3.14.25 \(\int \frac {(1+x^6) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx\)

Optimal. Leaf size=95 \[ -\frac {1}{4} \sqrt {\frac {1}{3} \left (1+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt {x^6-x^2-2}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (\sqrt {3}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {3}-1} x}{\sqrt {x^6-x^2-2}}\right ) \]

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Rubi [F]  time = 0.56, 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^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^6)*Sqrt[-2 - x^2 + x^6])/(4 - 3*x^4 - 4*x^6 + x^12),x]

[Out]

Defer[Int][Sqrt[-2 - x^2 + x^6]/(4 - 3*x^4 - 4*x^6 + x^12), x] + Defer[Int][(x^6*Sqrt[-2 - x^2 + x^6])/(4 - 3*
x^4 - 4*x^6 + x^12), x]

Rubi steps

\begin {align*} \int \frac {\left (1+x^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx &=\int \left (\frac {\sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}}+\frac {x^6 \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}}\right ) \, dx\\ &=\int \frac {\sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx+\int \frac {x^6 \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^6)*Sqrt[-2 - x^2 + x^6])/(4 - 3*x^4 - 4*x^6 + x^12),x]

[Out]

Integrate[((1 + x^6)*Sqrt[-2 - x^2 + x^6])/(4 - 3*x^4 - 4*x^6 + x^12), x]

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IntegrateAlgebraic [A]  time = 2.36, size = 95, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \sqrt {\frac {1}{3} \left (1+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (-1+\sqrt {3}\right )} \tanh ^{-1}\left (\frac {\sqrt {-1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^6)*Sqrt[-2 - x^2 + x^6])/(4 - 3*x^4 - 4*x^6 + x^12),x]

[Out]

-1/4*(Sqrt[(1 + Sqrt[3])/3]*ArcTan[(Sqrt[1 + Sqrt[3]]*x)/Sqrt[-2 - x^2 + x^6]]) - (Sqrt[(-1 + Sqrt[3])/3]*ArcT
anh[(Sqrt[-1 + Sqrt[3]]*x)/Sqrt[-2 - x^2 + x^6]])/4

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fricas [B]  time = 0.84, size = 438, normalized size = 4.61 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \sqrt {\sqrt {3} + 1} \arctan \left (\frac {4 \, {\left (x^{7} + \sqrt {3} x^{3} - 2 \, x^{3} - 2 \, x\right )} \sqrt {x^{6} - x^{2} - 2} \sqrt {\sqrt {3} + 1} - {\left (2 \, x^{12} - 10 \, x^{8} - 8 \, x^{6} + 12 \, x^{4} + 20 \, x^{2} - \sqrt {3} {\left (x^{12} - 6 \, x^{8} - 4 \, x^{6} + 7 \, x^{4} + 12 \, x^{2} + 4\right )} + 8\right )} \sqrt {6 \, \sqrt {3} + 10} \sqrt {\sqrt {3} + 1}}{2 \, {\left (x^{12} - 4 \, x^{8} - 4 \, x^{6} + x^{4} + 8 \, x^{2} + 4\right )}}\right ) + \frac {1}{48} \, \sqrt {3} \sqrt {\sqrt {3} - 1} \log \left (\frac {4 \, {\left (2 \, x^{7} - 3 \, x^{3} - \sqrt {3} {\left (x^{7} - 2 \, x^{3} - 2 \, x\right )} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 2} + {\left (x^{12} - 8 \, x^{8} - 4 \, x^{6} + 9 \, x^{4} + 16 \, x^{2} - \sqrt {3} {\left (x^{12} - 4 \, x^{8} - 4 \, x^{6} + 5 \, x^{4} + 8 \, x^{2} + 4\right )} + 4\right )} \sqrt {\sqrt {3} - 1}}{x^{12} - 4 \, x^{6} - 3 \, x^{4} + 4}\right ) - \frac {1}{48} \, \sqrt {3} \sqrt {\sqrt {3} - 1} \log \left (\frac {4 \, {\left (2 \, x^{7} - 3 \, x^{3} - \sqrt {3} {\left (x^{7} - 2 \, x^{3} - 2 \, x\right )} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 2} - {\left (x^{12} - 8 \, x^{8} - 4 \, x^{6} + 9 \, x^{4} + 16 \, x^{2} - \sqrt {3} {\left (x^{12} - 4 \, x^{8} - 4 \, x^{6} + 5 \, x^{4} + 8 \, x^{2} + 4\right )} + 4\right )} \sqrt {\sqrt {3} - 1}}{x^{12} - 4 \, x^{6} - 3 \, x^{4} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6-x^2-2)^(1/2)/(x^12-4*x^6-3*x^4+4),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*sqrt(sqrt(3) + 1)*arctan(1/2*(4*(x^7 + sqrt(3)*x^3 - 2*x^3 - 2*x)*sqrt(x^6 - x^2 - 2)*sqrt(sqrt(
3) + 1) - (2*x^12 - 10*x^8 - 8*x^6 + 12*x^4 + 20*x^2 - sqrt(3)*(x^12 - 6*x^8 - 4*x^6 + 7*x^4 + 12*x^2 + 4) + 8
)*sqrt(6*sqrt(3) + 10)*sqrt(sqrt(3) + 1))/(x^12 - 4*x^8 - 4*x^6 + x^4 + 8*x^2 + 4)) + 1/48*sqrt(3)*sqrt(sqrt(3
) - 1)*log((4*(2*x^7 - 3*x^3 - sqrt(3)*(x^7 - 2*x^3 - 2*x) - 4*x)*sqrt(x^6 - x^2 - 2) + (x^12 - 8*x^8 - 4*x^6
+ 9*x^4 + 16*x^2 - sqrt(3)*(x^12 - 4*x^8 - 4*x^6 + 5*x^4 + 8*x^2 + 4) + 4)*sqrt(sqrt(3) - 1))/(x^12 - 4*x^6 -
3*x^4 + 4)) - 1/48*sqrt(3)*sqrt(sqrt(3) - 1)*log((4*(2*x^7 - 3*x^3 - sqrt(3)*(x^7 - 2*x^3 - 2*x) - 4*x)*sqrt(x
^6 - x^2 - 2) - (x^12 - 8*x^8 - 4*x^6 + 9*x^4 + 16*x^2 - sqrt(3)*(x^12 - 4*x^8 - 4*x^6 + 5*x^4 + 8*x^2 + 4) +
4)*sqrt(sqrt(3) - 1))/(x^12 - 4*x^6 - 3*x^4 + 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - x^{2} - 2} {\left (x^{6} + 1\right )}}{x^{12} - 4 \, x^{6} - 3 \, x^{4} + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6-x^2-2)^(1/2)/(x^12-4*x^6-3*x^4+4),x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - x^2 - 2)*(x^6 + 1)/(x^12 - 4*x^6 - 3*x^4 + 4), x)

