Integrand size = 13, antiderivative size = 88 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=3 \sqrt [3]{-a+x}+\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{a}}\right )+\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right ) \]
3*(-a+x)^(1/3)+1/2*a^(1/3)*ln(x)-3/2*a^(1/3)*ln(a^(1/3)+(-a+x)^(1/3))+a^(1 /3)*arctan(1/3*(a^(1/3)-2*(-a+x)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=3 \sqrt [3]{-a+x}+\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right )+\frac {1}{2} \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \]
3*(-a + x)^(1/3) + Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*(-a + x)^(1/3))/a^(1/3)) /Sqrt[3]] - a^(1/3)*Log[a^(1/3) + (-a + x)^(1/3)] + (a^(1/3)*Log[a^(2/3) - a^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)])/2
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {60, 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.
\(\displaystyle \int \frac {\sqrt [3]{x-a}}{x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 3 \sqrt [3]{x-a}-a \int \frac {1}{x (x-a)^{2/3}}dx\) |
\(\Big \downarrow \) 70 |
\(\displaystyle 3 \sqrt [3]{x-a}-a \left (\frac {3 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{x-a}}d\sqrt [3]{x-a}}{2 a^{2/3}}+\frac {3 \int \frac {1}{a^{2/3}-\sqrt [3]{x-a} \sqrt [3]{a}+(x-a)^{2/3}}d\sqrt [3]{x-a}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 a^{2/3}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 3 \sqrt [3]{x-a}-a \left (\frac {3 \int \frac {1}{a^{2/3}-\sqrt [3]{x-a} \sqrt [3]{a}+(x-a)^{2/3}}d\sqrt [3]{x-a}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right )}{2 a^{2/3}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle 3 \sqrt [3]{x-a}-a \left (\frac {3 \int \frac {1}{-(x-a)^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{x-a}}{\sqrt [3]{a}}\right )}{a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right )}{2 a^{2/3}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 3 \sqrt [3]{x-a}-a \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-a}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right )}{2 a^{2/3}}\right )\) |
3*(-a + x)^(1/3) - a*(-((Sqrt[3]*ArcTan[(1 - (2*(-a + x)^(1/3))/a^(1/3))/S qrt[3]])/a^(2/3)) - Log[x]/(2*a^(2/3)) + (3*Log[a^(1/3) + (-a + x)^(1/3)]) /(2*a^(2/3)))
3.3.24.3.1 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_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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 = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(3 \left (-a +x \right )^{\frac {1}{3}}-3 \left (\frac {\ln \left (a^{\frac {1}{3}}+\left (-a +x \right )^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (-a +x \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (-a +x \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (-a +x \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\) | \(89\) |
default | \(3 \left (-a +x \right )^{\frac {1}{3}}-3 \left (\frac {\ln \left (a^{\frac {1}{3}}+\left (-a +x \right )^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (-a +x \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (-a +x \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (-a +x \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\) | \(89\) |
3*(-a+x)^(1/3)-3*(1/3/a^(2/3)*ln(a^(1/3)+(-a+x)^(1/3))-1/6/a^(2/3)*ln((-a+ x)^(2/3)-a^(1/3)*(-a+x)^(1/3)+a^(2/3))+1/3/a^(2/3)*3^(1/2)*arctan(1/3*3^(1 /2)*(2/a^(1/3)*(-a+x)^(1/3)-1)))*a
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=\sqrt {3} \left (-a\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} a - 2 \, \sqrt {3} \left (-a\right )^{\frac {2}{3}} {\left (-a + x\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \frac {1}{2} \, \left (-a\right )^{\frac {1}{3}} \log \left (\left (-a\right )^{\frac {2}{3}} + \left (-a\right )^{\frac {1}{3}} {\left (-a + x\right )}^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {2}{3}}\right ) + \left (-a\right )^{\frac {1}{3}} \log \left (-\left (-a\right )^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-a + x\right )}^{\frac {1}{3}} \]
sqrt(3)*(-a)^(1/3)*arctan(-1/3*(sqrt(3)*a - 2*sqrt(3)*(-a)^(2/3)*(-a + x)^ (1/3))/a) - 1/2*(-a)^(1/3)*log((-a)^(2/3) + (-a)^(1/3)*(-a + x)^(1/3) + (- a + x)^(2/3)) + (-a)^(1/3)*log(-(-a)^(1/3) + (-a + x)^(1/3)) + 3*(-a + x)^ (1/3)
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=\frac {4 \sqrt [3]{a} e^{- \frac {i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{- a + x} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} - \frac {4 \sqrt [3]{a} \log {\left (1 - \frac {\sqrt [3]{- a + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{a} e^{\frac {i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{- a + x} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{- a + x} \Gamma \left (\frac {4}{3}\right )}{\Gamma \left (\frac {7}{3}\right )} \]
4*a**(1/3)*exp(-I*pi/3)*log(1 - (-a + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3) )*gamma(4/3)/(3*gamma(7/3)) - 4*a**(1/3)*log(1 - (-a + x)**(1/3)*exp_polar (I*pi)/a**(1/3))*gamma(4/3)/(3*gamma(7/3)) + 4*a**(1/3)*exp(I*pi/3)*log(1 - (-a + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(3*gamma(7/3)) + 4*(-a + x)**(1/3)*gamma(4/3)/gamma(7/3)
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (a^{\frac {1}{3}} - 2 \, {\left (-a + x\right )}^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + \frac {1}{2} \, a^{\frac {1}{3}} \log \left (a^{\frac {2}{3}} - a^{\frac {1}{3}} {\left (-a + x\right )}^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {2}{3}}\right ) - a^{\frac {1}{3}} \log \left (a^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-a + x\right )}^{\frac {1}{3}} \]
-sqrt(3)*a^(1/3)*arctan(-1/3*sqrt(3)*(a^(1/3) - 2*(-a + x)^(1/3))/a^(1/3)) + 1/2*a^(1/3)*log(a^(2/3) - a^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)) - a^ (1/3)*log(a^(1/3) + (-a + x)^(1/3)) + 3*(-a + x)^(1/3)
Time = 0.49 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=-\sqrt {3} \left (-a\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-a\right )^{\frac {1}{3}} + 2 \, {\left (-a + x\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-a\right )^{\frac {1}{3}}}\right ) - \frac {1}{2} \, \left (-a\right )^{\frac {1}{3}} \log \left (\left (-a\right )^{\frac {2}{3}} + \left (-a\right )^{\frac {1}{3}} {\left (-a + x\right )}^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {2}{3}}\right ) + \left (-a\right )^{\frac {1}{3}} \log \left ({\left | -\left (-a\right )^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (-a + x\right )}^{\frac {1}{3}} \]
-sqrt(3)*(-a)^(1/3)*arctan(1/3*sqrt(3)*((-a)^(1/3) + 2*(-a + x)^(1/3))/(-a )^(1/3)) - 1/2*(-a)^(1/3)*log((-a)^(2/3) + (-a)^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)) + (-a)^(1/3)*log(abs(-(-a)^(1/3) + (-a + x)^(1/3))) + 3*(-a + x)^(1/3)
Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx={\left (-a\right )}^{1/3}\,\ln \left (-9\,{\left (-a\right )}^{4/3}-9\,a\,{\left (x-a\right )}^{1/3}\right )+3\,{\left (x-a\right )}^{1/3}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (\frac {9\,{\left (-a\right )}^{4/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+9\,a\,{\left (x-a\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (\frac {9\,{\left (-a\right )}^{4/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-9\,a\,{\left (x-a\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]