\(\int \frac {1}{a+\frac {b}{x^3}} \, dx\) [1970]

Optimal result

Integrand size = 9, antiderivative size = 119 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \]

[Out]

x/a-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)+1/
3*b^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)/b^(1/3)*3^(1/2))/a^(4/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {199, 327, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{4/3}}+\frac {x}{a} \]

[In]

Int[(a + b/x^3)^(-1),x]

[Out]

x/a + (b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(4/3)) - (b^(1/3)*Log[b^(1/3) + a
^(1/3)*x])/(3*a^(4/3)) + (b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(4/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[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]

Rule 210

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])

Rule 327

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] - Dist[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]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{b+a x^3} \, dx \\ & = \frac {x}{a}-\frac {b \int \frac {1}{b+a x^3} \, dx}{a} \\ & = \frac {x}{a}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a} \\ & = \frac {x}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{4/3}}-\frac {b^{2/3} \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a} \\ & = \frac {x}{a}-\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}}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{4/3}} \\ & = \frac {x}{a}+\frac {\sqrt [3]{b} \tan ^{-1}\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {6 \sqrt [3]{a} x+2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \]

[In]

Integrate[(a + b/x^3)^(-1),x]

[Out]

(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))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.02 (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	\(\frac {x}{a}-\frac {\left (\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right ) b}{a}\)	\(103\)

[In]

int(1/(a+b/x^3),x,method=_RETURNVERBOSE)

[Out]

x/a-1/3/a^2*b*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*a+b))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {2 \, \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 6 \, x}{6 \, a} \]

[In]

integrate(1/(a+b/x^3),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{4} + b, \left ( t \mapsto t \log {\left (- 3 t a + x \right )} \right )\right )} + \frac {x}{a} \]

[In]

integrate(1/(a+b/x**3),x)

[Out]

RootSum(27*_t**3*a**4 + b, Lambda(_t, _t*log(-3*_t*a + x))) + x/a

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a} - \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]

[In]

integrate(1/(a+b/x^3),x, algorithm="maxima")

[Out]

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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {\left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {x}{a} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} \]

[In]

integrate(1/(a+b/x^3),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a}+\frac {{\left (-b\right )}^{1/3}\,\ln \left ({\left (-b\right )}^{4/3}+a^{1/3}\,b\,x\right )}{3\,a^{4/3}}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (3\,a^{2/3}\,{\left (-b\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-3\,a\,b\,x\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (9\,a^{2/3}\,{\left (-b\right )}^{4/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+3\,a\,b\,x\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}} \]

[In]

int(1/(a + b/x^3),x)

[Out]

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)*1
i)/6 - 1/6) + 3*a*b*x)*((3^(1/2)*1i)/6 - 1/6))/a^(4/3)