\(\int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx\) [1342]

Optimal result

Integrand size = 13, antiderivative size = 97 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {\left (-3-x^3\right ) \sqrt [3]{1+x^3}}{18 x^6}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{27} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{54} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \] Output:

1/18*(-x^3-3)*(x^3+1)^(1/3)/x^6+1/27*arctan(1/3*3^(1/2)+2/3*(x^3+1)^(1/3)* 
3^(1/2))*3^(1/2)-1/27*ln(-1+(x^3+1)^(1/3))+1/54*ln(1+(x^3+1)^(1/3)+(x^3+1) 
^(2/3))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {1}{54} \left (-\frac {3 \sqrt [3]{1+x^3} \left (3+x^3\right )}{x^6}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{1+x^3}\right )+\log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \] Input:

Integrate[(1 + x^3)^(1/3)/x^7,x]
 

Output:

((-3*(1 + x^3)^(1/3)*(3 + x^3))/x^6 + 2*Sqrt[3]*ArcTan[(1 + 2*(1 + x^3)^(1 
/3))/Sqrt[3]] - 2*Log[-1 + (1 + x^3)^(1/3)] + Log[1 + (1 + x^3)^(1/3) + (1 
 + x^3)^(2/3)])/54
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {798, 51, 52, 69, 16, 1083, 217}

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.

\(\displaystyle \int \frac {\sqrt [3]{x^3+1}}{x^7} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{x^3+1}}{x^9}dx^3\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{3} \left (\frac {1}{6} \int \frac {1}{x^6 \left (x^3+1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3+1}}{2 x^6}\right )\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \int \frac {1}{x^3 \left (x^3+1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3+1}}{x^3}\right )-\frac {\sqrt [3]{x^3+1}}{2 x^6}\right )\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {1}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^3+1}}d\sqrt [3]{x^3+1}-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{x^3+1}+1}d\sqrt [3]{x^3+1}-\frac {1}{2} \log \left (x^3\right )\right )-\frac {\sqrt [3]{x^3+1}}{x^3}\right )-\frac {\sqrt [3]{x^3+1}}{2 x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{x^3+1}+1}d\sqrt [3]{x^3+1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^3+1}\right )\right )-\frac {\sqrt [3]{x^3+1}}{x^3}\right )-\frac {\sqrt [3]{x^3+1}}{2 x^6}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3+1}+1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^3+1}\right )\right )-\frac {\sqrt [3]{x^3+1}}{x^3}\right )-\frac {\sqrt [3]{x^3+1}}{2 x^6}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^3+1}\right )\right )-\frac {\sqrt [3]{x^3+1}}{x^3}\right )-\frac {\sqrt [3]{x^3+1}}{2 x^6}\right )\)

Input:

Int[(1 + x^3)^(1/3)/x^7,x]
 

Output:

(-1/2*(1 + x^3)^(1/3)/x^6 + (-((1 + x^3)^(1/3)/x^3) - (2*(-(Sqrt[3]*ArcTan 
[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]]) - Log[x^3]/2 + (3*Log[1 - (1 + x^3)^(1/ 
3)])/2))/3)/6)/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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

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 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 1.54 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62

method	result	size
meijerg	\(-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}\)	\(60\)
risch	\(-\frac {x^{6}+4 x^{3}+3}{18 x^{6} \left (x^{3}+1\right )^{\frac {2}{3}}}-\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27 \Gamma \left (\frac {2}{3}\right )}\)	\(69\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{6}+\ln \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right ) x^{6}-2 \ln \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right ) x^{6}-3 x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}-9 \left (x^{3}+1\right )^{\frac {1}{3}}}{54 {\left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )}^{2} {\left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right )}^{2}}\)	\(115\)
trager	\(-\frac {\left (x^{3}+3\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{18 x^{6}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-15 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {2}{3}}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{27}-\frac {\ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {2}{3}}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{27}+\frac {\ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {2}{3}}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{27}\)	\(406\)

Input:

int((x^3+1)^(1/3)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/9/GAMMA(2/3)*(-5/27*GAMMA(2/3)*x^3*hypergeom([1,1,8/3],[2,4],-x^3)+1/3* 
(1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*GAMMA(2/3)+3/2*GAMMA(2/3)/x^6+GAMMA(2/3 
)/x^3)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{3} + 3\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \] Input:

integrate((x^3+1)^(1/3)/x^7,x, algorithm="fricas")
 

Output:

1/54*(2*sqrt(3)*x^6*arctan(2/3*sqrt(3)*(x^3 + 1)^(1/3) + 1/3*sqrt(3)) + x^ 
6*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1) - 2*x^6*log((x^3 + 1)^(1/3) - 
 1) - 3*(x^3 + 3)*(x^3 + 1)^(1/3))/x^6
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=- \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} \] Input:

integrate((x**3+1)**(1/3)/x**7,x)
 

Output:

-gamma(5/3)*hyper((-1/3, 5/3), (8/3,), exp_polar(I*pi)/x**3)/(3*x**5*gamma 
(8/3))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{54} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate((x^3+1)^(1/3)/x^7,x, algorithm="maxima")
 

Output:

1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + 1/18*((x^3 + 1) 
^(4/3) + 2*(x^3 + 1)^(1/3))/(2*x^3 - (x^3 + 1)^2 + 1) + 1/54*log((x^3 + 1) 
^(2/3) + (x^3 + 1)^(1/3) + 1) - 1/27*log((x^3 + 1)^(1/3) - 1)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} + \frac {1}{54} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:

integrate((x^3+1)^(1/3)/x^7,x, algorithm="giac")
 

Output:

1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) - 1/18*((x^3 + 1) 
^(4/3) + 2*(x^3 + 1)^(1/3))/x^6 + 1/54*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/ 
3) + 1) - 1/27*log(abs((x^3 + 1)^(1/3) - 1))
 

Mupad [B] (verification not implemented)

Time = 8.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {\frac {{\left (x^3+1\right )}^{1/3}}{9}+\frac {{\left (x^3+1\right )}^{4/3}}{18}}{2\,x^3-{\left (x^3+1\right )}^2+1}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{81}-\frac {1}{81}\right )}{27}-\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{3}+\frac {1}{6}-\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{3}+\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \] Input:

int((x^3 + 1)^(1/3)/x^7,x)
 

Output:

((x^3 + 1)^(1/3)/9 + (x^3 + 1)^(4/3)/18)/(2*x^3 - (x^3 + 1)^2 + 1) - log(( 
x^3 + 1)^(1/3)/81 - 1/81)/27 - log((x^3 + 1)^(1/3)/3 - (3^(1/2)*1i)/6 + 1/ 
6)*((3^(1/2)*1i)/54 - 1/54) + log((3^(1/2)*1i)/6 + (x^3 + 1)^(1/3)/3 + 1/6 
)*((3^(1/2)*1i)/54 + 1/54)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {-\left (x^{3}+1\right )^{\frac {1}{3}} x^{3}-3 \left (x^{3}+1\right )^{\frac {1}{3}}-2 \left (\int \frac {\left (x^{3}+1\right )^{\frac {1}{3}}}{x^{4}+x}d x \right ) x^{6}}{18 x^{6}} \] Input:

int((x^3+1)^(1/3)/x^7,x)
 

Output:

( - (x**3 + 1)**(1/3)*x**3 - 3*(x**3 + 1)**(1/3) - 2*int((x**3 + 1)**(1/3) 
/(x**4 + x),x)*x**6)/(18*x**6)