3.22.27 \(\int \frac {-36-6 x^2+6 x^3+x^6}{x (-6+x^3) \sqrt [6]{\frac {6+x^3}{-6+x^3}} (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8)} \, dx\)

Optimal. Leaf size=154 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x^3+6}{x^3-6}}}{x-1}\right )-\frac {1}{6} \tan ^{-1}\left (\frac {\sqrt [3]{\frac {x^3+6}{x^3-6}}-x^2+2 x-1}{(x-1) \sqrt [6]{\frac {x^3+6}{x^3-6}}}\right )+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {3} x-\sqrt {3}\right ) \sqrt [6]{\frac {x^3+6}{x^3-6}}}{\sqrt [3]{\frac {x^3+6}{x^3-6}}+x^2-2 x+1}\right )}{2 \sqrt {3}} \]

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Rubi [F]  time = 10.60, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3))^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*
x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)),x]

[Out]

((6 + x^3)^(1/6)*ArcTan[(6 + x^3)^(1/6)/(-6 + x^3)^(1/6)])/(9*(-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6))
- ((6 + x^3)^(1/6)*ArcTan[Sqrt[3] - (2*(6 + x^3)^(1/6))/(-6 + x^3)^(1/6)])/(18*(-6 + x^3)^(1/6)*(-((6 + x^3)/
(6 - x^3)))^(1/6)) + ((6 + x^3)^(1/6)*ArcTan[Sqrt[3] + (2*(6 + x^3)^(1/6))/(-6 + x^3)^(1/6)])/(18*(-6 + x^3)^(
1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) + ((6 + x^3)^(1/6)*Log[1 - (Sqrt[3]*(6 + x^3)^(1/6))/(-6 + x^3)^(1/6) + (
6 + x^3)^(1/3)/(-6 + x^3)^(1/3)])/(12*Sqrt[3]*(-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) - ((6 + x^3)^(1
/6)*Log[1 + (Sqrt[3]*(6 + x^3)^(1/6))/(-6 + x^3)^(1/6) + (6 + x^3)^(1/3)/(-6 + x^3)^(1/3)])/(12*Sqrt[3]*(-6 +
x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) - (90*(6 + x^3)^(1/6)*Defer[Int][1/((-6 + x^3)^(5/6)*(6 + x^3)^(1/6
)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 + x^3)
/(6 - x^3)))^(1/6)) + (116*(6 + x^3)^(1/6)*Defer[Int][x/((-6 + x^3)^(5/6)*(6 + x^3)^(1/6)*(36 - 90*x + 122*x^2
- 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) -
(90*(6 + x^3)^(1/6)*Defer[Int][x^2/((-6 + x^3)^(5/6)*(6 + x^3)^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 -
26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) + (51*(6 + x^3)^(1/6)*D
efer[Int][x^3/((-6 + x^3)^(5/6)*(6 + x^3)^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x
^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) - (26*(6 + x^3)^(1/6)*Defer[Int][x^4/((-6 +
x^3)^(5/6)*(6 + x^3)^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6
+ x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)) + (16*(6 + x^3)^(1/6)*Defer[Int][x^5/((-6 + x^3)^(5/6)*(6 + x^3)^
(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 +
x^3)/(6 - x^3)))^(1/6)) - (6*(6 + x^3)^(1/6)*Defer[Int][x^6/((-6 + x^3)^(5/6)*(6 + x^3)^(1/6)*(36 - 90*x + 122
*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6)
) + ((6 + x^3)^(1/6)*Defer[Int][x^7/((-6 + x^3)^(5/6)*(6 + x^3)^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 -
26*x^5 + 15*x^6 - 6*x^7 + x^8)), x])/((-6 + x^3)^(1/6)*(-((6 + x^3)/(6 - x^3)))^(1/6))

Rubi steps

\begin {align*} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx &=\frac {\sqrt [6]{6+x^3} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \int \left (-\frac {1}{x \left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3}}+\frac {-90+116 x-90 x^2+51 x^3-26 x^4+16 x^5-6 x^6+x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}\right ) \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=-\frac {\sqrt [6]{6+x^3} \int \frac {1}{x \left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3}} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {-90+116 x-90 x^2+51 x^3-26 x^4+16 x^5-6 x^6+x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{(-6+x)^{5/6} x \sqrt [6]{6+x}} \, dx,x,x^3\right )}{3 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \left (-\frac {90}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {116 x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}-\frac {90 x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {51 x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}-\frac {26 x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {16 x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}-\frac {6 x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}\right ) \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (2 \sqrt [6]{6+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{-6-6 x^6} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{36 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{36 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \log \left (1-\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \log \left (1+\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \log \left (1-\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \log \left (1+\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ \end {align*}

