3.16.22 \(\int \frac {\sqrt {-1-x^2+x^6} (1+2 x^6)}{8-x^4-16 x^6+8 x^{12}} \, dx\)

Optimal. Leaf size=105 \[ \frac {1}{8} \sqrt {\frac {1}{2} \left (4-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4-\sqrt {2}} x}{2 \sqrt {x^6-x^2-1}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (4+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {x^6-x^2-1}}\right ) \]

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Rubi [F]  time = 0.61, 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 {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]

[Out]

Defer[Int][Sqrt[-1 - x^2 + x^6]/(8 - x^4 - 16*x^6 + 8*x^12), x] + 2*Defer[Int][(x^6*Sqrt[-1 - x^2 + x^6])/(8 -
x^4 - 16*x^6 + 8*x^12), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx &=\int \left (\frac {\sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}}+\frac {2 x^6 \sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}}\right ) \, dx\\ &=2 \int \frac {x^6 \sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]

[Out]

Integrate[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12), x]

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IntegrateAlgebraic [A]  time = 2.18, size = 105, normalized size = 1.00 \begin {gather*} \frac {1}{8} \sqrt {\frac {1}{2} \left (4-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4-\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (4+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]

[Out]

(Sqrt[(4 - Sqrt[2])/2]*ArcTan[(Sqrt[4 - Sqrt[2]]*x)/(2*Sqrt[-1 - x^2 + x^6])])/8 - (Sqrt[(4 + Sqrt[2])/2]*ArcT
an[(Sqrt[4 + Sqrt[2]]*x)/(2*Sqrt[-1 - x^2 + x^6])])/8

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fricas [B]  time = 0.71, size = 336, normalized size = 3.20 \begin {gather*} -\frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 4} \arctan \left (\frac {196 \, {\left (4 \, x^{7} + \sqrt {2} x^{3} - 8 \, x^{3} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 1} \sqrt {\sqrt {2} + 4} - {\left (72 \, x^{12} - 176 \, x^{8} - 144 \, x^{6} + 41 \, x^{4} + 176 \, x^{2} - 4 \, \sqrt {2} {\left (8 \, x^{12} - 25 \, x^{8} - 16 \, x^{6} + 10 \, x^{4} + 25 \, x^{2} + 8\right )} + 72\right )} \sqrt {50 \, \sqrt {2} + 88} \sqrt {\sqrt {2} + 4}}{98 \, {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 31 \, x^{4} + 32 \, x^{2} + 8\right )}}\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} + 4} \arctan \left (-\frac {196 \, {\left (4 \, x^{7} - \sqrt {2} x^{3} - 8 \, x^{3} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 1} \sqrt {-\sqrt {2} + 4} - {\left (72 \, x^{12} - 176 \, x^{8} - 144 \, x^{6} + 41 \, x^{4} + 176 \, x^{2} + 4 \, \sqrt {2} {\left (8 \, x^{12} - 25 \, x^{8} - 16 \, x^{6} + 10 \, x^{4} + 25 \, x^{2} + 8\right )} + 72\right )} \sqrt {-\sqrt {2} + 4} \sqrt {-50 \, \sqrt {2} + 88}}{98 \, {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 31 \, x^{4} + 32 \, x^{2} + 8\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x, algorithm="fricas")

[Out]

-1/16*sqrt(2)*sqrt(sqrt(2) + 4)*arctan(1/98*(196*(4*x^7 + sqrt(2)*x^3 - 8*x^3 - 4*x)*sqrt(x^6 - x^2 - 1)*sqrt(
sqrt(2) + 4) - (72*x^12 - 176*x^8 - 144*x^6 + 41*x^4 + 176*x^2 - 4*sqrt(2)*(8*x^12 - 25*x^8 - 16*x^6 + 10*x^4
+ 25*x^2 + 8) + 72)*sqrt(50*sqrt(2) + 88)*sqrt(sqrt(2) + 4))/(8*x^12 - 32*x^8 - 16*x^6 + 31*x^4 + 32*x^2 + 8))
- 1/16*sqrt(2)*sqrt(-sqrt(2) + 4)*arctan(-1/98*(196*(4*x^7 - sqrt(2)*x^3 - 8*x^3 - 4*x)*sqrt(x^6 - x^2 - 1)*s
qrt(-sqrt(2) + 4) - (72*x^12 - 176*x^8 - 144*x^6 + 41*x^4 + 176*x^2 + 4*sqrt(2)*(8*x^12 - 25*x^8 - 16*x^6 + 10
*x^4 + 25*x^2 + 8) + 72)*sqrt(-sqrt(2) + 4)*sqrt(-50*sqrt(2) + 88))/(8*x^12 - 32*x^8 - 16*x^6 + 31*x^4 + 32*x^
2 + 8))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - x^{2} - 1}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x, algorithm="giac")

