} \left (a e^2+c d^2\right )}-\frac {c^{2/3} \left (\sqrt {-a} e+\sqrt {c} d\right ) \log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{6 (-a)^{7/6} \left (a e^2+c d^2\right )}-\frac {e^{7/3} \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{4/3} \left (a e^2+c d^2\right )}+\frac {e^{7/3} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{4/3} \left (a e^2+c d^2\right )}-\frac {1}{a d x}\) |
Input:
Int[1/(x^2*(d + e*x^3)*(a + c*x^6)),x]
Output:
-(1/(a*d*x)) - (c^(2/3)*(Sqrt[c]*d + Sqrt[-a]*e)*ArcTan[(1 - (2*c^(1/6)*x) /(-a)^(1/6))/Sqrt[3]])/(2*Sqrt[3]*(-a)^(7/6)*(c*d^2 + a*e^2)) + (c^(2/3)*( Sqrt[c]*d - Sqrt[-a]*e)*ArcTan[(1 + (2*c^(1/6)*x)/(-a)^(1/6))/Sqrt[3]])/(2 *Sqrt[3]*(-a)^(7/6)*(c*d^2 + a*e^2)) + (e^(7/3)*ArcTan[(d^(1/3) - 2*e^(1/3 )*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(4/3)*(c*d^2 + a*e^2)) + (c^(2/3)*(Sqr t[c]*d - Sqrt[-a]*e)*Log[(-a)^(1/6) - c^(1/6)*x])/(6*(-a)^(7/6)*(c*d^2 + a *e^2)) - (c^(2/3)*(Sqrt[c]*d + Sqrt[-a]*e)*Log[(-a)^(1/6) + c^(1/6)*x])/(6 *(-a)^(7/6)*(c*d^2 + a*e^2)) + (e^(7/3)*Log[d^(1/3) + e^(1/3)*x])/(3*d^(4/ 3)*(c*d^2 + a*e^2)) + (c^(2/3)*(Sqrt[c]*d + Sqrt[-a]*e)*Log[(-a)^(1/3) - ( -a)^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*(-a)^(7/6)*(c*d^2 + a*e^2)) - (c^( 2/3)*(Sqrt[c]*d - Sqrt[-a]*e)*Log[(-a)^(1/3) + (-a)^(1/6)*c^(1/6)*x + c^(1 /3)*x^2])/(12*(-a)^(7/6)*(c*d^2 + a*e^2)) - (e^(7/3)*Log[d^(2/3) - d^(1/3) *e^(1/3)*x + e^(2/3)*x^2])/(6*d^(4/3)*(c*d^2 + a*e^2))
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.))/((a_) + (c_.)*(x_)^ (n2_.)), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^n)^q/(a + c*x^( 2*n))), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0 ] && IntegerQ[q] && IntegerQ[m]
Time = 0.22 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(-\frac {\left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right ) e^{3}}{d \left (a \,e^{2}+c \,d^{2}\right )}-\frac {1}{a d x}-\frac {\left (\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {4}{3}} e}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {c \left (\frac {a}{c}\right )^{\frac {4}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{3}-\frac {c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{3}} e}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6}+\frac {d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right ) c}{a \left (a \,e^{2}+c \,d^{2}\right )}\) | \(486\) |
| risch | \(-\frac {1}{a d x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} d^{4} e^{6}+3 a^{2} c \,d^{6} e^{4}+3 a \,c^{2} d^{8} e^{2}+c^{3} d^{10}\right ) \textit {\_Z}^{3}-e^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-256 e^{12} d^{4} a^{13}-1376 e^{10} d^{6} c \,a^{12}-3168 e^{8} d^{8} c^{2} a^{11}-4032 d^{10} e^{6} c^{3} a^{10}-3008 e^{4} d^{12} c^{4} a^{9}-1248 e^{2} d^{14} c^{5} a^{8}-224 d^{16} c^{6} a^{7}\right ) \textit {\_R}^{9}+\left (192 a^{10} e^{13}+544 a^{9} c \,d^{2} e^{11}+512 a^{8} c^{2} d^{4} e^{9}+260 a^{7} c^{3} d^{6} e^{7}+348 