3.184 \(\int \frac{x^3}{2+x^3+x^6} \, dx\)

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

[Out]

((-I)*((1 - I*Sqrt[7])/2)^(1/3)*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqr
t[3]])/Sqrt[21] + (I*((1 + I*Sqrt[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])*Lo
g[(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))

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Rubi [A]  time = 0.74257, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\left (7+i \sqrt{7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt{7}\right )} x+\left (1-i \sqrt{7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1-i \sqrt{7}\right )^{2/3}}-\frac{\left (7-i \sqrt{7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt{7}\right )} x+\left (1+i \sqrt{7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1+i \sqrt{7}\right )^{2/3}}+\frac{\left (7+i \sqrt{7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{7}}\right )}{21 \sqrt [3]{2} \left (1-i \sqrt{7}\right )^{2/3}}+\frac{\left (7-i \sqrt{7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{7}}\right )}{21 \sqrt [3]{2} \left (1+i \sqrt{7}\right )^{2/3}}-\frac{i \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{7}\right )} \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21}}+\frac{i \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{7}\right )} \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21}} \]

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))/Sqr
t[3]])/Sqrt[21] + (I*((1 + I*Sqrt[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])*Lo
g[(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))

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Rubi in Sympy [A]  time = 108.427, size = 345, normalized size = 0.86 \[ \frac{2^{\frac{2}{3}} \sqrt{7} i \sqrt [3]{1 - \sqrt{7} i} \log{\left (\sqrt [3]{2} x + \sqrt [3]{1 - \sqrt{7} i} \right )}}{42} - \frac{2^{\frac{2}{3}} \sqrt{7} i \sqrt [3]{1 + \sqrt{7} i} \log{\left (\sqrt [3]{2} x + \sqrt [3]{1 + \sqrt{7} i} \right )}}{42} - \frac{2^{\frac{2}{3}} \sqrt{7} i \sqrt [3]{1 - \sqrt{7} i} \log{\left (x^{2} - \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{7} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{7} i\right )^{\frac{2}{3}}}{2} \right )}}{84} + \frac{2^{\frac{2}{3}} \sqrt{7} i \sqrt [3]{1 + \sqrt{7} i} \log{\left (x^{2} - \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{7} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{7} i\right )^{\frac{2}{3}}}{2} \right )}}{84} - \frac{2^{\frac{2}{3}} \sqrt{21} i \sqrt [3]{1 - \sqrt{7} i} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{7} i}} + \frac{1}{3}\right ) \right )}}{42} + \frac{2^{\frac{2}{3}} \sqrt{21} i \sqrt [3]{1 + \sqrt{7} i} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{7} i}} + \frac{1}{3}\right ) \right )}}{42} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**6+x**3+2),x)

[Out]

2**(2/3)*sqrt(7)*I*(1 - sqrt(7)*I)**(1/3)*log(2**(1/3)*x + (1 - sqrt(7)*I)**(1/3
))/42 - 2**(2/3)*sqrt(7)*I*(1 + sqrt(7)*I)**(1/3)*log(2**(1/3)*x + (1 + sqrt(7)*
I)**(1/3))/42 - 2**(2/3)*sqrt(7)*I*(1 - sqrt(7)*I)**(1/3)*log(x**2 - 2**(2/3)*x*
(1 - sqrt(7)*I)**(1/3)/2 + 2**(1/3)*(1 - sqrt(7)*I)**(2/3)/2)/84 + 2**(2/3)*sqrt
(7)*I*(1 + sqrt(7)*I)**(1/3)*log(x**2 - 2**(2/3)*x*(1 + sqrt(7)*I)**(1/3)/2 + 2*
*(1/3)*(1 + sqrt(7)*I)**(2/3)/2)/84 - 2**(2/3)*sqrt(21)*I*(1 - sqrt(7)*I)**(1/3)
*atan(sqrt(3)*(-2*2**(1/3)*x/(3*(1 - sqrt(7)*I)**(1/3)) + 1/3))/42 + 2**(2/3)*sq
rt(21)*I*(1 + sqrt(7)*I)**(1/3)*atan(sqrt(3)*(-2*2**(1/3)*x/(3*(1 + sqrt(7)*I)**
(1/3)) + 1/3))/42

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Mathematica [C]  time = 0.0151688, size = 37, normalized size = 0.09 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6+\text{$\#$1}^3+2\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3+1}\&\right ] \]

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

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Maple [C]  time = 0.007, size = 36, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+{{\it \_Z}}^{3}+2 \right ) }{\frac{{{\it \_R}}^{3}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}+{{\it \_R}}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^6+x^3+2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{x^{6} + x^{3} + 2}\,{d x} \]

