Integrand size = 9, antiderivative size = 119 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \] Output:
x/a+1/3*b^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3^(1/2)/b^(1/3))*3^(1/2)/ a^(4/3)-1/3*b^(1/3)*ln(b^(1/3)+a^(1/3)*x)/a^(4/3)+1/6*b^(1/3)*ln(b^(2/3)-a ^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(4/3)
Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {6 \sqrt [3]{a} x+2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \] Input:
Integrate[(a + b/x^3)^(-1),x]
Output:
(6*a^(1/3)*x + 2*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3 ]] - 2*b^(1/3)*Log[b^(1/3) + a^(1/3)*x] + b^(1/3)*Log[b^(2/3) - a^(1/3)*b^ (1/3)*x + a^(2/3)*x^2])/(6*a^(4/3))
Time = 0.47 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {772, 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.
| \(\displaystyle \int \frac {1}{a+\frac {b}{x^3}} \, dx\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \int \frac {x^3}{a x^3+b}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{a x^3+b}dx}{a}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {-\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\) |
Input:
Int[(a + b/x^3)^(-1),x]
Output:
x/a - (b*(Log[b^(1/3) + a^(1/3)*x]/(3*a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTa n[(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
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*(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 = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29
| method | result | size |
| risch | \(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 a^{2}}\) | \(34\) |
| default | \(\frac {x}{a}-\frac {\left (\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right ) b}{a}\) | \(103\) |
Input:
int(1/(a+b/x^3),x,method=_RETURNVERBOSE)
Output:
x/a-1/3/a^2*b*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*a+b))
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {2 \, \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 6 \, x}{6 \, a} \] Input:
integrate(1/(a+b/x^3),x, algorithm="fricas")
Output:
1/6*(2*sqrt(3)*(-b/a)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b/a)^(2/3) - sqrt( 3)*b)/b) - (-b/a)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3)) + 2*(-b/a )^(1/3)*log(x - (-b/a)^(1/3)) + 6*x)/a
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{4} + b, \left ( t \mapsto t \log {\left (- 3 t a + x \right )} \right )\right )} + \frac {x}{a} \] Input:
integrate(1/(a+b/x**3),x)
Output:
RootSum(27*_t**3*a**4 + b, Lambda(_t, _t*log(-3*_t*a + x))) + x/a
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a} - \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(a+b/x^3),x, algorithm="maxima")
Output:
x/a - 1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/3))/(a ^2*(b/a)^(2/3)) + 1/6*b*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3))/(a^2*(b/a)^ (2/3)) - 1/3*b*log(x + (b/a)^(1/3))/(a^2*(b/a)^(2/3))
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {\left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {x}{a} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} \] Input:
integrate(1/(a+b/x^3),x, algorithm="giac")
Output:
1/3*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/a + x/a - 1/3*sqrt(3)*(-a^2*b) ^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/a^2 - 1/6*(-a ^2*b)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^2
Time = 0.52 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a}+\frac {{\left (-b\right )}^{1/3}\,\ln \left ({\left (-b\right )}^{4/3}+a^{1/3}\,b\,x\right )}{3\,a^{4/3}}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (3\,a^{2/3}\,{\left (-b\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-3\,a\,b\,x\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (9\,a^{2/3}\,{\left (-b\right )}^{4/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+3\,a\,b\,x\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}} \] Input:
int(1/(a + b/x^3),x)
Output:
x/a + ((-b)^(1/3)*log((-b)^(4/3) + a^(1/3)*b*x))/(3*a^(4/3)) - ((-b)^(1/3) *log(3*a^(2/3)*(-b)^(4/3)*((3^(1/2)*1i)/2 + 1/2) - 3*a*b*x)*((3^(1/2)*1i)/ 2 + 1/2))/(3*a^(4/3)) + ((-b)^(1/3)*log(9*a^(2/3)*(-b)^(4/3)*((3^(1/2)*1i) /6 - 1/6) + 3*a*b*x)*((3^(1/2)*1i)/6 - 1/6))/a^(4/3)
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {-2 b^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right )+6 a^{\frac {1}{3}} x +b^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right )-2 b^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right )}{6 a^{\frac {4}{3}}} \] Input:
int(1/(a+b/x^3),x)
Output:
( - 2*b**(1/3)*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3))) + 6*a**(1/3)*x + b**(1/3)*log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/ 3)) - 2*b**(1/3)*log(a**(1/3)*x + b**(1/3)))/(6*a**(1/3)*a)