Integrand size = 17, antiderivative size = 123 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{b}}+\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \]
1/3*arctan(-1/3*3^(1/2)+2/3*(a*x^3-b)^(1/3)*3^(1/2)/b^(1/3))*3^(1/2)/b^(1/ 3)-1/3*ln(b^(1/3)+(a*x^3-b)^(1/3))/b^(1/3)+1/6*ln(b^(2/3)-b^(1/3)*(a*x^3-b )^(1/3)+(a*x^3-b)^(2/3))/b^(1/3)
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )+\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \]
(-2*Sqrt[3]*ArcTan[(1 - (2*(-b + a*x^3)^(1/3))/b^(1/3))/Sqrt[3]] - 2*Log[b ^(1/3) + (-b + a*x^3)^(1/3)] + Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(6*b^(1/3))
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {798, 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.
\(\displaystyle \int \frac {1}{x \sqrt [3]{a x^3-b}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^3 \sqrt [3]{a x^3-b}}dx^3\) |
\(\Big \downarrow \) 68 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^6+b^{2/3}-\sqrt [3]{b} \sqrt [3]{a x^3-b}}d\sqrt [3]{a x^3-b}-\frac {3 \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a x^3-b}}d\sqrt [3]{a x^3-b}}{2 \sqrt [3]{b}}+\frac {\log \left (x^3\right )}{2 \sqrt [3]{b}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^6+b^{2/3}-\sqrt [3]{b} \sqrt [3]{a x^3-b}}d\sqrt [3]{a x^3-b}-\frac {3 \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}+\frac {\log \left (x^3\right )}{2 \sqrt [3]{b}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{-x^6-3}d\left (1-\frac {2 \sqrt [3]{a x^3-b}}{\sqrt [3]{b}}\right )}{\sqrt [3]{b}}-\frac {3 \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}+\frac {\log \left (x^3\right )}{2 \sqrt [3]{b}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a x^3-b}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {3 \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}+\frac {\log \left (x^3\right )}{2 \sqrt [3]{b}}\right )\) |
(-((Sqrt[3]*ArcTan[(1 - (2*(-b + a*x^3)^(1/3))/b^(1/3))/Sqrt[3]])/b^(1/3)) + Log[x^3]/(2*b^(1/3)) - (3*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(2*b^(1/3) ))/3
3.19.9.3.1 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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 = 0.60 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 \left (a \,x^{3}-b \right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}}}\right )+\ln \left (b^{\frac {1}{3}}+\left (a \,x^{3}-b \right )^{\frac {1}{3}}\right )-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (a \,x^{3}-b \right )^{\frac {1}{3}}+\left (a \,x^{3}-b \right )^{\frac {2}{3}}\right )}{2}}{3 b^{\frac {1}{3}}}\) | \(87\) |
-1/3/b^(1/3)*(3^(1/2)*arctan(1/3*3^(1/2)*(-2*(a*x^3-b)^(1/3)+b^(1/3))/b^(1 /3))+ln(b^(1/3)+(a*x^3-b)^(1/3))-1/2*ln(b^(2/3)-b^(1/3)*(a*x^3-b)^(1/3)+(a *x^3-b)^(2/3)))
Time = 0.33 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.50 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} + {\left (a x^{3} - b\right )}^{\frac {1}{3}} b + \left (-b\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} - 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} - 3 \, b}{x^{3}}\right ) + \left (-b\right )^{\frac {2}{3}} \log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} + {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} + \left (-b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-b\right )^{\frac {2}{3}} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} - \left (-b\right )^{\frac {1}{3}}\right )}{6 \, b}, \frac {6 \, \sqrt {\frac {1}{3}} b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} + \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}\right ) + \left (-b\right )^{\frac {2}{3}} \log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} + {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} + \left (-b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-b\right )^{\frac {2}{3}} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} - \left (-b\right )^{\frac {1}{3}}\right )}{6 \, b}\right ] \]
[1/6*(3*sqrt(1/3)*b*sqrt((-b)^(1/3)/b)*log((2*a*x^3 + 3*sqrt(1/3)*(2*(a*x^ 3 - b)^(2/3)*(-b)^(2/3) + (a*x^3 - b)^(1/3)*b + (-b)^(1/3)*b)*sqrt((-b)^(1 /3)/b) - 3*(a*x^3 - b)^(1/3)*(-b)^(2/3) - 3*b)/x^3) + (-b)^(2/3)*log((a*x^ 3 - b)^(2/3) + (a*x^3 - b)^(1/3)*(-b)^(1/3) + (-b)^(2/3)) - 2*(-b)^(2/3)*l og((a*x^3 - b)^(1/3) - (-b)^(1/3)))/b, 1/6*(6*sqrt(1/3)*b*sqrt(-(-b)^(1/3) /b)*arctan(sqrt(1/3)*(2*(a*x^3 - b)^(1/3) + (-b)^(1/3))*sqrt(-(-b)^(1/3)/b )) + (-b)^(2/3)*log((a*x^3 - b)^(2/3) + (a*x^3 - b)^(1/3)*(-b)^(1/3) + (-b )^(2/3)) - 2*(-b)^(2/3)*log((a*x^3 - b)^(1/3) - (-b)^(1/3)))/b]
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \]
-gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*exp_polar(2*I*pi)/(a*x**3))/(3*a** (1/3)*x*gamma(4/3))
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{3 \, b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{6 \, b^{\frac {1}{3}}} - \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{3 \, b^{\frac {1}{3}}} \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(a*x^3 - b)^(1/3) - b^(1/3))/b^(1/3))/b^ (1/3) + 1/6*log((a*x^3 - b)^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/b ^(1/3) - 1/3*log((a*x^3 - b)^(1/3) + b^(1/3))/b^(1/3)
Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=-\frac {\sqrt {3} \left (-b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} + \left (-b\right )^{\frac {1}{3}}\right )}}{3 \, \left (-b\right )^{\frac {1}{3}}}\right )}{3 \, b} + \frac {\left (-b\right )^{\frac {2}{3}} \log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} + {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} + \left (-b\right )^{\frac {2}{3}}\right )}{6 \, b} - \frac {\left (-b\right )^{\frac {2}{3}} \log \left ({\left | {\left (a x^{3} - b\right )}^{\frac {1}{3}} - \left (-b\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} \]
-1/3*sqrt(3)*(-b)^(2/3)*arctan(1/3*sqrt(3)*(2*(a*x^3 - b)^(1/3) + (-b)^(1/ 3))/(-b)^(1/3))/b + 1/6*(-b)^(2/3)*log((a*x^3 - b)^(2/3) + (a*x^3 - b)^(1/ 3)*(-b)^(1/3) + (-b)^(2/3))/b - 1/3*(-b)^(2/3)*log(abs((a*x^3 - b)^(1/3) - (-b)^(1/3)))/b
Time = 6.01 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {\ln \left ({\left (a\,x^3-b\right )}^{1/3}-{\left (-b\right )}^{1/3}\right )}{3\,{\left (-b\right )}^{1/3}}+\frac {\ln \left ({\left (a\,x^3-b\right )}^{1/3}-\frac {{\left (-b\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}}-\frac {\ln \left ({\left (a\,x^3-b\right )}^{1/3}-\frac {{\left (-b\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}} \]