3.29.5 \(\int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} (1+x^6)} \, dx\)
Optimal. Leaf size=274 \[ -\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt [3]{x^4-x^2}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4-x^2}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \tan ^{-1}\left (\frac {x \sqrt [3]{x^4-x^2}}{\left (x^4-x^2\right )^{2/3}-x^2}\right )-\frac {\tan ^{-1}\left (\frac {2^{2/3} x \sqrt [3]{x^4-x^2}}{\sqrt [3]{2} \left (x^4-x^2\right )^{2/3}-2 x^2}\right )}{6 \sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {\left (x^4-x^2\right )^{2/3}}{\sqrt {3}}}{x \sqrt [3]{x^4-x^2}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [3]{2} x^2}{\sqrt {3}}+\frac {\left (x^4-x^2\right )^{2/3}}{\sqrt [3]{2} \sqrt {3}}}{x \sqrt [3]{x^4-x^2}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \]
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Rubi [F] time = 2.73, 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 {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]
[Out]
((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) -
x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subs
t][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 -
I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(1/9)*x), x],
x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defe
r[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^(1/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x
^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(2/9)*x), x], x, x^(1/3)]
)/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1
)^(1/18) + (-1)^(2/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 +
x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(1/3)*x), x], x, x^(1/3)])/(6*(-x^2 +
x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/
((-1)^(1/18) + (-1)^(1/3)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(
-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(-x
^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2
/3)/((-1)^(1/18) + (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1
/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(5/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1
/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^(5/9
)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Su
bst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(2/3)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)
^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^
(2/3)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defe
r[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(7/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + (
(-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (
-1)^(7/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Def
er[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(8/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*
x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^(8/9)*x), x], x, x^(1/3)
])/(6*(-x^2 + x^4)^(1/3))
Rubi steps
