3.20.11 \(\int \frac {(-1+x^2) \sqrt {1+x^2+x^4}}{(1+x^2) (1+x+x^2+x^3+x^4)} \, dx\) [1911]

Optimal. Leaf size=133 \[ -2 \text {ArcTan}\left (\frac {\sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {1+x^2+x^4}}{\sqrt {2+\sqrt {5}} \left (1-x+x^2\right )}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {1+x^2+x^4}}{\sqrt {-2+\sqrt {5}} \left (1-x+x^2\right )}\right ) \]

[Out]

-2*arctan((x^4+x^2+1)^(1/2)/(x^2-x+1))+1/5*(10+10*5^(1/2))^(1/2)*arctan((x^4+x^2+1)^(1/2)/(2+5^(1/2))^(1/2)/(x
^2-x+1))+1/5*(-10+10*5^(1/2))^(1/2)*arctanh((x^4+x^2+1)^(1/2)/(-2+5^(1/2))^(1/2)/(x^2-x+1))

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Rubi [F]
time = 0.63, 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 {\left (-1+x^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^2\right ) \left (1+x+x^2+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^2)*Sqrt[1 + x^2 + x^4])/((1 + x^2)*(1 + x + x^2 + x^3 + x^4)),x]

[Out]

(-2*x*Sqrt[1 + x^2 + x^4])/(1 + x^2) - ArcTan[x/Sqrt[1 + x^2 + x^4]] + (2*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 +
x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/Sqrt[1 + x^2 + x^4] - (3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*Elli
pticF[2*ArcTan[x], 1/4])/(2*Sqrt[1 + x^2 + x^4]) + Defer[Int][Sqrt[1 + x^2 + x^4]/(1 + x + x^2 + x^3 + x^4), x
] + 2*Defer[Int][(x*Sqrt[1 + x^2 + x^4])/(1 + x + x^2 + x^3 + x^4), x] + 2*Defer[Int][(x^2*Sqrt[1 + x^2 + x^4]
)/(1 + x + x^2 + x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^2\right ) \left (1+x+x^2+x^3+x^4\right )} \, dx &=\int \left (-\frac {2 \sqrt {1+x^2+x^4}}{1+x^2}+\frac {\left (1+2 x+2 x^2\right ) \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2+x^4}}{1+x^2} \, dx\right )+\int \frac {\left (1+2 x+2 x^2\right ) \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ &=-\left (2 \int \frac {x^2}{\sqrt {1+x^2+x^4}} \, dx\right )-2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \left (\frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}+\frac {2 x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}+\frac {2 x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx\right )+2 \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx-\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ &=-\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}-\tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 129, normalized size = 0.97 \begin {gather*} -2 \text {ArcTan}\left (\frac {\sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {2}{5} \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-2+\sqrt {5}} \sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {2}{5} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {5}} \sqrt {1+x^2+x^4}}{1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^2)*Sqrt[1 + x^2 + x^4])/((1 + x^2)*(1 + x + x^2 + x^3 + x^4)),x]

[Out]

-2*ArcTan[Sqrt[1 + x^2 + x^4]/(1 - x + x^2)] + Sqrt[(2*(1 + Sqrt[5]))/5]*ArcTan[(Sqrt[-2 + Sqrt[5]]*Sqrt[1 + x
^2 + x^4])/(1 - x + x^2)] + Sqrt[(2*(-1 + Sqrt[5]))/5]*ArcTanh[(Sqrt[2 + Sqrt[5]]*Sqrt[1 + x^2 + x^4])/(1 - x
+ x^2)]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.73, size = 712, normalized size = 5.35

method	result	size
elliptic	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (\sqrt {x^{4}+x^{2}+1}-x^{2}\right )^{2}+\left (-15 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} \right ) \left (\sqrt {x^{4}+x^{2}+1}-x^{2}\right )-15 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +2\right )\right )}{2}+\frac {\left (-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {2 \sqrt {5}\, \arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(186\)
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (6 \underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}-6 \underline {\hspace {1.25 ex}}\alpha +4\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha }}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{10}+\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticE \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(712\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)*(x^4+x^2+1)^(1/2)/(x^2+1)/(x^4+x^3+x^2+x+1),x,method=_RETURNVERBOSE)

[Out]

