Integrand size = 14, antiderivative size = 123 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {1}{a x}-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}} \] Output:
-1/a/x-1/3*b^(1/3)*arctan(1/3*(a^(1/3)+2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/ 2)/a^(4/3)-1/3*b^(1/3)*ln(a^(1/3)-b^(1/3)*x)/a^(4/3)+1/6*b^(1/3)*ln(a^(2/3 )+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {6 \sqrt [3]{a}+2 \sqrt {3} \sqrt [3]{b} x \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-\sqrt [3]{b} x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} x} \] Input:
Integrate[1/(x^2*(a - b*x^3)),x]
Output:
-1/6*(6*a^(1/3) + 2*Sqrt[3]*b^(1/3)*x*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/S qrt[3]] + 2*b^(1/3)*x*Log[a^(1/3) - b^(1/3)*x] - b^(1/3)*x*Log[a^(2/3) + a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(4/3)*x)
Time = 0.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {847, 821, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {b \int \frac {x}{a-b x^3}dx}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {b \left (\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b} x}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {b \left (-\frac {\int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {b \left (-\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{b} x+\sqrt [3]{a}\right )}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \left (-\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {b \left (-\frac {-\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {b \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {b \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\) |
Input:
Int[1/(x^2*(a - b*x^3)),x]
Output:
-(1/(a*x)) + (b*(-1/3*Log[a^(1/3) - b^(1/3)*x]/(a^(1/3)*b^(2/3)) - ((Sqrt[ 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^(1/3)*b^(1/3))))/a
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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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 = 0.46 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.41
method | result | size |
risch | \(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{3}+b \right )}{\sum }\textit {\_R} \ln \left (\left (4 \textit {\_R}^{3} a^{4}+3 b \right ) x +a^{3} \textit {\_R}^{2}\right )\right )}{3}\) | \(50\) |
default | \(-\frac {1}{a x}-\frac {\left (\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(107\) |
Input:
int(1/x^2/(-b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-1/a/x+1/3*sum(_R*ln((4*_R^3*a^4+3*b)*x+a^3*_R^2),_R=RootOf(_Z^3*a^4+b))
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {2 \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} + a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, a x} \] Input:
integrate(1/x^2/(-b*x^3+a),x, algorithm="fricas")
Output:
-1/6*(2*sqrt(3)*x*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) - 1/3*sqr t(3)) + x*(-b/a)^(1/3)*log(b*x^2 + a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) - 2* x*(-b/a)^(1/3)*log(b*x - a*(-b/a)^(2/3)) + 6)/(a*x)
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=- \operatorname {RootSum} {\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (- \frac {9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac {1}{a x} \] Input:
integrate(1/x**2/(-b*x**3+a),x)
Output:
-RootSum(27*_t**3*a**4 - b, Lambda(_t, _t*log(-9*_t**2*a**3/b + x))) - 1/( a*x)
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {1}{a x} \] Input:
integrate(1/x^2/(-b*x^3+a),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(a*(a/b)^ (1/3)) + 1/6*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^(1/3)) - 1/3* log(x - (a/b)^(1/3))/(a*(a/b)^(1/3)) - 1/(a*x)
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {b \left (\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} - \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {\left (a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} - \frac {1}{a x} \] Input:
integrate(1/x^2/(-b*x^3+a),x, algorithm="giac")
Output:
-1/3*b*(a/b)^(2/3)*log(abs(x - (a/b)^(1/3)))/a^2 - 1/3*sqrt(3)*(a*b^2)^(2/ 3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b) + 1/6*(a*b^ 2)^(2/3)*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b) - 1/(a*x)
Time = 0.34 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=\frac {{\left (-b\right )}^{1/3}\,\ln \left (b^3\,x-a^{1/3}\,{\left (-b\right )}^{8/3}\right )}{3\,a^{4/3}}-\frac {1}{a\,x}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (a\,b^3\,x-a^{4/3}\,{\left (-b\right )}^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (a\,b^3\,x-9\,a^{4/3}\,{\left (-b\right )}^{8/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}} \] Input:
int(1/(x^2*(a - b*x^3)),x)
Output:
((-b)^(1/3)*log(b^3*x - a^(1/3)*(-b)^(8/3)))/(3*a^(4/3)) - 1/(a*x) - ((-b) ^(1/3)*log(a*b^3*x - a^(4/3)*(-b)^(8/3)*((3^(1/2)*1i)/2 + 1/2)^2)*((3^(1/2 )*1i)/2 + 1/2))/(3*a^(4/3)) + ((-b)^(1/3)*log(a*b^3*x - 9*a^(4/3)*(-b)^(8/ 3)*((3^(1/2)*1i)/6 - 1/6)^2)*((3^(1/2)*1i)/6 - 1/6))/a^(4/3)
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}+2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b x -6 b^{\frac {2}{3}} a^{\frac {1}{3}}+\mathrm {log}\left (a^{\frac {2}{3}}+b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b x -2 \,\mathrm {log}\left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) b x}{6 b^{\frac {2}{3}} a^{\frac {4}{3}} x} \] Input:
int(1/x^2/(-b*x^3+a),x)
Output:
( - 2*sqrt(3)*atan((a**(1/3) + 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*x - 6*b **(2/3)*a**(1/3) + log(a**(2/3) + b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b*x - 2*log(a**(1/3) - b**(1/3)*x)*b*x)/(6*b**(2/3)*a**(1/3)*a*x)