Integrand size = 19, antiderivative size = 74 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3+b^3 x}}{\sqrt {3} a}\right )}{a}+\frac {\log (x)}{2 a}-\frac {3 \log \left (a+\sqrt [3]{-a^3+b^3 x}\right )}{2 a} \]
1/2*ln(x)/a-3/2*ln(a+(b^3*x-a^3)^(1/3))/a-arctan(1/3*(a-2*(b^3*x-a^3)^(1/3 ))/a*3^(1/2))*3^(1/2)/a
Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3+b^3 x}}{\sqrt {3} a}\right )-2 \log \left (a+\sqrt [3]{-a^3+b^3 x}\right )+\log \left (a^2-a \sqrt [3]{-a^3+b^3 x}+\left (-a^3+b^3 x\right )^{2/3}\right )}{2 a} \]
(-2*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)])/(2*a)
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {68, 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.
\(\displaystyle \int \frac {1}{x \sqrt [3]{b^3 x-a^3}} \, dx\) |
\(\Big \downarrow \) 68 |
\(\displaystyle -\frac {3 \int \frac {1}{a+\sqrt [3]{b^3 x-a^3}}d\sqrt [3]{b^3 x-a^3}}{2 a}+\frac {3}{2} \int \frac {1}{a^2-\sqrt [3]{b^3 x-a^3} a+\left (b^3 x-a^3\right )^{2/3}}d\sqrt [3]{b^3 x-a^3}+\frac {\log (x)}{2 a}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {3}{2} \int \frac {1}{a^2-\sqrt [3]{b^3 x-a^3} a+\left (b^3 x-a^3\right )^{2/3}}d\sqrt [3]{b^3 x-a^3}-\frac {3 \log \left (\sqrt [3]{b^3 x-a^3}+a\right )}{2 a}+\frac {\log (x)}{2 a}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {3 \int \frac {1}{-\left (b^3 x-a^3\right )^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b^3 x-a^3}}{a}\right )}{a}-\frac {3 \log \left (\sqrt [3]{b^3 x-a^3}+a\right )}{2 a}+\frac {\log (x)}{2 a}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b^3 x-a^3}}{a}}{\sqrt {3}}\right )}{a}-\frac {3 \log \left (\sqrt [3]{b^3 x-a^3}+a\right )}{2 a}+\frac {\log (x)}{2 a}\) |
-((Sqrt[3]*ArcTan[(1 - (2*(-a^3 + b^3*x)^(1/3))/a)/Sqrt[3]])/a) + Log[x]/( 2*a) - (3*Log[a + (-a^3 + b^3*x)^(1/3)])/(2*a)
3.5.22.3.1 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_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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 = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.22
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a -2 \left (b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )-2 \ln \left (a +\left (b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )+\ln \left (a^{2}-a \left (b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2 a}\) | \(90\) |
derivativedivides | \(-\frac {\ln \left (a +\left (b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {\frac {\ln \left (a^{2}-a \left (b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 \left (b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\) | \(96\) |
default | \(-\frac {\ln \left (a +\left (b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {\frac {\ln \left (a^{2}-a \left (b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 \left (b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\) | \(96\) |
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
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} a - 2 \, \sqrt {3} {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}}}{3 \, a}\right ) + \log \left (a^{2} - {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right ) - 2 \, \log \left (a + {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}{2 \, a} \]
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
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.81 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=- \frac {e^{- \frac {i \pi }{3}} \log {\left (- \frac {a e^{\frac {i \pi }{3}}}{b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} + \frac {\log {\left (- \frac {a e^{i \pi }}{b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} - \frac {e^{\frac {i \pi }{3}} \log {\left (- \frac {a e^{\frac {5 i \pi }{3}}}{b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} \]
-exp(-I*pi/3)*log(-a*exp_polar(I*pi/3)/(b*(-a**3/b**3 + x)**(1/3)) + 1)*ga mma(-1/3)/(3*a*gamma(2/3)) + log(-a*exp_polar(I*pi)/(b*(-a**3/b**3 + x)**( 1/3)) + 1)*gamma(-1/3)/(3*a*gamma(2/3)) - exp(I*pi/3)*log(-a*exp_polar(5*I *pi/3)/(b*(-a**3/b**3 + x)**(1/3)) + 1)*gamma(-1/3)/(3*a*gamma(2/3))
Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} + \frac {\log \left (a^{2} - {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a} - \frac {\log \left (a + {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}{a} \]
sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*(b^3*x - a^3)^(1/3))/a)/a + 1/2*log(a^2 - (b^3*x - a^3)^(1/3)*a + (b^3*x - a^3)^(2/3))/a - log(a + (b^3*x - a^3)^ (1/3))/a
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} + \frac {\log \left (a^{2} - {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a} - \frac {\log \left ({\left | a + {\left (b^{3} x - a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{a} \]
sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*(b^3*x - a^3)^(1/3))/a)/a + 1/2*log(a^2 - (b^3*x - a^3)^(1/3)*a + (b^3*x - a^3)^(2/3))/a - log(abs(a + (b^3*x - a ^3)^(1/3)))/a
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51 \[ \int \frac {1}{x \sqrt [3]{-a^3+b^3 x}} \, dx=-\frac {\ln \left (9\,a+9\,{\left (b^3\,x-a^3\right )}^{1/3}\right )}{a}-\frac {\ln \left (\frac {9\,a\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}+9\,{\left (b^3\,x-a^3\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a}+\frac {\ln \left (\frac {9\,a\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}+9\,{\left (b^3\,x-a^3\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a} \]