Integrand size = 13, antiderivative size = 124 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=-\frac {1}{2 a x^2}+\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}} \] Output:
-1/2/a/x^2+1/3*b^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3 ^(1/2)/a^(5/3)-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)
Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=\frac {-3 a^{2/3}+2 \sqrt {3} b^{2/3} x^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^{2/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} x^2} \] Input:
Integrate[1/(x^3*(a + b*x^3)),x]
Output:
(-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 = 0.45 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {847, 750, 16, 1142, 25, 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 |
| \(\displaystyle -\frac {b \int \frac {1}{b x^3+a}dx}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\) |
Input:
Int[1/(x^3*(a + b*x^3)),x]
Output:
-1/2*1/(a*x^2) - (b*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (-((Sq rt[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
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 = 0.44 (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 | \(-\frac {1}{2 a \,x^{2}}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}\) | \(106\) |
Input:
int(1/x^3/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-1/2/a/x^2+1/3*sum(_R*ln((-4*_R^3*a^5-3*b^2)*x-a^2*b*_R),_R=RootOf(_Z^3*a^ 5+b^2))
Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3}{6 \, a x^{2}} \] Input:
integrate(1/x^3/(b*x^3+a),x, algorithm="fricas")
Output:
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 = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{5} + b^{2}, \left ( t \mapsto t \log {\left (- \frac {3 t a^{2}}{b} + x \right )} \right )\right )} - \frac {1}{2 a x^{2}} \] Input:
integrate(1/x**3/(b*x**3+a),x)
Output:
RootSum(27*_t**3*a**5 + b**2, Lambda(_t, _t*log(-3*_t*a**2/b + x))) - 1/(2 *a*x**2)
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1}{2 \, a x^{2}} \] Input:
integrate(1/x^3/(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)^ (2/3)) + 1/6*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^(2/3)) - 1/3* log(x + (a/b)^(1/3))/(a*(a/b)^(2/3)) - 1/2/(a*x^2)
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} - \frac {1}{2 \, a x^{2}} \] Input:
integrate(1/x^3/(b*x^3+a),x, algorithm="giac")
Output:
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 = 0.32 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=\frac {b^{2/3}\,\ln \left ({\left (-a\right )}^{7/3}-a^2\,b^{1/3}\,x\right )}{3\,{\left (-a\right )}^{5/3}}-\frac {1}{2\,a\,x^2}-\frac {b^{2/3}\,\ln \left (3\,a^2\,b^3\,x+3\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-a\right )}^{5/3}}+\frac {b^{2/3}\,\ln \left (3\,a^2\,b^3\,x-9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{{\left (-a\right )}^{5/3}} \] Input:
int(1/(x^3*(a + b*x^3)),x)
Output:
(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)
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx=\frac {2 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{2}+a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{2}-2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{2}-3 b^{\frac {1}{3}} a}{6 b^{\frac {1}{3}} a^{2} x^{2}} \] Input:
int(1/x^3/(b*x^3+a),x)
Output:
(2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*x **2 + a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b*x**2 - 2*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*b*x**2 - 3*b**(1/3)*a)/(6*b**(1/3) *a**2*x**2)