\begin {align*} \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {-1+x^6}{x^{2/3} \sqrt [3]{-1+x^2} \left (1+x^6\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {\left (-1+x^2\right )^{2/3} \left (1+x^2+x^4\right )}{x^{2/3} \left (1+x^6\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6+x^{12}\right )}{1+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [9]{-1} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+\sqrt [9]{-1} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{2/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{2/9} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [3]{-1} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+\sqrt [3]{-1} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{4/9} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{4/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{5/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{5/9} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{2/3} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{2/3} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{7/9} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{7/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{8/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{8/9} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (\sqrt [18]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{2/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+(-1)^{2/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{5/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+(-1)^{5/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{8/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+(-1)^{8/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-\sqrt [9]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+\sqrt [9]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{4/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+(-1)^{4/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{7/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+(-1)^{7/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [18]{-1} \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{\sqrt [18]{-1}+(-1)^{2/3} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{-x^2+x^4}}\\ \end {align*}
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Mathematica [F] time = 0.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]
[Out]
Integrate[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)), x]
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IntegrateAlgebraic [C] time = 0.87, size = 269, normalized size = 0.98 \begin {gather*} -\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt [3]{-x^2+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{-x^2+x^4}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} i \left (-i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1-i \sqrt {3}\right ) x}{2 \sqrt [3]{-x^2+x^4}}\right )+\frac {1}{3} i \left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt [3]{-x^2+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {2^{2/3} x \sqrt [3]{-x^2+x^4}}{-2 x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}}\right )}{6 \sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [3]{2} x^2}{\sqrt {3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{\sqrt [3]{2} \sqrt {3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]
[Out]
(-2*ArcTan[x/(-x^2 + x^4)^(1/3)])/3 - ArcTan[(2^(1/3)*x)/(-x^2 + x^4)^(1/3)]/(3*2^(1/3)) - (I/3)*(-I + Sqrt[3]
)*ArcTan[((1 - I*Sqrt[3])*x)/(2*(-x^2 + x^4)^(1/3))] + (I/3)*(I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/(2*(-x^2
+ x^4)^(1/3))] - ArcTan[(2^(2/3)*x*(-x^2 + x^4)^(1/3))/(-2*x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3))]/(6*2^(1/3)) - A
rcTanh[((2^(1/3)*x^2)/Sqrt[3] + (-x^2 + x^4)^(2/3)/(2^(1/3)*Sqrt[3]))/(x*(-x^2 + x^4)^(1/3))]/(2*2^(1/3)*Sqrt[
3])
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fricas [B] time = 14.52, size = 3724, normalized size = 13.59
