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

Optimal. Leaf size=79 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{x^6+x^2-1}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{x^6-x^2-1}\right )}{2 \sqrt {2}} \]

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Rubi [F]  time = 0.51, 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^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.12, size = 91, normalized size = 1.15 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1+x^2+x^6}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^6}{\sqrt {2}}}{x \sqrt {1-x^6}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.66, size = 512, normalized size = 6.48 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} + x^{3} - x\right )} \sqrt {-x^{6} + 1} - {\left (4 \, \sqrt {-x^{6} + 1} x^{3} - \sqrt {2} {\left (x^{12} + 2 \, x^{8} - 2 \, x^{6} - x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1}{x^{12} - 2 \, x^{6} + x^{4} + 1}} + 1}{x^{12} + 4 \, x^{8} - 2 \, x^{6} + x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} + x^{3} - x\right )} \sqrt {-x^{6} + 1} - {\left (4 \, \sqrt {-x^{6} + 1} x^{3} + \sqrt {2} {\left (x^{12} + 2 \, x^{8} - 2 \, x^{6} - x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1}{x^{12} - 2 \, x^{6} + x^{4} + 1}} + 1}{x^{12} + 4 \, x^{8} - 2 \, x^{6} + x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1\right )}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1\right )}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*arctan(-(x^12 - 2*x^6 + x^4 + 2*sqrt(2)*(x^7 + x^3 - x)*sqrt(-x^6 + 1) - (4*sqrt(-x^6 + 1)*x^3 - s
qrt(2)*(x^12 + 2*x^8 - 2*x^6 - x^4 - 2*x^2 + 1))*sqrt((x^12 - 4*x^8 - 2*x^6 + x^4 + 2*sqrt(2)*(x^7 - x^3 - x)*
sqrt(-x^6 + 1) + 4*x^2 + 1)/(x^12 - 2*x^6 + x^4 + 1)) + 1)/(x^12 + 4*x^8 - 2*x^6 + x^4 - 4*x^2 + 1)) - 1/4*sqr
t(2)*arctan(-(x^12 - 2*x^6 + x^4 - 2*sqrt(2)*(x^7 + x^3 - x)*sqrt(-x^6 + 1) - (4*sqrt(-x^6 + 1)*x^3 + sqrt(2)*
(x^12 + 2*x^8 - 2*x^6 - x^4 - 2*x^2 + 1))*sqrt((x^12 - 4*x^8 - 2*x^6 + x^4 - 2*sqrt(2)*(x^7 - x^3 - x)*sqrt(-x
^6 + 1) + 4*x^2 + 1)/(x^12 - 2*x^6 + x^4 + 1)) + 1)/(x^12 + 4*x^8 - 2*x^6 + x^4 - 4*x^2 + 1)) - 1/16*sqrt(2)*l
og(4*(x^12 - 4*x^8 - 2*x^6 + x^4 + 2*sqrt(2)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 4*x^2 + 1)/(x^12 - 2*x^6 + x^4 +
1)) + 1/16*sqrt(2)*log(4*(x^12 - 4*x^8 - 2*x^6 + x^4 - 2*sqrt(2)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 4*x^2 + 1)/
(x^12 - 2*x^6 + x^4 + 1))

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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} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.50, size = 157, normalized size = 1.99

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{6}+1}\, x}{x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {-x^{6}+1}\, x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{-x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{4}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^6-RootOf(_Z^4+1)^5*x^2-RootOf(_Z^4+1)^3+2*(-x^6+1)^(1/2)*x)/(x^6
+RootOf(_Z^4+1)^2*x^2-1))-1/4*RootOf(_Z^4+1)*ln(-(RootOf(_Z^4+1)*x^6+RootOf(_Z^4+1)^3*x^2+2*(-x^6+1)^(1/2)*x-R
ootOf(_Z^4+1))/(-x^6+RootOf(_Z^4+1)^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} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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