Integrand size = 9, antiderivative size = 140 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {x}{a^2}+\frac {b x}{3 a^2 \left (b+a x^3\right )}+\frac {4 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{7/3}} \] Output:
x/a^2+1/3*b*x/a^2/(a*x^3+b)+4/9*b^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3 ^(1/2)/b^(1/3))*3^(1/2)/a^(7/3)-4/9*b^(1/3)*ln(b^(1/3)+a^(1/3)*x)/a^(7/3)+ 2/9*b^(1/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(7/3)
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {9 \sqrt [3]{a} x+\frac {3 \sqrt [3]{a} b x}{b+a x^3}+4 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-4 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+2 \sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{7/3}} \] Input:
Integrate[(a + b/x^3)^(-2),x]
Output:
(9*a^(1/3)*x + (3*a^(1/3)*b*x)/(b + a*x^3) + 4*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] - 4*b^(1/3)*Log[b^(1/3) + a^(1/3)*x] + 2* b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(9*a^(7/3))
Time = 0.50 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {772, 817, 843, 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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \int \frac {x^6}{\left (a x^3+b\right )^2}dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {4 \int \frac {x^3}{a x^3+b}dx}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \int \frac {1}{a x^3+b}dx}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 \left (\frac {x}{a}-\frac {b \left (\frac {-\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{3 a}-\frac {x^4}{3 a \left (a x^3+b\right )}\) |
Input:
Int[(a + b/x^3)^(-2),x]
Output:
-1/3*x^4/(a*(b + a*x^3)) + (4*(x/a - (b*(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))))/a))/(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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && 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 = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(\frac {x}{a^{2}}+\frac {b x}{3 a^{2} \left (a \,x^{3}+b \right )}-\frac {4 b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 a^{3}}\) | \(50\) |
| default | \(\frac {x}{a^{2}}-\frac {b \left (-\frac {x}{3 \left (a \,x^{3}+b \right )}+\frac {4 \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {2 \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right )}{a^{2}}\) | \(115\) |
Input:
int(1/(a+b/x^3)^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*x+1/3*b*x/a^2/(a*x^3+b)-4/9/a^3*b*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3 *a+b))
Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {9 \, a x^{4} + 4 \, \sqrt {3} {\left (a x^{3} + b\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 2 \, {\left (a x^{3} + b\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 4 \, {\left (a x^{3} + b\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 12 \, b x}{9 \, {\left (a^{3} x^{3} + a^{2} b\right )}} \] Input:
integrate(1/(a+b/x^3)^2,x, algorithm="fricas")
Output:
1/9*(9*a*x^4 + 4*sqrt(3)*(a*x^3 + b)*(-b/a)^(1/3)*arctan(1/3*(2*sqrt(3)*a* x*(-b/a)^(2/3) - sqrt(3)*b)/b) - 2*(a*x^3 + b)*(-b/a)^(1/3)*log(x^2 + x*(- b/a)^(1/3) + (-b/a)^(2/3)) + 4*(a*x^3 + b)*(-b/a)^(1/3)*log(x - (-b/a)^(1/ 3)) + 12*b*x)/(a^3*x^3 + a^2*b)
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {b x}{3 a^{3} x^{3} + 3 a^{2} b} + \operatorname {RootSum} {\left (729 t^{3} a^{7} + 64 b, \left ( t \mapsto t \log {\left (- \frac {9 t a^{2}}{4} + x \right )} \right )\right )} + \frac {x}{a^{2}} \] Input:
integrate(1/(a+b/x**3)**2,x)
Output:
b*x/(3*a**3*x**3 + 3*a**2*b) + RootSum(729*_t**3*a**7 + 64*b, Lambda(_t, _ t*log(-9*_t*a**2/4 + x))) + x/a**2
Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {b x}{3 \, {\left (a^{3} x^{3} + a^{2} b\right )}} + \frac {x}{a^{2}} - \frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {2 \, b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {4 \, b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(a+b/x^3)^2,x, algorithm="maxima")
Output:
1/3*b*x/(a^3*x^3 + a^2*b) + x/a^2 - 4/9*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/3))/(a^3*(b/a)^(2/3)) + 2/9*b*log(x^2 - x*(b/a)^(1 /3) + (b/a)^(2/3))/(a^3*(b/a)^(2/3)) - 4/9*b*log(x + (b/a)^(1/3))/(a^3*(b/ a)^(2/3))
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {4 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {x}{a^{2}} + \frac {b x}{3 \, {\left (a x^{3} + b\right )} a^{2}} - \frac {4 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {2 \, \left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{3}} \] Input:
integrate(1/(a+b/x^3)^2,x, algorithm="giac")
Output:
4/9*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/a^2 + x/a^2 + 1/3*b*x/((a*x^3 + b)*a^2) - 4/9*sqrt(3)*(-a^2*b)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1 /3))/(-b/a)^(1/3))/a^3 - 2/9*(-a^2*b)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b /a)^(2/3))/a^3
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {x}{a^2}+\frac {4\,{\left (-b\right )}^{1/3}\,\ln \left ({\left (-b\right )}^{4/3}+a^{1/3}\,b\,x\right )}{9\,a^{7/3}}+\frac {b\,x}{3\,\left (a^3\,x^3+b\,a^2\right )}-\frac {4\,{\left (-b\right )}^{1/3}\,\ln \left (4\,b\,x-\frac {4\,{\left (-b\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{7/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (4\,b\,x+\frac {9\,{\left (-b\right )}^{4/3}\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{a^{1/3}}\right )\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{a^{7/3}} \] Input:
int(1/(a + b/x^3)^2,x)
Output:
x/a^2 + (4*(-b)^(1/3)*log((-b)^(4/3) + a^(1/3)*b*x))/(9*a^(7/3)) + (b*x)/( 3*(a^2*b + a^3*x^3)) - (4*(-b)^(1/3)*log(4*b*x - (4*(-b)^(4/3)*((3^(1/2)*1 i)/2 + 1/2))/a^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*a^(7/3)) + ((-b)^(1/3)*lo g(4*b*x + (9*(-b)^(4/3)*((3^(1/2)*2i)/9 - 2/9))/a^(1/3))*((3^(1/2)*2i)/9 - 2/9))/a^(7/3)
Time = 0.24 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {-4 b^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a \,x^{3}-4 b^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right )+9 a^{\frac {4}{3}} x^{4}+12 a^{\frac {1}{3}} b x +2 b^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) a \,x^{3}+2 b^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right )-4 b^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) a \,x^{3}-4 b^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right )}{9 a^{\frac {7}{3}} \left (a \,x^{3}+b \right )} \] Input:
int(1/(a+b/x^3)^2,x)
Output:
( - 4*b**(1/3)*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))* a*x**3 - 4*b**(1/3)*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt( 3)))*b + 9*a**(1/3)*a*x**4 + 12*a**(1/3)*b*x + 2*b**(1/3)*log(a**(2/3)*x** 2 - b**(1/3)*a**(1/3)*x + b**(2/3))*a*x**3 + 2*b**(1/3)*log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3))*b - 4*b**(1/3)*log(a**(1/3)*x + b**(1/3) )*a*x**3 - 4*b**(1/3)*log(a**(1/3)*x + b**(1/3))*b)/(9*a**(1/3)*a**2*(a*x* *3 + b))