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x, algorithm="fricas")
[Out]
-1/96*sqrt(3)*2^(2/3)*log(8500000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(2/3)*(x^2 -
1) + 6*2^(2/3)*x) + 2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(3*x^3 + 2*sqrt(3)*x^2 - 3*x))/(x^5 + 2*x
^3 + x)) - 1/96*sqrt(3)*2^(2/3)*log(2125000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(2
/3)*(x^2 - 1) + 6*2^(2/3)*x) + 2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(3*x^3 + 2*sqrt(3)*x^2 - 3*x))/
(x^5 + 2*x^3 + x)) + 1/96*sqrt(3)*2^(2/3)*log(-2125000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(s
qrt(3)*2^(2/3)*(x^2 - 1) - 6*2^(2/3)*x) - 2^(1/3)*(x^5 + 2*x^3 + x) - 4*(x^4 - x^2)^(1/3)*(3*x^3 - 2*sqrt(3)*x
^2 - 3*x))/(x^5 + 2*x^3 + x)) + 1/96*sqrt(3)*2^(2/3)*log(-8500000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^
2)^(2/3)*(sqrt(3)*2^(2/3)*(x^2 - 1) - 6*2^(2/3)*x) - 2^(1/3)*(x^5 + 2*x^3 + x) - 4*(x^4 - x^2)^(1/3)*(3*x^3 -
2*sqrt(3)*x^2 - 3*x))/(x^5 + 2*x^3 + x)) + 1/12*2^(2/3)*arctan(-(74071498415429632*x^9 + 1645279755446275808*x
^8 - 2346817955632029696*x^7 - 11516958288123930656*x^6 + 5730636889080074240*x^5 + 11516958288123930656*x^4 -
2346817955632029696*x^3 - 1645279755446275808*x^2 - 125*sqrt(34)*(4*sqrt(3)*2^(1/3)*(78465570355328*x^9 - 330
1419835659*x^8 + 1100839094578688*x^7 - 595767752585659*x^6 - 3614058455553280*x^5 + 595767752585659*x^4 + 110
0839094578688*x^3 + 3301419835659*x^2 + 78465570355328*x) + 16*(x^4 - x^2)^(2/3)*(4*sqrt(3)*2^(2/3)*(151368856
3712*x^6 + 57183135266496*x^5 - 26977277846305*x^4 - 167158338888320*x^3 + 26977277846305*x^2 + 57183135266496
*x - 1513688563712) - 2^(2/3)*(79163286177664*x^6 - 56815411732213*x^5 - 187311276664960*x^4 + 112551186315710
*x^3 + 187311276664960*x^2 - 56815411732213*x - 79163286177664)) - 2^(1/3)*(36167723835659*x^9 + 4738598437685
248*x^8 - 1343569332842636*x^7 - 16069401562314752*x^6 + 2036119636643410*x^5 + 16069401562314752*x^4 - 134356
9332842636*x^3 - 4738598437685248*x^2 + 36167723835659*x) - 4*(183204669874443*x^7 + 4116235393055744*x^6 - 22
25700627116645*x^5 - 10698715224852480*x^4 + 2225700627116645*x^3 + 4116235393055744*x^2 - 531250*sqrt(3)*(100
9306368*x^7 - 511421263*x^6 - 4316628224*x^5 + 1207618962*x^4 + 4316628224*x^3 - 511421263*x^2 - 1009306368*x)
- 183204669874443*x)*(x^4 - x^2)^(1/3))*sqrt((8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^
(2/3)*(x^2 - 1) + 6*2^(2/3)*x) + 2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(3*x^3 + 2*sqrt(3)*x^2 - 3*x)
)/(x^5 + 2*x^3 + x)) - 1062500*(x^4 - x^2)^(2/3)*(2*sqrt(3)*2^(1/3)*(23651383808*x^6 + 470146644789*x^5 - 2263
86757120*x^4 - 71809982630*x^3 + 226386757120*x^2 + 470146644789*x - 23651383808) - 2^(1/3)*(618463173263*x^6
- 733160605696*x^5 - 6989546598945*x^4 + 2615047352320*x^3 + 6989546598945*x^2 - 733160605696*x - 618463173263
)) - 265625*sqrt(3)*(613012268401*x^9 - 500076281856*x^8 - 1596364015228*x^7 + 3500533972992*x^6 + 11774899788
070*x^5 - 3500533972992*x^4 - 1596364015228*x^3 + 500076281856*x^2 + 613012268401*x) - 1062500*(x^4 - x^2)^(1/
3)*(sqrt(3)*2^(2/3)*(217120826737*x^7 + 155432605696*x^6 + 1229224098945*x^5 - 689287352320*x^4 - 122922409894
5*x^3 + 155432605696*x^2 - 217120826737*x) - 2*2^(2/3)*(71795383808*x^7 + 1283539269789*x^6 - 948546757120*x^5
- 5040931232630*x^4 + 948546757120*x^3 + 1283539269789*x^2 - 71795383808*x)) + 74071498415429632*x)/(47995856
8556831351*x^9 - 1202832749691437056*x^8 - 12744795130528777828*x^7 + 8419829247840059392*x^6 + 32209010220853