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maple [C]  time = 4.14, size = 578, normalized size = 6.08

method	result	size
trager	\(\RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-384 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{3} x^{6}-73728 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{5} x^{2}+4 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right ) x^{6}+1152 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{3} x^{2}+192 \sqrt {x^{6}-x^{2}-2}\, \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} x +768 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{3}-4 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right ) x^{2}-\sqrt {x^{6}-x^{2}-2}\, x -8 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )}{-x^{6}+192 x^{2} \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+x^{2}+2}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) \ln \left (\frac {-48 \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} x^{6}+9216 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{4} \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{6}+336 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{2}+576 \sqrt {x^{6}-x^{2}-2}\, \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} x +96 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right )+3 \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{2}+9 \sqrt {x^{6}-x^{2}-2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right )}{x^{6}+192 x^{2} \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+x^{2}-2}\right )}{24}\)	\(578\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)*(x^6-x^2-2)^(1/2)/(x^12-4*x^6-3*x^4+4),x,method=_RETURNVERBOSE)

[Out]

RootOf(18432*_Z^4+192*_Z^2-1)*ln(-(-384*RootOf(18432*_Z^4+192*_Z^2-1)^3*x^6-73728*RootOf(18432*_Z^4+192*_Z^2-1
)^5*x^2+4*RootOf(18432*_Z^4+192*_Z^2-1)*x^6+1152*RootOf(18432*_Z^4+192*_Z^2-1)^3*x^2+192*(x^6-x^2-2)^(1/2)*Roo
tOf(18432*_Z^4+192*_Z^2-1)^2*x+768*RootOf(18432*_Z^4+192*_Z^2-1)^3-4*RootOf(18432*_Z^4+192*_Z^2-1)*x^2-(x^6-x^
2-2)^(1/2)*x-8*RootOf(18432*_Z^4+192*_Z^2-1))/(-x^6+192*x^2*RootOf(18432*_Z^4+192*_Z^2-1)^2+x^2+2))+1/24*RootO
f(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2-1)^2+6)*ln((-48*RootOf(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2-1)^2+6)*RootO
f(18432*_Z^4+192*_Z^2-1)^2*x^6+9216*RootOf(18432*_Z^4+192*_Z^2-1)^4*RootOf(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2
-1)^2+6)*x^2-RootOf(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2-1)^2+6)*x^6+336*RootOf(18432*_Z^4+192*_Z^2-1)^2*RootOf
(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2-1)^2+6)*x^2+576*(x^6-x^2-2)^(1/2)*RootOf(18432*_Z^4+192*_Z^2-1)^2*x+96*Ro
otOf(18432*_Z^4+192*_Z^2-1)^2*RootOf(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2-1)^2+6)+3*RootOf(_Z^2+576*RootOf(1843
2*_Z^4+192*_Z^2-1)^2+6)*x^2+9*(x^6-x^2-2)^(1/2)*x+2*RootOf(_Z^2+576*RootOf(18432*_Z^4+192*_Z^2-1)^2+6))/(x^6+1
92*x^2*RootOf(18432*_Z^4+192*_Z^2-1)^2+x^2-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - x^{2} - 2} {\left (x^{6} + 1\right )}}{x^{12} - 4 \, x^{6} - 3 \, x^{4} + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6-x^2-2)^(1/2)/(x^12-4*x^6-3*x^4+4),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - x^2 - 2)*(x^6 + 1)/(x^12 - 4*x^6 - 3*x^4 + 4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^6+1\right )\,\sqrt {x^6-x^2-2}}{-x^{12}+4\,x^6+3\,x^4-4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^6 + 1)*(x^6 - x^2 - 2)^(1/2))/(3*x^4 + 4*x^6 - x^12 - 4),x)

[Out]

int(-((x^6 + 1)*(x^6 - x^2 - 2)^(1/2))/(3*x^4 + 4*x^6 - x^12 - 4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right ) \sqrt {x^{6} - x^{2} - 2}}{x^{12} - 4 x^{6} - 3 x^{4} + 4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)*(x**6-x**2-2)**(1/2)/(x**12-4*x**6-3*x**4+4),x)

[Out]

Integral((x**2 + 1)*(x**4 - x**2 + 1)*sqrt(x**6 - x**2 - 2)/(x**12 - 4*x**6 - 3*x**4 + 4), x)

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