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Mathematica [F]  time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3))^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3
+ 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)),x]

[Out]

Integrate[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3))^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3
+ 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x]

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IntegrateAlgebraic [A]  time = 3.11, size = 154, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt [6]{\frac {6+x^3}{-6+x^3}}}{-1+x}\right )-\frac {1}{6} \tan ^{-1}\left (\frac {-1+2 x-x^2+\sqrt [3]{\frac {6+x^3}{-6+x^3}}}{(-1+x) \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\right )+\frac {\tanh ^{-1}\left (\frac {\left (-\sqrt {3}+\sqrt {3} x\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}}}{1-2 x+x^2+\sqrt [3]{\frac {6+x^3}{-6+x^3}}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3))^(1/6)*(36 - 90*x + 122*x^2
- 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)),x]

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^
2-90*x+36),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 6 \, x^{3} - 6 \, x^{2} - 36}{{\left (x^{8} - 6 \, x^{7} + 15 \, x^{6} - 26 \, x^{5} + 51 \, x^{4} - 96 \, x^{3} + 122 \, x^{2} - 90 \, x + 36\right )} {\left (x^{3} - 6\right )} x \left (\frac {x^{3} + 6}{x^{3} - 6}\right )^{\frac {1}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^
2-90*x+36),x, algorithm="giac")

[Out]

integrate((x^6 + 6*x^3 - 6*x^2 - 36)/((x^8 - 6*x^7 + 15*x^6 - 26*x^5 + 51*x^4 - 96*x^3 + 122*x^2 - 90*x + 36)*
(x^3 - 6)*x*((x^3 + 6)/(x^3 - 6))^(1/6)), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{6}+6 x^{3}-6 x^{2}-36}{x \left (x^{3}-6\right ) \left (\frac {x^{3}+6}{x^{3}-6}\right )^{\frac {1}{6}} \left (x^{8}-6 x^{7}+15 x^{6}-26 x^{5}+51 x^{4}-96 x^{3}+122 x^{2}-90 x +36\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x
+36),x)

[Out]

int((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x
+36),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 6 \, x^{3} - 6 \, x^{2} - 36}{{\left (x^{8} - 6 \, x^{7} + 15 \, x^{6} - 26 \, x^{5} + 51 \, x^{4} - 96 \, x^{3} + 122 \, x^{2} - 90 \, x + 36\right )} {\left (x^{3} - 6\right )} x \left (\frac {x^{3} + 6}{x^{3} - 6}\right )^{\frac {1}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^
2-90*x+36),x, algorithm="maxima")

[Out]

integrate((x^6 + 6*x^3 - 6*x^2 - 36)/((x^8 - 6*x^7 + 15*x^6 - 26*x^5 + 51*x^4 - 96*x^3 + 122*x^2 - 90*x + 36)*
(x^3 - 6)*x*((x^3 + 6)/(x^3 - 6))^(1/6)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {-x^6-6\,x^3+6\,x^2+36}{x\,{\left (\frac {x^3+6}{x^3-6}\right )}^{1/6}\,\left (x^3-6\right )\,\left (x^8-6\,x^7+15\,x^6-26\,x^5+51\,x^4-96\,x^3+122\,x^2-90\,x+36\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x^2 - 6*x^3 - x^6 + 36)/(x*((x^3 + 6)/(x^3 - 6))^(1/6)*(x^3 - 6)*(122*x^2 - 90*x - 96*x^3 + 51*x^4 - 2
6*x^5 + 15*x^6 - 6*x^7 + x^8 + 36)),x)

[Out]

-int((6*x^2 - 6*x^3 - x^6 + 36)/(x*((x^3 + 6)/(x^3 - 6))^(1/6)*(x^3 - 6)*(122*x^2 - 90*x - 96*x^3 + 51*x^4 - 2
6*x^5 + 15*x^6 - 6*x^7 + x^8 + 36)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+6*x**3-6*x**2-36)/x/(x**3-6)/((x**3+6)/(x**3-6))**(1/6)/(x**8-6*x**7+15*x**6-26*x**5+51*x**4-9
6*x**3+122*x**2-90*x+36),x)

[Out]

Timed out

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