Integrand size = 17, antiderivative size = 145 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=\frac {a x}{c}-\frac {(b c-a d) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{c} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} c^{4/3} d^{2/3}}+\frac {(b c-a d) \log \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{3 c^{4/3} d^{2/3}}-\frac {(b c-a d) \log \left (d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2\right )}{6 c^{4/3} d^{2/3}} \]
a*x/c+1/3*(-a*d+b*c)*ln(d^(1/3)+c^(1/3)*x)/c^(4/3)/d^(2/3)-1/6*(-a*d+b*c)* ln(d^(2/3)-c^(1/3)*d^(1/3)*x+c^(2/3)*x^2)/c^(4/3)/d^(2/3)-1/3*(-a*d+b*c)*a rctan(1/3*(d^(1/3)-2*c^(1/3)*x)/d^(1/3)*3^(1/2))/c^(4/3)/d^(2/3)*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.89 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=\frac {6 a \sqrt [3]{c} d^{2/3} x-2 \sqrt {3} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{c} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )+2 (b c-a d) \log \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )-(b c-a d) \log \left (d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2\right )}{6 c^{4/3} d^{2/3}} \]
(6*a*c^(1/3)*d^(2/3)*x - 2*Sqrt[3]*(b*c - a*d)*ArcTan[(1 - (2*c^(1/3)*x)/d ^(1/3))/Sqrt[3]] + 2*(b*c - a*d)*Log[d^(1/3) + c^(1/3)*x] - (b*c - a*d)*Lo g[d^(2/3) - c^(1/3)*d^(1/3)*x + c^(2/3)*x^2])/(6*c^(4/3)*d^(2/3))
Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {898, 913, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx\) |
| \(\Big \downarrow \) 898 |
| \(\displaystyle \int \frac {a x^3+b}{c x^3+d}dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {(b c-a d) \int \frac {1}{c x^3+d}dx}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{c} x+\sqrt [3]{d}}dx}{3 d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{c} \left (\sqrt [3]{d}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx}{2 \sqrt [3]{c}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx+\frac {\int \frac {\sqrt [3]{c} \left (\sqrt [3]{d}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx}{2 \sqrt [3]{c}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt [3]{d}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{c}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{c} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{c} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )}{2 \sqrt [3]{c}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{c}+\frac {a x}{c}\) |
(a*x)/c + ((b*c - a*d)*(Log[d^(1/3) + c^(1/3)*x]/(3*c^(1/3)*d^(2/3)) + (-( (Sqrt[3]*ArcTan[(1 - (2*c^(1/3)*x)/d^(1/3))/Sqrt[3]])/c^(1/3)) - Log[d^(2/ 3) - c^(1/3)*d^(1/3)*x + c^(2/3)*x^2]/(2*c^(1/3)))/(3*d^(2/3))))/c
3.3.75.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> Int[x^(n*(p + q))*(b + a/x^n)^p*(d + c/x^n)^q, x] /; FreeQ[{a, b, c, d , n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[p, q] && NegQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.29
| method | result | size |
| risch | \(\frac {a x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (-a d +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 c^{2}}\) | \(42\) |
| default | \(\frac {a x}{c}+\frac {\left (\frac {\ln \left (x +\left (\frac {d}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {d}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{c}\right )^{\frac {1}{3}} x +\left (\frac {d}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {d}{c}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {d}{c}\right )^{\frac {2}{3}}}\right ) \left (-a d +b c \right )}{c}\) | \(110\) |
Time = 0.26 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.69 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=\left [\frac {6 \, a c d^{2} x - 3 \, \sqrt {\frac {1}{3}} {\left (b c^{2} d - a c d^{2}\right )} \sqrt {\frac {\left (-c d^{2}\right )^{\frac {1}{3}}}{c}} \log \left (\frac {2 \, c d x^{3} + 3 \, \left (-c d^{2}\right )^{\frac {1}{3}} d x - d^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d x^{2} + \left (-c d^{2}\right )^{\frac {2}{3}} x + \left (-c d^{2}\right )^{\frac {1}{3}} d\right )} \sqrt {\frac {\left (-c d^{2}\right )^{\frac {1}{3}}}{c}}}{c x^{3} + d}\right ) - \left (-c d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x^{2} - \left (-c d^{2}\right )^{\frac {2}{3}} x - \left (-c d^{2}\right )^{\frac {1}{3}} d\right ) + 2 \, \left (-c d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x + \left (-c d^{2}\right )^{\frac {2}{3}}\right )}{6 \, c^{2} d^{2}}, \frac {6 \, a c d^{2} x + 6 \, \sqrt {\frac {1}{3}} {\left (b c^{2} d - a c d^{2}\right )} \sqrt {-\frac {\left (-c d^{2}\right )^{\frac {1}{3}}}{c}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-c d^{2}\right )^{\frac {2}{3}} x + \left (-c d^{2}\right )^{\frac {1}{3}} d\right )} \sqrt {-\frac {\left (-c d^{2}\right )^{\frac {1}{3}}}{c}}}{d^{2}}\right ) - \left (-c d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x^{2} - \left (-c d^{2}\right )^{\frac {2}{3}} x - \left (-c d^{2}\right )^{\frac {1}{3}} d\right ) + 2 \, \left (-c d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x + \left (-c d^{2}\right )^{\frac {2}{3}}\right )}{6 \, c^{2} d^{2}}\right ] \]
[1/6*(6*a*c*d^2*x - 3*sqrt(1/3)*(b*c^2*d - a*c*d^2)*sqrt((-c*d^2)^(1/3)/c) *log((2*c*d*x^3 + 3*(-c*d^2)^(1/3)*d*x - d^2 - 3*sqrt(1/3)*(2*c*d*x^2 + (- c*d^2)^(2/3)*x + (-c*d^2)^(1/3)*d)*sqrt((-c*d^2)^(1/3)/c))/(c*x^3 + d)) - (-c*d^2)^(2/3)*(b*c - a*d)*log(c*d*x^2 - (-c*d^2)^(2/3)*x - (-c*d^2)^(1/3) *d) + 2*(-c*d^2)^(2/3)*(b*c - a*d)*log(c*d*x + (-c*d^2)^(2/3)))/(c^2*d^2), 1/6*(6*a*c*d^2*x + 6*sqrt(1/3)*(b*c^2*d - a*c*d^2)*sqrt(-(-c*d^2)^(1/3)/c )*arctan(sqrt(1/3)*(2*(-c*d^2)^(2/3)*x + (-c*d^2)^(1/3)*d)*sqrt(-(-c*d^2)^ (1/3)/c)/d^2) - (-c*d^2)^(2/3)*(b*c - a*d)*log(c*d*x^2 - (-c*d^2)^(2/3)*x - (-c*d^2)^(1/3)*d) + 2*(-c*d^2)^(2/3)*(b*c - a*d)*log(c*d*x + (-c*d^2)^(2 /3)))/(c^2*d^2)]
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.49 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=\frac {a x}{c} + \operatorname {RootSum} {\left (27 t^{3} c^{4} d^{2} + a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log {\left (- \frac {3 t c d}{a d - b c} + x \right )} \right )\right )} \]
a*x/c + RootSum(27*_t**3*c**4*d**2 + a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b** 2*c**2*d - b**3*c**3, Lambda(_t, _t*log(-3*_t*c*d/(a*d - b*c) + x)))
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=\frac {a x}{c} + \frac {\sqrt {3} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{c}\right )^{\frac {1}{3}}}\right )}{3 \, c^{2} \left (\frac {d}{c}\right )^{\frac {2}{3}}} - \frac {{\left (b c - a d\right )} \log \left (x^{2} - x \left (\frac {d}{c}\right )^{\frac {1}{3}} + \left (\frac {d}{c}\right )^{\frac {2}{3}}\right )}{6 \, c^{2} \left (\frac {d}{c}\right )^{\frac {2}{3}}} + \frac {{\left (b c - a d\right )} \log \left (x + \left (\frac {d}{c}\right )^{\frac {1}{3}}\right )}{3 \, c^{2} \left (\frac {d}{c}\right )^{\frac {2}{3}}} \]
a*x/c + 1/3*sqrt(3)*(b*c - a*d)*arctan(1/3*sqrt(3)*(2*x - (d/c)^(1/3))/(d/ c)^(1/3))/(c^2*(d/c)^(2/3)) - 1/6*(b*c - a*d)*log(x^2 - x*(d/c)^(1/3) + (d /c)^(2/3))/(c^2*(d/c)^(2/3)) + 1/3*(b*c - a*d)*log(x + (d/c)^(1/3))/(c^2*( d/c)^(2/3))
Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=-\frac {\sqrt {3} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{c}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-c^{2} d\right )^{\frac {2}{3}}} - \frac {{\left (b c - a d\right )} \log \left (x^{2} + x \left (-\frac {d}{c}\right )^{\frac {1}{3}} + \left (-\frac {d}{c}\right )^{\frac {2}{3}}\right )}{6 \, \left (-c^{2} d\right )^{\frac {2}{3}}} + \frac {a x}{c} - \frac {{\left (b c - a d\right )} \left (-\frac {d}{c}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{c}\right )^{\frac {1}{3}} \right |}\right )}{3 \, c d} \]
-1/3*sqrt(3)*(b*c - a*d)*arctan(1/3*sqrt(3)*(2*x + (-d/c)^(1/3))/(-d/c)^(1 /3))/(-c^2*d)^(2/3) - 1/6*(b*c - a*d)*log(x^2 + x*(-d/c)^(1/3) + (-d/c)^(2 /3))/(-c^2*d)^(2/3) + a*x/c - 1/3*(b*c - a*d)*(-d/c)^(1/3)*log(abs(x - (-d /c)^(1/3)))/(c*d)
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.85 \[ \int \frac {a+\frac {b}{x^3}}{c+\frac {d}{x^3}} \, dx=\frac {a\,x}{c}-\frac {\ln \left (c^{1/3}\,x+d^{1/3}\right )\,\left (a\,d-b\,c\right )}{3\,c^{4/3}\,d^{2/3}}+\frac {\ln \left (d^{1/3}-2\,c^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )}{3\,c^{4/3}\,d^{2/3}}-\frac {\ln \left (2\,c^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )}{3\,c^{4/3}\,d^{2/3}} \]