[Out]

integrate((2*x^6 + 1)*sqrt(x^6 - x^2 - 1)/(8*x^12 - 16*x^6 - x^4 + 8), x)

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maple [C]  time = 4.32, size = 577, normalized size = 5.50

method	result	size
trager	\(\RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) \ln \left (\frac {16384 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3} x^{6}+2097152 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{5} x^{2}+112 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) x^{6}-2048 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3} x^{2}-2048 \sqrt {x^{6}-x^{2}-1}\, \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x -16384 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3}-112 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) x^{2}-7 \sqrt {x^{6}-x^{2}-1}\, x -112 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )}{-x^{6}+128 x^{2} \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+x^{2}+1}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \ln \left (\frac {-2048 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x^{6}+262144 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{4} x^{2}-18 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{6}+8448 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{2}+2048 \sqrt {x^{6}-x^{2}-1}\, \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x +2048 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right )+54 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{2}+25 \sqrt {x^{6}-x^{2}-1}\, x +18 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right )}{x^{6}+128 x^{2} \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+x^{2}-1}\right )}{8}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x,method=_RETURNVERBOSE)

[Out]

RootOf(131072*_Z^4+2048*_Z^2+7)*ln((16384*RootOf(131072*_Z^4+2048*_Z^2+7)^3*x^6+2097152*RootOf(131072*_Z^4+204
8*_Z^2+7)^5*x^2+112*RootOf(131072*_Z^4+2048*_Z^2+7)*x^6-2048*RootOf(131072*_Z^4+2048*_Z^2+7)^3*x^2-2048*(x^6-x
^2-1)^(1/2)*RootOf(131072*_Z^4+2048*_Z^2+7)^2*x-16384*RootOf(131072*_Z^4+2048*_Z^2+7)^3-112*RootOf(131072*_Z^4
+2048*_Z^2+7)*x^2-7*(x^6-x^2-1)^(1/2)*x-112*RootOf(131072*_Z^4+2048*_Z^2+7))/(-x^6+128*x^2*RootOf(131072*_Z^4+
2048*_Z^2+7)^2+x^2+1))+1/8*RootOf(_Z^2+64*RootOf(131072*_Z^4+2048*_Z^2+7)^2+1)*ln((-2048*RootOf(_Z^2+64*RootOf
(131072*_Z^4+2048*_Z^2+7)^2+1)*RootOf(131072*_Z^4+2048*_Z^2+7)^2*x^6+262144*RootOf(_Z^2+64*RootOf(131072*_Z^4+
2048*_Z^2+7)^2+1)*RootOf(131072*_Z^4+2048*_Z^2+7)^4*x^2-18*RootOf(_Z^2+64*RootOf(131072*_Z^4+2048*_Z^2+7)^2+1)
*x^6+8448*RootOf(131072*_Z^4+2048*_Z^2+7)^2*RootOf(_Z^2+64*RootOf(131072*_Z^4+2048*_Z^2+7)^2+1)*x^2+2048*(x^6-
x^2-1)^(1/2)*RootOf(131072*_Z^4+2048*_Z^2+7)^2*x+2048*RootOf(131072*_Z^4+2048*_Z^2+7)^2*RootOf(_Z^2+64*RootOf(
131072*_Z^4+2048*_Z^2+7)^2+1)+54*RootOf(_Z^2+64*RootOf(131072*_Z^4+2048*_Z^2+7)^2+1)*x^2+25*(x^6-x^2-1)^(1/2)*
x+18*RootOf(_Z^2+64*RootOf(131072*_Z^4+2048*_Z^2+7)^2+1))/(x^6+128*x^2*RootOf(131072*_Z^4+2048*_Z^2+7)^2+x^2-1
))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - x^{2} - 1}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x, algorithm="maxima")

[Out]

integrate((2*x^6 + 1)*sqrt(x^6 - x^2 - 1)/(8*x^12 - 16*x^6 - x^4 + 8), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (2\,x^6+1\right )\,\sqrt {x^6-x^2-1}}{-8\,x^{12}+16\,x^6+x^4-8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^6 + 1)*(x^6 - x^2 - 1)^(1/2))/(x^4 + 16*x^6 - 8*x^12 - 8),x)

[Out]

int(-((2*x^6 + 1)*(x^6 - x^2 - 1)^(1/2))/(x^4 + 16*x^6 - 8*x^12 - 8), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{6} + 1\right ) \sqrt {x^{6} - x^{2} - 1}}{8 x^{12} - 16 x^{6} - x^{4} + 8}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-x**2-1)**(1/2)*(2*x**6+1)/(8*x**12-16*x**6-x**4+8),x)

[Out]

Integral((2*x**6 + 1)*sqrt(x**6 - x**2 - 1)/(8*x**12 - 16*x**6 - x**4 + 8), x)

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