3.2.84 \(\int \frac {x^3}{2+x^3+x^6} \, dx\) [184]

Optimal. Leaf size=399 \[ -\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}} \]

[Out]

1/42*ln(2^(1/3)*x+(1+I*7^(1/2))^(1/3))*(7-I*7^(1/2))*2^(2/3)/(1+I*7^(1/2))^(2/3)-1/84*ln(2^(2/3)*x^2-2^(1/3)*x
*(1+I*7^(1/2))^(1/3)+(1+I*7^(1/2))^(2/3))*(7-I*7^(1/2))*2^(2/3)/(1+I*7^(1/2))^(2/3)+1/42*ln(2^(1/3)*x+(1-I*7^(
1/2))^(1/3))*(7+I*7^(1/2))*2^(2/3)/(1-I*7^(1/2))^(2/3)-1/84*ln(2^(2/3)*x^2-2^(1/3)*x*(1-I*7^(1/2))^(1/3)+(1-I*
7^(1/2))^(2/3))*(7+I*7^(1/2))*2^(2/3)/(1-I*7^(1/2))^(2/3)-1/42*I*arctan(1/3*(1-2*2^(1/3)*x/(1-I*7^(1/2))^(1/3)
)*3^(1/2))*(1-I*7^(1/2))^(1/3)*2^(2/3)*21^(1/2)+1/42*I*arctan(1/3*(1-2*2^(1/3)*x/(1+I*7^(1/2))^(1/3))*3^(1/2))
*(1+I*7^(1/2))^(1/3)*2^(2/3)*21^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1388, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+\left (1-i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+\left (1+i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(2 + x^3 + x^6),x]

[Out]

((-I)*((1 - I*Sqrt[7])/2)^(1/3)*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/Sqrt[21] + (I*((1 + I*S
qrt[7])/2)^(1/3)*ArcTan[(1 - (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/Sqrt[21] + ((7 + I*Sqrt[7])*Log[(1 - I
*Sqrt[7])^(1/3) + 2^(1/3)*x])/(21*2^(1/3)*(1 - I*Sqrt[7])^(2/3)) + ((7 - I*Sqrt[7])*Log[(1 + I*Sqrt[7])^(1/3)
+ 2^(1/3)*x])/(21*2^(1/3)*(1 + I*Sqrt[7])^(2/3)) - ((7 + I*Sqrt[7])*Log[(1 - I*Sqrt[7])^(2/3) - (2*(1 - I*Sqrt
[7]))^(1/3)*x + 2^(2/3)*x^2])/(42*2^(1/3)*(1 - I*Sqrt[7])^(2/3)) - ((7 - I*Sqrt[7])*Log[(1 + I*Sqrt[7])^(2/3)
- (2*(1 + I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2])/(42*2^(1/3)*(1 + I*Sqrt[7])^(2/3))

Rule 31

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

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 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]

Rule 1388

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n/2)*(b/q + 1), Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n/2)*(b/q - 1), Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rubi steps

\begin {align*} \int \frac {x^3}{2+x^3+x^6} \, dx &=\frac {1}{14} \left (7-i \sqrt {7}\right ) \int \frac {1}{\frac {1}{2}+\frac {i \sqrt {7}}{2}+x^3} \, dx+\frac {1}{14} \left (7+i \sqrt {7}\right ) \int \frac {1}{\frac {1}{2}-\frac {i \sqrt {7}}{2}+x^3} \, dx\\ &=\frac {\left (7-i \sqrt {7}\right ) \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}+x} \, dx}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}+x} \, dx}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}\\ &=\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{14\ 2^{2/3} \sqrt [3]{1+i \sqrt {7}}}-\frac {\left (7+i \sqrt {7}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{14\ 2^{2/3} \sqrt [3]{1-i \sqrt {7}}}\\ &=\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{7 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{7 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}\\ &=-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 37, normalized size = 0.09 \begin {gather*} \frac {1}{3} \text {RootSum}\left [2+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(2 + x^3 + x^6),x]

[Out]

RootSum[2 + #1^3 + #1^6 & , (Log[x - #1]*#1)/(1 + 2*#1^3) & ]/3

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.02, size = 36, normalized size = 0.09

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\)	\(36\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^6+x^3+2),x,method=_RETURNVERBOSE)

[Out]

1/3*sum(_R^3/(2*_R^5+_R^2)*ln(x-_R),_R=RootOf(_Z^6+_Z^3+2))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+x^3+2),x, algorithm="maxima")

[Out]

integrate(x^3/(x^6 + x^3 + 2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1048 vs. \(2 (263) = 526\).
time = 0.43, size = 1048, normalized size = 2.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+x^3+2),x, algorithm="fricas")

[Out]