a^{6} c^{4} d^{8} e^{5}+396 a^{5} c^{5} d^{10} e^{3}+148 a^{4} c^{6} d^{12} e \right ) \textit {\_R}^{6}+\left (48 a^{5} c^{2} e^{10}-148 a^{4} c^{3} d^{2} e^{8}-4 a^{3} c^{4} d^{4} e^{6}-10 a^{2} c^{5} d^{6} e^{4}-9 a \,c^{6} d^{8} e^{2}-3 c^{7} d^{10}\right ) \textit {\_R}^{3}+3 c^{4} e^{7}\right ) x +\left (-64 a^{12} d^{3} e^{12}-192 a^{11} c \,d^{5} e^{10}-144 a^{10} c^{2} d^{7} e^{8}+64 a^{9} c^{3} d^{9} e^{6}+96 a^{8} c^{4} d^{11} e^{4}-16 a^{6} c^{6} d^{15}\right ) \textit {\_R}^{8}+\left (-64 a^{7} c^{2} d^{3} e^{9}+64 a^{6} c^{3} d^{5} e^{7}\right ) \textit {\_R}^{5}-a^{2} c^{4} d^{3} e^{6} \textit {\_R}^{2}\right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{10} e^{6}+3 a^{9} c \,d^{2} e^{4}+3 a^{8} c^{2} d^{4} e^{2}+a^{7} c^{3} d^{6}\right ) \textit {\_Z}^{6}+\left (2 a^{5} c^{2} e^{3}-6 a^{4} c^{3} d^{2} e \right ) \textit {\_Z}^{3}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-8 e^{12} d^{4} a^{13}-43 e^{10} d^{6} c \,a^{12}-99 e^{8} d^{8} c^{2} a^{11}-126 d^{10} e^{6} c^{3} a^{10}-94 e^{4} d^{12} c^{4} a^{9}-39 e^{2} d^{14} c^{5} a^{8}-7 d^{16} c^{6} a^{7}\right ) \textit {\_R}^{9}+\left (48 a^{10} e^{13}+136 a^{9} c \,d^{2} e^{11}+128 a^{8} c^{2} d^{4} e^{9}+65 a^{7} c^{3} d^{6} e^{7}+87 a^{6} c^{4} d^{8} e^{5}+99 a^{5} c^{5} d^{10} e^{3}+37 a^{4} c^{6} d^{12} e \right ) \textit {\_R}^{6}+\left (96 a^{5} c^{2} e^{10}-296 a^{4} c^{3} d^{2} e^{8}-8 a^{3} c^{4} d^{4} e^{6}-20 a^{2} c^{5} d^{6} e^{4}-18 a \,c^{6} d^{8} e^{2}-6 c^{7} d^{10}\right ) \textit {\_R}^{3}+48 c^{4} e^{7}\right ) x +\left (-4 a^{12} d^{3} e^{12}-12 a^{11} c \,d^{5} e^{10}-9 a^{10} c^{2} d^{7} e^{8}+4 a^{9} c^{3} d^{9} e^{6}+6 a^{8} c^{4} d^{11} e^{4}-a^{6} c^{6} d^{15}\right ) \textit {\_R}^{8}+\left (-32 a^{7} c^{2} d^{3} e^{9}+32 a^{6} c^{3} d^{5} e^{7}\right ) \textit {\_R}^{5}-4 a^{2} c^{4} d^{3} e^{6} \textit {\_R}^{2}\right )\right )}{6}\) | \(981\) |
Input:
int(1/x^2/(e*x^3+d)/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-(-1/3/e/(d/e)^(1/3)*ln(x+(d/e)^(1/3))+1/6/e/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3 )*x+(d/e)^(2/3))+1/3*3^(1/2)/e/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/ 3)*x-1)))*e^3/d/(a*e^2+c*d^2)-1/a/d/x-(1/12*c/a*ln(x^2-3^(1/2)*(a/c)^(1/6) *x+(a/c)^(1/3))*3^(1/2)*(a/c)^(5/6)*d-1/12*c/a*ln(x^2-3^(1/2)*(a/c)^(1/6)* x+(a/c)^(1/3))*(a/c)^(4/3)*e+1/6/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6)-3^(1/2 ))*d-1/6*c/a*(a/c)^(4/3)*arctan(2*x/(a/c)^(1/6)-3^(1/2))*3^(1/2)*e+1/3*(a/ c)^(1/3)*arctan(2*x/(a/c)^(1/6)-3^(1/2))*3^(1/2)*e-1/12*c/a*ln(x^2+3^(1/2) *(a/c)^(1/6)*x+(a/c)^(1/3))*3^(1/2)*(a/c)^(5/6)*d-1/12*ln(x^2+3^(1/2)*(a/c )^(1/6)*x+(a/c)^(1/3))*(a/c)^(1/3)*e+1/6/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6 )+3^(1/2))*d-1/6*(a/c)^(1/3)*arctan(2*x/(a/c)^(1/6)+3^(1/2))*3^(1/2)*e+1/6 *(a/c)^(1/3)*e*ln(x^2+(a/c)^(1/3))+1/3*d/(a/c)^(1/6)*arctan(x/(a/c)^(1/6)) )*c/a/(a*e^2+c*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 4880 vs. \(2 (412) = 824\).