194570*x^5 - 8419829247840059392*x^4 - 12744795130528777828*x^3 + 1202832749691437056*x^2 + 479958568556831351
*x)) - 1/12*2^(2/3)*arctan((74071498415429632*x^9 + 1645279755446275808*x^8 - 2346817955632029696*x^7 - 115169
58288123930656*x^6 + 5730636889080074240*x^5 + 11516958288123930656*x^4 - 2346817955632029696*x^3 - 1645279755
446275808*x^2 + 125*sqrt(34)*(4*sqrt(3)*2^(1/3)*(78465570355328*x^9 - 3301419835659*x^8 + 1100839094578688*x^7
- 595767752585659*x^6 - 3614058455553280*x^5 + 595767752585659*x^4 + 1100839094578688*x^3 + 3301419835659*x^2
+ 78465570355328*x) + 16*(x^4 - x^2)^(2/3)*(4*sqrt(3)*2^(2/3)*(1513688563712*x^6 + 57183135266496*x^5 - 26977
277846305*x^4 - 167158338888320*x^3 + 26977277846305*x^2 + 57183135266496*x - 1513688563712) + 2^(2/3)*(791632
86177664*x^6 - 56815411732213*x^5 - 187311276664960*x^4 + 112551186315710*x^3 + 187311276664960*x^2 - 56815411
732213*x - 79163286177664)) + 2^(1/3)*(36167723835659*x^9 + 4738598437685248*x^8 - 1343569332842636*x^7 - 1606
9401562314752*x^6 + 2036119636643410*x^5 + 16069401562314752*x^4 - 1343569332842636*x^3 - 4738598437685248*x^2
+ 36167723835659*x) + 4*(183204669874443*x^7 + 4116235393055744*x^6 - 2225700627116645*x^5 - 1069871522485248
0*x^4 + 2225700627116645*x^3 + 4116235393055744*x^2 + 531250*sqrt(3)*(1009306368*x^7 - 511421263*x^6 - 4316628
224*x^5 + 1207618962*x^4 + 4316628224*x^3 - 511421263*x^2 - 1009306368*x) - 183204669874443*x)*(x^4 - x^2)^(1/
3))*sqrt(-(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(2/3)*(x^2 - 1) - 6*2^(2/3)*x) - 2^(
1/3)*(x^5 + 2*x^3 + x) - 4*(x^4 - x^2)^(1/3)*(3*x^3 - 2*sqrt(3)*x^2 - 3*x))/(x^5 + 2*x^3 + x)) + 1062500*(x^4
- x^2)^(2/3)*(2*sqrt(3)*2^(1/3)*(23651383808*x^6 + 470146644789*x^5 - 226386757120*x^4 - 71809982630*x^3 + 226
386757120*x^2 + 470146644789*x - 23651383808) + 2^(1/3)*(618463173263*x^6 - 733160605696*x^5 - 6989546598945*x
^4 + 2615047352320*x^3 + 6989546598945*x^2 - 733160605696*x - 618463173263)) + 265625*sqrt(3)*(613012268401*x^
9 - 500076281856*x^8 - 1596364015228*x^7 + 3500533972992*x^6 + 11774899788070*x^5 - 3500533972992*x^4 - 159636
4015228*x^3 + 500076281856*x^2 + 613012268401*x) + 1062500*(x^4 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*(217120826737*x^
7 + 155432605696*x^6 + 1229224098945*x^5 - 689287352320*x^4 - 1229224098945*x^3 + 155432605696*x^2 - 217120826
737*x) + 2*2^(2/3)*(71795383808*x^7 + 1283539269789*x^6 - 948546757120*x^5 - 5040931232630*x^4 + 948546757120*
x^3 + 1283539269789*x^2 - 71795383808*x)) + 74071498415429632*x)/(479958568556831351*x^9 - 1202832749691437056
*x^8 - 12744795130528777828*x^7 + 8419829247840059392*x^6 + 32209010220853194570*x^5 - 8419829247840059392*x^4
- 12744795130528777828*x^3 + 1202832749691437056*x^2 + 479958568556831351*x)) + 1/6*2^(2/3)*arctan(-1/2*(3564
544*x^5 + 249106968*x^4 - 21387264*x^3 + 2125000*2^(2/3)*(x^4 - x^2)^(1/3)*(512*x^3 + 59*x^2 - 512*x) + 106250
0*2^(1/3)*(x^4 - x^2)^(2/3)*(59*x^2 - 2048*x - 59) - 249106968*x^2 - 125*sqrt(34)*2^(1/6)*(4*2^(2/3)*(x^4 - x^
2)^(2/3)*(15104*x^2 + 527769*x - 15104) + 3481*2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(527769*x^3 - 6
0416*x^2 - 527769*x)) + 3564544*x)/(205379*x^5 - 2168870912*x^4 - 1232274*x^3 + 2168870912*x^2 + 205379*x)) -
1/12*sqrt(3)*log(26618852*(x^5 - x^3 + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*(x^2 - 1) + 3*x) + 4*sqrt(3)*(x^4 - x^2) +
2*(x^4 - x^2)^(1/3)*(3*x^3 + sqrt(3)*x^2 - 3*x) + x)/(x^5 - x^3 + x)) + 1/12*sqrt(3)*log(26618852*(x^5 - x^3
