Integrand size = 11, antiderivative size = 115 \[ \int \frac {1}{-b+a x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b}+2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \]
1/3*ln(b^(1/3)-a^(1/3)*x)/a^(1/3)/b^(2/3)-1/6*ln(b^(2/3)+a^(1/3)*b^(1/3)*x +a^(2/3)*x^2)/a^(1/3)/b^(2/3)-1/3*arctan(1/3*(b^(1/3)+2*a^(1/3)*x)/b^(1/3) *3^(1/2))/a^(1/3)/b^(2/3)*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {1}{-b+a x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )+\log \left (b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \]
-1/6*(2*Sqrt[3]*ArcTan[(1 + (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] - 2*Log[b^(1/3 ) - a^(1/3)*x] + Log[b^(2/3) + a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(a^(1/3)* b^(2/3))
Time = 0.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {750, 16, 25, 1142, 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.
\(\displaystyle \int \frac {1}{a x^3-b} \, dx\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {\int -\frac {\sqrt [3]{a} x+2 \sqrt [3]{b}}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a} x-\sqrt [3]{b}}dx}{3 b^{2/3}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\int -\frac {\sqrt [3]{a} x+2 \sqrt [3]{b}}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\int \frac {\sqrt [3]{a} x+2 \sqrt [3]{b}}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{a} x+\sqrt [3]{b}\right )}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {1}{2} \int \frac {2 \sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}+1\right )}{\sqrt [3]{a}}}{3 b^{2/3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}\) |
Log[b^(1/3) - a^(1/3)*x]/(3*a^(1/3)*b^(2/3)) - ((Sqrt[3]*ArcTan[(1 + (2*a^ (1/3)*x)/b^(1/3))/Sqrt[3]])/a^(1/3) + Log[b^(2/3) + a^(1/3)*b^(1/3)*x + a^ (2/3)*x^2]/(2*a^(1/3)))/(3*b^(2/3))
3.1.35.3.1 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_) + (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 = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 a}\) | \(29\) |
default | \(\frac {\ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\) | \(92\) |
Time = 0.27 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.61 \[ \int \frac {1}{-b+a x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b x + b^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a x^{3} - b}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x - \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) + \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) - 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x - \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}\right ] \]
[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a*b^2)^(1/3)/a)*log((2*a*b*x^3 - 3*(a*b^2)^(1 /3)*b*x + b^2 - 3*sqrt(1/3)*(2*a*b*x^2 - (a*b^2)^(2/3)*x - (a*b^2)^(1/3)*b )*sqrt(-(a*b^2)^(1/3)/a))/(a*x^3 - b)) - (a*b^2)^(2/3)*log(a*b*x^2 + (a*b^ 2)^(2/3)*x + (a*b^2)^(1/3)*b) + 2*(a*b^2)^(2/3)*log(a*b*x - (a*b^2)^(2/3)) )/(a*b^2), -1/6*(6*sqrt(1/3)*a*b*sqrt((a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2 *(a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b)*sqrt((a*b^2)^(1/3)/a)/b^2) + (a*b^2)^( 2/3)*log(a*b*x^2 + (a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b) - 2*(a*b^2)^(2/3)*lo g(a*b*x - (a*b^2)^(2/3)))/(a*b^2)]
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.17 \[ \int \frac {1}{-b+a x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a b^{2} - 1, \left ( t \mapsto t \log {\left (- 3 t b + x \right )} \right )\right )} \]
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {1}{-b+a x^3} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} + x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (b/a)^(1/3))/(b/a)^(1/3))/(a*(b/a)^ (2/3)) - 1/6*log(x^2 + x*(b/a)^(1/3) + (b/a)^(2/3))/(a*(b/a)^(2/3)) + 1/3* log(x - (b/a)^(1/3))/(a*(b/a)^(2/3))
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \frac {1}{-b+a x^3} \, dx=\frac {\left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a b} - \frac {\left (a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a b} \]
1/3*(b/a)^(1/3)*log(abs(x - (b/a)^(1/3)))/b - 1/3*sqrt(3)*(a^2*b)^(1/3)*ar ctan(1/3*sqrt(3)*(2*x + (b/a)^(1/3))/(b/a)^(1/3))/(a*b) - 1/6*(a^2*b)^(1/3 )*log(x^2 + x*(b/a)^(1/3) + (b/a)^(2/3))/(a*b)
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int \frac {1}{-b+a x^3} \, dx=\frac {\ln \left (a^{1/3}\,x-b^{1/3}\right )}{3\,a^{1/3}\,b^{2/3}}+\frac {\ln \left (3\,a^2\,x-\frac {3\,a^{5/3}\,b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}}-\frac {\ln \left (3\,a^2\,x+\frac {3\,a^{5/3}\,b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}} \]