Integrand size = 20, antiderivative size = 76 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt {3} a}\right )}{a^2}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )}{2 a^2} \]
-1/2*ln(x)/a^2+3/2*ln(a+(-b^3*x-a^3)^(1/3))/a^2-arctan(1/3*(a-2*(-b^3*x-a^ 3)^(1/3))/a*3^(1/2))*3^(1/2)/a^2
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt {3} a}\right )-2 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )+\log \left (a^2-a \sqrt [3]{-a^3-b^3 x}+\left (-a^3-b^3 x\right )^{2/3}\right )}{2 a^2} \]
-1/2*(2*Sqrt[3]*ArcTan[(a - 2*(-a^3 - b^3*x)^(1/3))/(Sqrt[3]*a)] - 2*Log[a + (-a^3 - b^3*x)^(1/3)] + Log[a^2 - a*(-a^3 - b^3*x)^(1/3) + (-a^3 - b^3* x)^(2/3)])/a^2
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {70, 16, 1082, 217}
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.
\(\displaystyle \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 70 |
\(\displaystyle \frac {3 \int \frac {1}{a+\sqrt [3]{-a^3-b^3 x}}d\sqrt [3]{-a^3-b^3 x}}{2 a^2}+\frac {3 \int \frac {1}{a^2-\sqrt [3]{-a^3-b^3 x} a+\left (-a^3-b^3 x\right )^{2/3}}d\sqrt [3]{-a^3-b^3 x}}{2 a}-\frac {\log (x)}{2 a^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {3 \int \frac {1}{a^2-\sqrt [3]{-a^3-b^3 x} a+\left (-a^3-b^3 x\right )^{2/3}}d\sqrt [3]{-a^3-b^3 x}}{2 a}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {3 \int \frac {1}{-\left (-a^3-b^3 x\right )^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{-a^3-b^3 x}}{a}\right )}{a^2}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {\log (x)}{2 a^2}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a^3-b^3 x}}{a}}{\sqrt {3}}\right )}{a^2}+\frac {3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}\) |
-((Sqrt[3]*ArcTan[(1 - (2*(-a^3 - b^3*x)^(1/3))/a)/Sqrt[3]])/a^2) - Log[x] /(2*a^2) + (3*Log[a + (-a^3 - b^3*x)^(1/3)])/(2*a^2)
3.5.27.3.1 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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) , x] + (Simp[3/(2*b*q) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 /3)], x] + Simp[3/(2*b*q^2) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
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_) + (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]
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.26
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a -2 \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )+2 \ln \left (a +\left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )-\ln \left (a^{2}-a \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2 a^{2}}\) | \(96\) |
derivativedivides | \(\frac {-\frac {\ln \left (a^{2}-a \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}+\frac {\ln \left (a +\left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}\) | \(99\) |
default | \(\frac {-\frac {\ln \left (a^{2}-a \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}+\frac {\ln \left (a +\left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}\) | \(99\) |
1/2*(-2*3^(1/2)*arctan(1/3*(a-2*(-b^3*x-a^3)^(1/3))/a*3^(1/2))+2*ln(a+(-b^ 3*x-a^3)^(1/3))-ln(a^2-a*(-b^3*x-a^3)^(1/3)+(-b^3*x-a^3)^(2/3)))/a^2
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} a - 2 \, \sqrt {3} {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \log \left (a^{2} - {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right ) + 2 \, \log \left (a + {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}{2 \, a^{2}} \]
1/2*(2*sqrt(3)*arctan(-1/3*(sqrt(3)*a - 2*sqrt(3)*(-b^3*x - a^3)^(1/3))/a) - log(a^2 - (-b^3*x - a^3)^(1/3)*a + (-b^3*x - a^3)^(2/3)) + 2*log(a + (- b^3*x - a^3)^(1/3)))/a^2
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} + \frac {\log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} \]
exp(-2*I*pi/3)*log(1 - b*(a**3/b**3 + x)**(1/3)/a)*gamma(1/3)/(3*a**2*gamm a(4/3)) - exp(-I*pi/3)*log(1 - b*(a**3/b**3 + x)**(1/3)*exp_polar(2*I*pi/3 )/a)*gamma(1/3)/(3*a**2*gamma(4/3)) + log(1 - b*(a**3/b**3 + x)**(1/3)*exp _polar(4*I*pi/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3))
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac {\log \left (a^{2} - {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\log \left (a + {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}{a^{2}} \]
sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*(-b^3*x - a^3)^(1/3))/a)/a^2 - 1/2*log( a^2 - (-b^3*x - a^3)^(1/3)*a + (-b^3*x - a^3)^(2/3))/a^2 + log(a + (-b^3*x - a^3)^(1/3))/a^2
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac {\log \left (a^{2} - {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\log \left ({\left | a + {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{a^{2}} \]
sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*(-b^3*x - a^3)^(1/3))/a)/a^2 - 1/2*log( a^2 - (-b^3*x - a^3)^(1/3)*a + (-b^3*x - a^3)^(2/3))/a^2 + log(abs(a + (-b ^3*x - a^3)^(1/3)))/a^2
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {\ln \left (9\,a+9\,{\left (-a^3-x\,b^3\right )}^{1/3}\right )}{a^2}+\frac {\ln \left (9\,{\left (-a^3-x\,b^3\right )}^{1/3}+\frac {9\,a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2}-\frac {\ln \left (9\,{\left (-a^3-x\,b^3\right )}^{1/3}-\frac {9\,a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2} \]