\(\int x^3 \sqrt [3]{1+x^6} \, dx\) [1489]

Optimal result

Integrand size = 13, antiderivative size = 104 \[ \int x^3 \sqrt [3]{1+x^6} \, dx=\frac {1}{6} x^4 \sqrt [3]{1+x^6}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )}{6 \sqrt {3}}-\frac {1}{18} \log \left (-x^2+\sqrt [3]{1+x^6}\right )+\frac {1}{36} \log \left (x^4+x^2 \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]

[Out]

1/6*x^4*(x^6+1)^(1/3)-1/18*arctan(3^(1/2)*x^2/(x^2+2*(x^6+1)^(1/3)))*3^(1/2)-1/18*ln(-x^2+(x^6+1)^(1/3))+1/36*
ln(x^4+x^2*(x^6+1)^(1/3)+(x^6+1)^(2/3))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 285, 337} \[ \int x^3 \sqrt [3]{1+x^6} \, dx=-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {1}{6} \sqrt [3]{x^6+1} x^4-\frac {1}{12} \log \left (x^2-\sqrt [3]{x^6+1}\right ) \]

[In]

Int[x^3*(1 + x^6)^(1/3),x]

[Out]

(x^4*(1 + x^6)^(1/3))/6 - ArcTan[(1 + (2*x^2)/(1 + x^6)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[x^2 - (1 + x^6)^(1/3
)]/12

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \sqrt [3]{1+x^3} \, dx,x,x^2\right ) \\ & = \frac {1}{6} x^4 \sqrt [3]{1+x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^2\right ) \\ & = \frac {1}{6} x^4 \sqrt [3]{1+x^6}-\frac {\arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{1+x^6}}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {1}{12} \log \left (x^2-\sqrt [3]{1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.94 \[ \int x^3 \sqrt [3]{1+x^6} \, dx=\frac {1}{36} \left (6 x^4 \sqrt [3]{1+x^6}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )-2 \log \left (-x^2+\sqrt [3]{1+x^6}\right )+\log \left (x^4+x^2 \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right )\right ) \]

[In]

Integrate[x^3*(1 + x^6)^(1/3),x]

[Out]

(6*x^4*(1 + x^6)^(1/3) - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))] - 2*Log[-x^2 + (1 + x^6)^(1
/3)] + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)])/36

Maple [C] (verified)

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

Time = 1.85 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.16

method	result	size
meijerg	\(\frac {x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{6}\right )}{4}\)	\(17\)
risch	\(\frac {x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}}{6}+\frac {x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{6}\right )}{12}\)	\(30\)
pseudoelliptic	\(\frac {6 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}+1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )+\ln \left (\frac {x^{4}+x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{4}}\right )-2 \ln \left (\frac {-x^{2}+\left (x^{6}+1\right )^{\frac {1}{3}}}{x^{2}}\right )}{36 \left (x^{4}+x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}\right ) \left (-x^{2}+\left (x^{6}+1\right )^{\frac {1}{3}}\right )}\)	\(129\)
trager	\(\frac {x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}}{6}-\frac {\ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{18}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{4}-2 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}+3 x^{2} \left (x^{6}+1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{18}\)	\(209\)

[In]

int(x^3*(x^6+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/4*x^4*hypergeom([-1/3,2/3],[5/3],-x^6)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt [3]{1+x^6} \, dx=\frac {1}{6} \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{4} + \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{18} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{36} \, \log \left (\frac {x^{4} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]

[In]

integrate(x^3*(x^6+1)^(1/3),x, algorithm="fricas")

[Out]

1/6*(x^6 + 1)^(1/3)*x^4 + 1/18*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 + 1)^(1/3))/x^2) - 1/18*log(-(
x^2 - (x^6 + 1)^(1/3))/x^2) + 1/36*log((x^4 + (x^6 + 1)^(1/3)*x^2 + (x^6 + 1)^(2/3))/x^4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.30 \[ \int x^3 \sqrt [3]{1+x^6} \, dx=\frac {x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} \]

[In]

integrate(x**3*(x**6+1)**(1/3),x)

[Out]

x**4*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), x**6*exp_polar(I*pi))/(6*gamma(5/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt [3]{1+x^6} \, dx=\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) + \frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{6 \, x^{2} {\left (\frac {x^{6} + 1}{x^{6}} - 1\right )}} + \frac {1}{36} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{18} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]

[In]

integrate(x^3*(x^6+1)^(1/3),x, algorithm="maxima")

[Out]

1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3)/x^2 + 1)) + 1/6*(x^6 + 1)^(1/3)/(x^2*((x^6 + 1)/x^6 - 1)) +
1/36*log((x^6 + 1)^(1/3)/x^2 + (x^6 + 1)^(2/3)/x^4 + 1) - 1/18*log((x^6 + 1)^(1/3)/x^2 - 1)

Giac [F]

\[ \int x^3 \sqrt [3]{1+x^6} \, dx=\int { {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{3} \,d x } \]

[In]

integrate(x^3*(x^6+1)^(1/3),x, algorithm="giac")

[Out]

integrate((x^6 + 1)^(1/3)*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \sqrt [3]{1+x^6} \, dx=\int x^3\,{\left (x^6+1\right )}^{1/3} \,d x \]

[In]

int(x^3*(x^6 + 1)^(1/3),x)

[Out]

int(x^3*(x^6 + 1)^(1/3), x)