Time = 10.76 (sec) , antiderivative size = 4880, normalized size of antiderivative = 9.16 \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(e*x^3+d)/(c*x^6+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(e*x**3+d)/(c*x**6+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^2/(e*x^3+d)/(c*x^6+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.18 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/(e*x^3+d)/(c*x^6+a),x, algorithm="giac")
Output:
1/3*e^3*(-d/e)^(2/3)*log(abs(x - (-d/e)^(1/3)))/(c*d^4 + a*d^2*e^2) - 1/3* (a*c^5)^(5/6)*d*arctan(x/(a/c)^(1/6))/(a^2*c^4*d^2 + a^3*c^3*e^2) + (-d*e^ 2)^(2/3)*e*arctan(1/3*sqrt(3)*(2*x + (-d/e)^(1/3))/(-d/e)^(1/3))/(sqrt(3)* c*d^4 + sqrt(3)*a*d^2*e^2) - 1/6*(-d*e^2)^(2/3)*e*log(x^2 + x*(-d/e)^(1/3) + (-d/e)^(2/3))/(c*d^4 + a*d^2*e^2) - 1/6*(a*c^5)^(1/3)*e*log(x^2 + (a/c) ^(1/3))/(a*c^2*d^2 + a^2*c*e^2) + 1/6*(sqrt(3)*(a*c^5)^(1/3)*a*c^2*e - (a* c^5)^(5/6)*d)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^4*d^2 + a^3*c^3*e^2) - 1/6*(sqrt(3)*(a*c^5)^(1/3)*a*c^2*e + (a*c^5)^(5/6)*d)*ar ctan((2*x - sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^4*d^2 + a^3*c^3*e^2) + 1/12*((a*c^5)^(1/3)*a*c^2*e + sqrt(3)*(a*c^5)^(5/6)*d)*log(x^2 + sqrt(3) *x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^4*d^2 + a^3*c^3*e^2) + 1/12*((a*c^5)^ (1/3)*a*c^2*e - sqrt(3)*(a*c^5)^(5/6)*d)*log(x^2 - sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^4*d^2 + a^3*c^3*e^2) - 1/(a*d*x)
Time = 61.33 (sec) , antiderivative size = 5967, normalized size of antiderivative = 11.20 \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a + c*x^6)*(d + e*x^3)),x)
Output:
log(((((-(a^5*c^2*e^3 + c*d^3*(-a^7*c^5)^(1/2) - 3*a^4*c^3*d^2*e - 3*a*d*e ^2*(-a^7*c^5)^(1/2))/(a^7*(a*e^2 + c*d^2)^3))^(2/3)*(((5832*a^16*c^6*d^13* e^3*x*(a*e^2 + c*d^2)^2*(8*a^4*e^8 + c^4*d^8 - 6*a*c^3*d^6*e^2 + a^2*c^2*d ^4*e^4) + 11664*a^19*c^6*d^16*e^4*(a*e^2 + c*d^2)^4*(a*e^2 - c*d^2)*(-(a^5 *c^2*e^3 + c*d^3*(-a^7*c^5)^(1/2) - 3*a^4*c^3*d^2*e - 3*a*d*e^2*(-a^7*c^5) ^(1/2))/(a^7*(a*e^2 + c*d^2)^3))^(2/3))*(-(a^5*c^2*e^3 + c*d^3*(-a^7*c^5)^ (1/2) - 3*a^4*c^3*d^2*e - 3*a*d*e^2*(-a^7*c^5)^(1/2))/(a^7*(a*e^2 + c*d^2) ^3))^(1/3))/6 + 972*a^15*c^12*d^24*e^3 + 2916*a^16*c^11*d^22*e^5 - 8748*a^ 17*c^10*d^20*e^7 - 6804*a^18*c^9*d^18*e^9 + 19440*a^19*c^8*d^16*e^11 + 155 52*a^20*c^7*d^14*e^13))/36 + 27*a^13*c^8*d^13*e^4*x*(8*a^4*e^8 + c^4*d^8 + 2*a*c^3*d^6*e^2 - 8*a^3*c*d^2*e^6 + a^2*c^2*d^4*e^4))*(-(a^5*c^2*e^3 + c* d^3*(-a^7*c^5)^(1/2) - 3*a^4*c^3*d^2*e - 3*a*d*e^2*(-a^7*c^5)^(1/2))/(a^7* (a*e^2 + c*d^2)^3))^(1/3))/6 + 9*a^13*c^11*d^18*e^6 + 9*a^14*c^10*d^16*e^8 - 36*a^15*c^9*d^14*e^10)*(-(a^5*c^2*e^3 + c*d^3*(-a^7*c^5)^(1/2) - 3*a^4* c^3*d^2*e - 3*a*d*e^2*(-a^7*c^5)^(1/2))/(216*(a^10*e^6 + a^7*c^3*d^6 + 3*a ^9*c*d^2*e^4 + 3*a^8*c^2*d^4*e^2)))^(1/3) + log(((((-(a^5*c^2*e^3 - c*d^3* (-a^7*c^5)^(1/2) - 3*a^4*c^3*d^2*e + 3*a*d*e^2*(-a^7*c^5)^(1/2))/(a^7*(a*e ^2 + c*d^2)^3))^(2/3)*(((5832*a^16*c^6*d^13*e^3*x*(a*e^2 + c*d^2)^2*(8*a^4 *e^8 + c^4*d^8 - 6*a*c^3*d^6*e^2 + a^2*c^2*d^4*e^4) + 11664*a^19*c^6*d^16* e^4*(a*e^2 + c*d^2)^4*(a*e^2 - c*d^2)*(-(a^5*c^2*e^3 - c*d^3*(-a^7*c^5)...
Time = 0.25 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (d+e x^3\right ) \left (a+c x^6\right )} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(e*x^3+d)/(c*x^6+a),x)
Output:
(4*e**(1/3)*c**(1/3)*a**(2/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**( 1/3)*sqrt(3)))*a*e**2*x + 2*d**(1/3)*sqrt(c)*sqrt(a)*atan((c**(1/6)*a**(1/ 6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d**2*x + 2*d**(1/3)*sqrt (3)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a *c*d*e*x - 2*d**(1/3)*sqrt(c)*sqrt(a)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2* c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d**2*x + 2*d**(1/3)*sqrt(3)*atan((c**(1 /6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*c*d*e*x - 4*d* *(1/3)*sqrt(c)*sqrt(a)*atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d**2*x - d **(1/3)*sqrt(c)*sqrt(a)*sqrt(3)*log( - c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1 /3) + c**(1/3)*x**2)*c*d**2*x + d**(1/3)*sqrt(c)*sqrt(a)*sqrt(3)*log(c**(1 /6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*c*d**2*x - 12*d**(1/3)* c**(1/3)*a**(2/3)*a*e**2 - 12*d**(1/3)*c**(1/3)*a**(2/3)*c*d**2 - 2*e**(1/ 3)*c**(1/3)*a**(2/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*a *e**2*x + 4*e**(1/3)*c**(1/3)*a**(2/3)*log(d**(1/3) + e**(1/3)*x)*a*e**2*x - 2*d**(1/3)*log(a**(1/3) + c**(1/3)*x**2)*a*c*d*e*x + d**(1/3)*log( - c* *(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*a*c*d*e*x + d**(1/3) *log(c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*a*c*d*e*x)/(1 2*d**(1/3)*c**(1/3)*a**(2/3)*a*d*x*(a*e**2 + c*d**2))