\(\int \frac {1}{(a+\frac {b}{x^3}) x^3} \, dx\) [437]

Optimal result

Integrand size = 13, antiderivative size = 115 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \] Output:

-1/3*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3^(1/2)/b^(1/3))*3^(1/2)/a^(1/3)/b^( 
2/3)+1/3*ln(b^(1/3)+a^(1/3)*x)/a^(1/3)/b^(2/3)-1/6*ln(b^(2/3)-a^(1/3)*b^(1 
/3)*x+a^(2/3)*x^2)/a^(1/3)/b^(2/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \] Input:

Integrate[1/((a + b/x^3)*x^3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {795, 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.

\(\displaystyle \int \frac {1}{x^3 \left (a+\frac {b}{x^3}\right )} \, dx\)

\(\Big \downarrow \) 795

\(\displaystyle \int \frac {1}{a x^3+b}dx\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 b^{2/3}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {-\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\)

Input:

Int[1/((a + b/x^3)*x^3),x]
 

Output:

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))
 

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 750

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]
 

rule 795

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

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

Maple [C] (verified)

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

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 a}\)	\(27\)
default	\(\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\)	\(91\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b x - b^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a x^{3} + b}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}\right ] \] Input:

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

Output:

[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a*b^2)^(1/3)/a)*log((2*a*b*x^3 - 3*(a*b^2)^(1 
/3)*b*x - b^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a*b^2)^(2/3)*x - (a*b^2)^(1/3)*b 
)*sqrt(-(a*b^2)^(1/3)/a))/(a*x^3 + b)) - (a*b^2)^(2/3)*log(a*b*x^2 - (a*b^ 
2)^(2/3)*x + (a*b^2)^(1/3)*b) + 2*(a*b^2)^(2/3)*log(a*b*x + (a*b^2)^(2/3)) 
)/(a*b^2), 1/6*(6*sqrt(1/3)*a*b*sqrt((a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2* 
(a*b^2)^(2/3)*x - (a*b^2)^(1/3)*b)*sqrt((a*b^2)^(1/3)/a)/b^2) - (a*b^2)^(2 
/3)*log(a*b*x^2 - (a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b) + 2*(a*b^2)^(2/3)*log 
(a*b*x + (a*b^2)^(2/3)))/(a*b^2)]
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a b^{2} - 1, \left ( t \mapsto t \log {\left (3 t b + x \right )} \right )\right )} \] Input:

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

Output:

RootSum(27*_t**3*a*b**2 - 1, Lambda(_t, _t*log(3*_t*b + x)))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} \] Input:

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

Output:

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/3))/(a*(b/a)^( 
2/3)) - 1/6*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3))/(a*(b/a)^(2/3)) + 1/3*l 
og(x + (b/a)^(1/3))/(a*(b/a)^(2/3))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=-\frac {\left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a b} + \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a b} \] Input:

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

Output:

-1/3*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/b + 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*b) + 1/6*(-a^2* 
b)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/(a*b)
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=\frac {\ln \left (a^{1/3}\,x+b^{1/3}\right )}{3\,a^{1/3}\,b^{2/3}}+\frac {\ln \left (3\,a^2\,x+\frac {3\,a^{5/3}\,b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}}-\frac {\ln \left (3\,a^2\,x-\frac {3\,a^{5/3}\,b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}} \] Input:

int(1/(x^3*(a + b/x^3)),x)
 

Output:

log(a^(1/3)*x + b^(1/3))/(3*a^(1/3)*b^(2/3)) + (log(3*a^2*x + (3*a^(5/3)*b 
^(1/3)*(3^(1/2)*1i - 1))/2)*(3^(1/2)*1i - 1))/(6*a^(1/3)*b^(2/3)) - (log(3 
*a^2*x - (3*a^(5/3)*b^(1/3)*(3^(1/2)*1i + 1))/2)*(3^(1/2)*1i + 1))/(6*a^(1 
/3)*b^(2/3))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right )-\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right )+2 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right )}{6 b^{\frac {2}{3}} a^{\frac {1}{3}}} \] Input:

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

Output:

(b**(1/3)*(2*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3))) - 
log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3)) + 2*log(a**(1/3)*x + b 
**(1/3))))/(6*a**(1/3)*b)