- 2*(x^4 - x^2)^(2/3)*(sqrt(3)*(x^2 - 1) - 3*x) - 4*sqrt(3)*(x^4 - x^2) + 2*(x^4 - x^2)^(1/3)*(3*x^3 - sqrt(3)
*x^2 - 3*x) + x)/(x^5 - x^3 + x)) + 1/3*arctan(-(163348821309602766976*x^9 + 3887432402679837751952*x^8 - 3793
551880416319588608*x^7 - 15549729610719351007808*x^6 + 7423754939523036410240*x^5 + 15549729610719351007808*x^
4 - 3793551880416319588608*x^3 - 3887432402679837751952*x^2 - 338*sqrt(233)*(67166456130593243*x^9 - 173187348
9534746816*x^8 - 8262322488125426948*x^7 + 5402376118068558976*x^6 + 16323145607859074167*x^5 - 54023761180685
58976*x^4 - 8262322488125426948*x^3 + 1731873489534746816*x^2 + 8*(11634681213606448*x^6 + 88410267443510747*x
^5 - 141607113799927264*x^4 - 304974921996124561*x^3 + 141607113799927264*x^2 + 2*sqrt(3)*(2398449325331968*x^
6 + 66317101349416968*x^5 + 306343852456405393*x^4 - 162717914879099272*x^3 - 306343852456405393*x^2 + 6631710
1349416968*x - 2398449325331968) + 88410267443510747*x - 11634681213606448)*(x^4 - x^2)^(2/3) + 2*sqrt(3)*(814
4636084443696*x^9 + 17828539163673051*x^8 - 307821488566558016*x^7 + 605786626407170998*x^6 + 5912090688797849
44*x^5 - 605786626407170998*x^4 - 307821488566558016*x^3 - 17828539163673051*x^2 + 8144636084443696*x) + 2*(19
5992612428698075*x^7 + 72793982843270240*x^6 - 2656296320238012626*x^5 - 50181424325452512*x^4 + 2656296320238
012626*x^3 + 72793982843270240*x^2 + 6654713*sqrt(3)*(14835265056*x^7 + 109365761599*x^6 + 11690073152*x^5 - 2
05755148299*x^4 - 11690073152*x^3 + 109365761599*x^2 - 14835265056*x) - 195992612428698075*x)*(x^4 - x^2)^(1/3
) + 67166456130593243*x)*sqrt((x^5 - x^3 + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*(x^2 - 1) + 3*x) + 4*sqrt(3)*(x^4 - x^
2) + 2*(x^4 - x^2)^(1/3)*(3*x^3 + sqrt(3)*x^2 - 3*x) + x)/(x^5 - x^3 + x)) + 26618852*(38271977676337*x^6 - 23
6679798487232*x^5 - 918515085244470*x^4 + 579302159719616*x^3 + 918515085244470*x^2 + sqrt(3)*(15779271877184*
x^6 + 170585488729845*x^5 - 175823521011328*x^4 - 273378905866169*x^3 + 175823521011328*x^2 + 170585488729845*
x - 15779271877184) - 236679798487232*x - 38271977676337)*(x^4 - x^2)^(2/3) + 6654713*sqrt(3)*(26791566067249*
x^9 - 288529954513920*x^8 - 464756800679794*x^7 + 1154119818055680*x^6 + 902722035292339*x^5 - 115411981805568
0*x^4 - 464756800679794*x^3 + 288529954513920*x^2 + 26791566067249*x) + 26618852*(105942562745152*x^7 + 803699
152215459*x^6 - 554507486722688*x^5 - 1645670282107255*x^4 + 554507486722688*x^3 + 803699152215459*x^2 + sqrt(
3)*(67792071593521*x^7 + 128485705379776*x^6 - 32790726050718*x^5 - 272750682636736*x^4 + 32790726050718*x^3 +
128485705379776*x^2 - 67792071593521*x) - 105942562745152*x)*(x^4 - x^2)^(1/3) + 163348821309602766976*x)/(13
8674673083346657193*x^9 - 6293534081666240678912*x^8 - 30464519046388862556482*x^7 + 25174136326664962715648*x
^6 + 60790363419694378455771*x^5 - 25174136326664962715648*x^4 - 30464519046388862556482*x^3 + 629353408166624
0678912*x^2 + 138674673083346657193*x)) - 1/3*arctan((163348821309602766976*x^9 + 3887432402679837751952*x^8 -
3793551880416319588608*x^7 - 15549729610719351007808*x^6 + 7423754939523036410240*x^5 + 155497296107193510078
08*x^4 - 3793551880416319588608*x^3 - 3887432402679837751952*x^2 - 338*sqrt(233)*(67166456130593243*x^9 - 1731
873489534746816*x^8 - 8262322488125426948*x^7 + 5402376118068558976*x^6 + 16323145607859074167*x^5 - 540237611
8068558976*x^4 - 8262322488125426948*x^3 + 1731873489534746816*x^2 + 8*(11634681213606448*x^6 + 88410267443510
747*x^5 - 141607113799927264*x^4 - 304974921996124561*x^3 + 141607113799927264*x^2 - 2*sqrt(3)*(23984493253319