2/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/
2)*EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-8/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*
3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)/(1+I*3^(1/2))*(EllipticF(1/2*x*(-2+2*
I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2)
))+1/10*sum((-2*_alpha^2-_alpha-2)*(-1/(-_alpha^3-_alpha)^(1/2)*arctanh(1/22*(2*_alpha^2+1)*(6*_alpha^3-3*_alp
ha^2+11*x^2-6*_alpha+4)/(-_alpha^3-_alpha)^(1/2)/(x^4+x^2+1)^(1/2))-2^(1/2)*(-_alpha^3-_alpha^2-_alpha-1)/(-1+
I*3^(1/2))^(1/2)*(x^2+2-I*3^(1/2)*x^2)^(1/2)*(x^2+2+I*3^(1/2)*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/
2*I*3^(1/2))^(1/2)*x,-1/2*I*_alpha^3*3^(1/2)-1/2*_alpha^3,(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*3^(1/2))^(1/2
))),_alpha=RootOf(_Z^4+_Z^3+_Z^2+_Z+1))+8/(-2+2*I*3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^
2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)^(1/2)/(1+I*3^(1/2))*EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*
3^(1/2))^(1/2))-8/(-2+2*I*3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))^(1/
2)/(x^4+x^2+1)^(1/2)/(1+I*3^(1/2))*EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-2/(-1/2+
1/2*I*3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)^(1/2)
*EllipticPi((-1/2+1/2*I*3^(1/2))^(1/2)*x,-1/(-1/2+1/2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*3^(1/2
))^(1/2))

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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^2-1)*(x^4+x^2+1)^(1/2)/(x^2+1)/(x^4+x^3+x^2+x+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + x^2 + 1)*(x^2 - 1)/((x^4 + x^3 + x^2 + x + 1)*(x^2 + 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (105) = 210\).
time = 0.51, size = 306, normalized size = 2.30 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{4} + \sqrt {5} x^{2} + x^{2} + 2\right )} \sqrt {2 \, \sqrt {5} + 2} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x - x + 2\right )} \sqrt {2 \, \sqrt {5} + 2}}{8 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} + {\left (x^{4} + 3 \, x^{2} + \sqrt {5} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} - {\left (x^{4} + 3 \, x^{2} + \sqrt {5} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4+x^2+1)^(1/2)/(x^2+1)/(x^4+x^3+x^2+x+1),x, algorithm="fricas")

[Out]

1/5*sqrt(5)*sqrt(2*sqrt(5) + 2)*arctan(1/8*(sqrt(2)*(2*x^4 + sqrt(5)*x^2 + x^2 + 2)*sqrt(2*sqrt(5) + 2)*sqrt(s
qrt(5) + 1) + 2*sqrt(x^4 + x^2 + 1)*(2*x^2 + sqrt(5)*x - x + 2)*sqrt(2*sqrt(5) + 2))/(x^4 - x^3 + x^2 - x + 1)
) + 1/20*sqrt(5)*sqrt(2*sqrt(5) - 2)*log(-(2*sqrt(x^4 + x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) + (x^4 + 3*x^2 +
sqrt(5)*(x^4 + x^2 + 1) + 1)*sqrt(2*sqrt(5) - 2))/(x^4 + x^3 + x^2 + x + 1)) - 1/20*sqrt(5)*sqrt(2*sqrt(5) - 2
)*log(-(2*sqrt(x^4 + x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) - (x^4 + 3*x^2 + sqrt(5)*(x^4 + x^2 + 1) + 1)*sqrt(2
*sqrt(5) - 2))/(x^4 + x^3 + x^2 + x + 1)) - arctan(x/sqrt(x^4 + x^2 + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)*(x**4+x**2+1)**(1/2)/(x**2+1)/(x**4+x**3+x**2+x+1),x)

[Out]

Integral(sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x - 1)*(x + 1)/((x**2 + 1)*(x**4 + x**3 + x**2 + x + 1)), x)

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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^2-1)*(x^4+x^2+1)^(1/2)/(x^2+1)/(x^4+x^3+x^2+x+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + x^2 + 1)*(x^2 - 1)/((x^4 + x^3 + x^2 + x + 1)*(x^2 + 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+x^2+1}}{\left (x^2+1\right )\,\left (x^4+x^3+x^2+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 1)*(x^2 + x^4 + 1)^(1/2))/((x^2 + 1)*(x + x^2 + x^3 + x^4 + 1)),x)

[Out]

int(((x^2 - 1)*(x^2 + x^4 + 1)^(1/2))/((x^2 + 1)*(x + x^2 + x^3 + x^4 + 1)), x)

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