Integrand size = 14, antiderivative size = 124 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=-\frac {1}{2 a x^2}+\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}} \]
-1/2/a/x^2-1/3*b^(2/3)*ln(a^(1/3)-b^(1/3)*x)/a^(5/3)+1/6*b^(2/3)*ln(a^(2/3 )+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)+1/3*b^(2/3)*arctan(1/3*(a^(1/3)+2 *b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/3)*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {-3 a^{2/3}+2 \sqrt {3} b^{2/3} x^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^{2/3} x^2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+b^{2/3} x^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} x^2} \]
(-3*a^(2/3) + 2*Sqrt[3]*b^(2/3)*x^2*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqr t[3]] - 2*b^(2/3)*x^2*Log[a^(1/3) - b^(1/3)*x] + b^(2/3)*x^2*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(5/3)*x^2)
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {847, 750, 16, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {b \int \frac {1}{a-b x^3}dx}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+2 \sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b} x}dx}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+2 \sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{b} x+\sqrt [3]{a}\right )}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {b \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {b \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {b \left (\frac {\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
-1/2*1/(a*x^2) + (b*(-1/3*Log[a^(1/3) - b^(1/3)*x]/(a^(2/3)*b^(1/3)) + ((S qrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/a
3.4.62.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , 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 = 3.65 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {1}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (4 \textit {\_R}^{3} a^{5}+3 b^{2}\right ) x -a^{2} b \textit {\_R} \right )\right )}{3}\) | \(54\) |
default | \(-\frac {\left (\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}-\frac {1}{2 a \,x^{2}}\) | \(107\) |
Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + \sqrt {3} b}{3 \, b}\right ) + x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 2 \, x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 3}{6 \, a x^{2}} \]
-1/6*(2*sqrt(3)*x^2*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^2)^ (2/3) + sqrt(3)*b)/b) + x^2*(-b^2/a^2)^(1/3)*log(b^2*x^2 - a*b*x*(-b^2/a^2 )^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 2*x^2*(-b^2/a^2)^(1/3)*log(b*x + a*(-b^2 /a^2)^(1/3)) + 3)/(a*x^2)
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=- \operatorname {RootSum} {\left (27 t^{3} a^{5} - b^{2}, \left ( t \mapsto t \log {\left (- \frac {3 t a^{2}}{b} + x \right )} \right )\right )} - \frac {1}{2 a x^{2}} \]
Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1}{2 \, a x^{2}} \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(a*(a/b)^( 2/3)) + 1/6*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^(2/3)) - 1/3*l og(x - (a/b)^(1/3))/(a*(a/b)^(2/3)) - 1/2/(a*x^2)
Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} + \frac {\left (a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} - \frac {1}{2 \, a x^{2}} \]
-1/3*b*(a/b)^(1/3)*log(abs(x - (a/b)^(1/3)))/a^2 + 1/3*sqrt(3)*(a*b^2)^(1/ 3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/a^2 + 1/6*(a*b^2)^( 1/3)*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/a^2 - 1/2/(a*x^2)
Time = 5.42 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {b^{2/3}\,\ln \left ({\left (-a\right )}^{7/3}+a^2\,b^{1/3}\,x\right )}{3\,{\left (-a\right )}^{5/3}}-\frac {1}{2\,a\,x^2}-\frac {b^{2/3}\,\ln \left (3\,a^2\,b^3\,x-3\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-a\right )}^{5/3}}+\frac {b^{2/3}\,\ln \left (3\,a^2\,b^3\,x+9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{{\left (-a\right )}^{5/3}} \]
(b^(2/3)*log((-a)^(7/3) + a^2*b^(1/3)*x))/(3*(-a)^(5/3)) - 1/(2*a*x^2) - ( b^(2/3)*log(3*a^2*b^3*x - 3*(-a)^(7/3)*b^(8/3)*((3^(1/2)*1i)/2 + 1/2))*((3 ^(1/2)*1i)/2 + 1/2))/(3*(-a)^(5/3)) + (b^(2/3)*log(3*a^2*b^3*x + 9*(-a)^(7 /3)*b^(8/3)*((3^(1/2)*1i)/6 - 1/6))*((3^(1/2)*1i)/6 - 1/6))/(-a)^(5/3)