68*x^6 + 66317101349416968*x^5 + 306343852456405393*x^4 - 162717914879099272*x^3 - 306343852456405393*x^2 + 66
317101349416968*x - 2398449325331968) + 88410267443510747*x - 11634681213606448)*(x^4 - x^2)^(2/3) - 2*sqrt(3)
*(8144636084443696*x^9 + 17828539163673051*x^8 - 307821488566558016*x^7 + 605786626407170998*x^6 + 59120906887
9784944*x^5 - 605786626407170998*x^4 - 307821488566558016*x^3 - 17828539163673051*x^2 + 8144636084443696*x) +
2*(195992612428698075*x^7 + 72793982843270240*x^6 - 2656296320238012626*x^5 - 50181424325452512*x^4 + 26562963
20238012626*x^3 + 72793982843270240*x^2 - 6654713*sqrt(3)*(14835265056*x^7 + 109365761599*x^6 + 11690073152*x^
5 - 205755148299*x^4 - 11690073152*x^3 + 109365761599*x^2 - 14835265056*x) - 195992612428698075*x)*(x^4 - x^2)
^(1/3) + 67166456130593243*x)*sqrt((x^5 - x^3 - 2*(x^4 - x^2)^(2/3)*(sqrt(3)*(x^2 - 1) - 3*x) - 4*sqrt(3)*(x^4
- x^2) + 2*(x^4 - x^2)^(1/3)*(3*x^3 - sqrt(3)*x^2 - 3*x) + x)/(x^5 - x^3 + x)) + 26618852*(38271977676337*x^6
- 236679798487232*x^5 - 918515085244470*x^4 + 579302159719616*x^3 + 918515085244470*x^2 - sqrt(3)*(1577927187
7184*x^6 + 170585488729845*x^5 - 175823521011328*x^4 - 273378905866169*x^3 + 175823521011328*x^2 + 17058548872
9845*x - 15779271877184) - 236679798487232*x - 38271977676337)*(x^4 - x^2)^(2/3) - 6654713*sqrt(3)*(2679156606
7249*x^9 - 288529954513920*x^8 - 464756800679794*x^7 + 1154119818055680*x^6 + 902722035292339*x^5 - 1154119818
055680*x^4 - 464756800679794*x^3 + 288529954513920*x^2 + 26791566067249*x) + 26618852*(105942562745152*x^7 + 8
03699152215459*x^6 - 554507486722688*x^5 - 1645670282107255*x^4 + 554507486722688*x^3 + 803699152215459*x^2 -
sqrt(3)*(67792071593521*x^7 + 128485705379776*x^6 - 32790726050718*x^5 - 272750682636736*x^4 + 32790726050718*
x^3 + 128485705379776*x^2 - 67792071593521*x) - 105942562745152*x)*(x^4 - x^2)^(1/3) + 163348821309602766976*x
)/(138674673083346657193*x^9 - 6293534081666240678912*x^8 - 30464519046388862556482*x^7 + 25174136326664962715
648*x^6 + 60790363419694378455771*x^5 - 25174136326664962715648*x^4 - 30464519046388862556482*x^3 + 6293534081
666240678912*x^2 + 138674673083346657193*x)) - 1/3*arctan(-2*(1910654896*x^5 - 17610113139*x^4 - 5731964688*x^
3 + 17610113139*x^2 - 6654713*(x^4 - x^2)^(2/3)*(2507*x^2 + 1216*x - 2507) + 6654713*(x^4 - x^2)^(1/3)*(1216*x
^3 - 2507*x^2 - 1216*x) + 1910654896*x)/(15756617843*x^5 + 24725904448*x^4 - 47269853529*x^3 - 24725904448*x^2
+ 15756617843*x))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x, algorithm="giac")
[Out]
integrate((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)), x)
________________________________________________________________________________________
maple [C] time = 172.65, size = 12539, normalized size = 45.76
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(12539\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x,method=_RETURNVERBOSE)
[Out]
result too large to display
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x, algorithm="maxima")
[Out]
integrate((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6-1}{\left (x^6+1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)),x)
[Out]
int((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**6-1)/(x**4-x**2)**(1/3)/(x**6+1),x)
[Out]
Integral((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)/((x**2*(x - 1)*(x + 1))**(1/3)*(x**2 + 1)*(x**4 - x**2
+ 1)), x)
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