1/294*98^(2/3)*56^(1/6)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))*log(-196*98^(2/3)*56^(1/6)*sqrt(7)*x*s
in(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 9604*x^2 + 1372*98^(1/3)*7^(1/3)) - 2/147*98^(2/3)*56^(1/6)*a
rctan(-1/38416*(98*98^(1/3)*56^(5/6)*sqrt(7)*x - 98^(1/3)*56^(5/6)*sqrt(7)*sqrt(-196*98^(2/3)*56^(1/6)*sqrt(7)
*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 9604*x^2 + 1372*98^(1/3)*7^(1/3)) - 38416*sin(2/3*arctan(
2/7*sqrt(14)*sqrt(7) + sqrt(7))))/cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*sin(2/3*arctan(2/7*sqrt(14)
*sqrt(7) + sqrt(7))) - 1/147*(98^(2/3)*56^(1/6)*sqrt(3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) - 98^(
2/3)*56^(1/6)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*arctan(1/38416*(196*98^(1/3)*56^(5/6)*sqrt(7)*x
*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) - 196*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(3)*x - 784*cos(2/3*arct
an(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + sqrt(-392*98^(2/3)*56^(
1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 392*98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2
/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 38416*x^2 + 5488*98^(1/3)*7^(1/3))*(98^(1/3)*56^(5/6)*sqrt(7)*sqr
t(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) - 98^(1/3)*56^(5/6)*sqrt(7)*cos(2/3*arctan(2/7*sqrt(14)*s
qrt(7) + sqrt(7)))) - 38416*sqrt(3))/(4*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 - 3)) - 1/147*(98^(2
/3)*56^(1/6)*sqrt(3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(2/3)*56^(1/6)*sin(2/3*arctan(2/7*sq
rt(14)*sqrt(7) + sqrt(7))))*arctan(-1/2744*(14*98^(1/3)*56^(5/6)*sqrt(7)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7)
+ sqrt(7))) - sqrt(2)*sqrt(98^(2/3)*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))
) + 98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98*x^2 + 14*98^(1/3)*7^(1/3)
)*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(1/3)*56^(5/6)*sqrt(
7)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))) + 14*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(3)*x + 784*cos(2/3*ar
ctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 2744*sqrt(3))/(4*cos(
2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 - 3)) + 1/588*(98^(2/3)*56^(1/6)*sqrt(3)*sin(2/3*arctan(2/7*sqrt
(14)*sqrt(7) + sqrt(7))) - 98^(2/3)*56^(1/6)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*log(392*98^(2/3)
*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 392*98^(2/3)*56^(1/6)*sqrt(7)*x*
sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 38416*x^2 + 5488*98^(1/3)*7^(1/3)) - 1/588*(98^(2/3)*56^(1/6
)*sqrt(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(2/3)*56^(1/6)*cos(2/3*arctan(2/7*sqrt(14)*sqrt
(7) + sqrt(7))))*log(-392*98^(2/3)*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))
+ 392*98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 38416*x^2 + 5488*98^(1/3)*
7^(1/3))

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 24, normalized size = 0.06 \begin {gather*} \operatorname {RootSum} {\left (250047 t^{6} + 1323 t^{3} + 2, \left ( t \mapsto t \log {\left (7938 t^{4} + 21 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**6+x**3+2),x)

[Out]

RootSum(250047*_t**6 + 1323*_t**3 + 2, Lambda(_t, _t*log(7938*_t**4 + 21*_t + x)))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+x^3+2),x, algorithm="giac")

[Out]

integrate(x^3/(x^6 + x^3 + 2), x)

________________________________________________________________________________________

Mupad [B]
time = 2.61, size = 351, normalized size = 0.88 \begin {gather*} \frac {\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-196-\sqrt {7}\,28{}\mathrm {i}\right )}^{1/3}}{42}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{42}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^3 + x^6 + 2),x)

[Out]

(log(x - (2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i)/14)*(- 7^(1/2)*28i - 196)^(1/3))/42 + (2^(2/3)*7^(1/3)*
log(x + (2^(2/3)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3)*1i)/14)*(7^(1/2)*1i - 7)^(1/3))/42 - (2^(2/3)*7^(1/3)*log(x +
(2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i)/28 - (2^(2/3)*3^(1/2)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3))/28)*(3^(
1/2)*1i + 1)*(- 7^(1/2)*1i - 7)^(1/3))/84 + (2^(2/3)*7^(1/3)*log(x + (2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)
*1i)/28 + (2^(2/3)*3^(1/2)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i - 1)*(- 7^(1/2)*1i - 7)^(1/3))/84
+ (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3)*1i)/28 - (2^(2/3)*3^(1/2)*7^(5/6)*(7^(1/2)
*1i - 7)^(1/3))/28)*(3^(1/2)*1i - 1)*(7^(1/2)*1i - 7)^(1/3))/84 - (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(5/6)*(7
^(1/2)*1i - 7)^(1/3)*1i)/28 + (2^(2/3)*3^(1/2)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i + 1)*(7^(1/2)*1
i - 7)^(1/3))/84

________________________________________________________________________________________