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)

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Fricas [A]  time = 0.277145, size = 1454, normalized size = 3.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2058*343^(5/6)*(2*2^(1/6)*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))*log(2*343^(
1/6)*sqrt(7)*2^(1/6)*x*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 7*2^(1/3)*cos(
2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + 7*2^(1/3)*sin(2/3*arctan(1/(sqrt(7) - 2
*sqrt(2))))^2 + 7*x^2) + 8*2^(1/6)*arctan(7*2^(1/6)*cos(2/3*arctan(1/(sqrt(7) -
2*sqrt(2))))/(343^(1/6)*sqrt(7)*x + 343^(1/6)*sqrt(7)*sqrt(2/7*343^(1/6)*sqrt(7)
*2^(1/6)*x*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 2^(1/3)*cos(2/3*arctan(1/(
sqrt(7) - 2*sqrt(2))))^2 + 2^(1/3)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 +
x^2) + 7*2^(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))))*sin(2/3*arctan(1/(sq
rt(7) - 2*sqrt(2)))) - 4*(sqrt(3)*2^(1/6)*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))
)) - 2^(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))))*arctan(7*(sqrt(3)*2^(1/6)
*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 2^(1/6)*cos(2/3*arctan(1/(sqrt(7) -
2*sqrt(2)))))/(2*343^(1/6)*sqrt(7)*x + 7*sqrt(3)*2^(1/6)*cos(2/3*arctan(1/(sqrt(
7) - 2*sqrt(2)))) + 2*343^(1/6)*sqrt(7)*sqrt(1/7*343^(1/6)*sqrt(7)*sqrt(3)*2^(1/
6)*x*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) - 1/7*343^(1/6)*sqrt(7)*2^(1/6)*x*
sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 2^(1/3)*cos(2/3*arctan(1/(sqrt(7) - 2
*sqrt(2))))^2 + 2^(1/3)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + x^2) - 7*2^
(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))))) - 4*(sqrt(3)*2^(1/6)*cos(2/3*ar
ctan(1/(sqrt(7) - 2*sqrt(2)))) + 2^(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))
))*arctan(7*(sqrt(3)*2^(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) - 2^(1/6)*
cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))))/(2*343^(1/6)*sqrt(7)*x - 7*sqrt(3)*2^(
1/6)*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 2*343^(1/6)*sqrt(7)*sqrt(-1/7*34
3^(1/6)*sqrt(7)*sqrt(3)*2^(1/6)*x*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) - 1/7
*343^(1/6)*sqrt(7)*2^(1/6)*x*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 2^(1/3)*
cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + 2^(1/3)*sin(2/3*arctan(1/(sqrt(7) -
2*sqrt(2))))^2 + x^2) - 7*2^(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))))) -
(sqrt(3)*2^(1/6)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) + 2^(1/6)*cos(2/3*arct
an(1/(sqrt(7) - 2*sqrt(2)))))*log(343^(1/6)*sqrt(7)*sqrt(3)*2^(1/6)*x*cos(2/3*ar
ctan(1/(sqrt(7) - 2*sqrt(2)))) - 343^(1/6)*sqrt(7)*2^(1/6)*x*sin(2/3*arctan(1/(s
qrt(7) - 2*sqrt(2)))) + 7*2^(1/3)*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + 7
*2^(1/3)*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + 7*x^2) + (sqrt(3)*2^(1/6)*
sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(2)))) - 2^(1/6)*cos(2/3*arctan(1/(sqrt(7) - 2
*sqrt(2)))))*log(-343^(1/6)*sqrt(7)*sqrt(3)*2^(1/6)*x*cos(2/3*arctan(1/(sqrt(7)
- 2*sqrt(2)))) - 343^(1/6)*sqrt(7)*2^(1/6)*x*sin(2/3*arctan(1/(sqrt(7) - 2*sqrt(
2)))) + 7*2^(1/3)*cos(2/3*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + 7*2^(1/3)*sin(2/3
*arctan(1/(sqrt(7) - 2*sqrt(2))))^2 + 7*x^2))

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Sympy [A]  time = 0.359883, size = 24, normalized size = 0.06 \[ \operatorname{RootSum}{\left (250047 t^{6} + 1323 t^{3} + 2, \left ( t \mapsto t \log{\left (7938 t^{4} + 21 t + x \right )} \right )\right )} \]

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)
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{x^{6} + x^{3} + 2}\,{d x} \]

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)