3.14.24 \(\int \frac {\sqrt {-1-2 x^2-2 x^3-x^8} (-1+x^3+3 x^8)}{(1+2 x^3+x^8) (1+x^2+2 x^3+x^8)} \, dx\)

Optimal. Leaf size=95 \[ \tan ^{-1}\left (\frac {x \sqrt {-x^8-2 x^3-2 x^2-1}}{x^8+2 x^3+2 x^2+1}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-x^8-2 x^3-2 x^2-1}}{x^8+2 x^3+2 x^2+1}\right ) \]

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Rubi [F]  time = 4.34, 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 {-1-2 x^2-2 x^3-x^8} \left (-1+x^3+3 x^8\right )}{\left (1+2 x^3+x^8\right ) \left (1+x^2+2 x^3+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]*(-1 + x^3 + 3*x^8))/((1 + 2*x^3 + x^8)*(1 + x^2 + 2*x^3 + x^8)),x]

[Out]

-1/2*Defer[Int][Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]/(1 + x), x] + Defer[Int][Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]/(1 - x
+ x^2 + x^3 - x^4 + x^5 - x^6 + x^7), x]/2 + 2*Defer[Int][(x*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 - x + x^2 + x^
3 - x^4 + x^5 - x^6 + x^7), x] - (3*Defer[Int][(x^2*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 - x + x^2 + x^3 - x^4 +
x^5 - x^6 + x^7), x])/2 + 2*Defer[Int][(x^3*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 - x + x^2 + x^3 - x^4 + x^5 -
x^6 + x^7), x] - (5*Defer[Int][(x^4*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 - x + x^2 + x^3 - x^4 + x^5 - x^6 + x^7
), x])/2 + 3*Defer[Int][(x^5*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 - x + x^2 + x^3 - x^4 + x^5 - x^6 + x^7), x] +
Defer[Int][(x^6*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 - x + x^2 + x^3 - x^4 + x^5 - x^6 + x^7), x]/2 + Defer[Int
][Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]/(-1 - x^2 - 2*x^3 - x^8), x] - 3*Defer[Int][(x*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]
)/(1 + x^2 + 2*x^3 + x^8), x] - 4*Defer[Int][(x^6*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 + x^2 + 2*x^3 + x^8), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8} \left (-1+x^3+3 x^8\right )}{\left (1+2 x^3+x^8\right ) \left (1+x^2+2 x^3+x^8\right )} \, dx &=\int \left (-\frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{2 (1+x)}+\frac {\left (1+4 x-3 x^2+4 x^3-5 x^4+6 x^5+x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{2 \left (1-x+x^2+x^3-x^4+x^5-x^6+x^7\right )}+\frac {\left (-1-3 x-4 x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1+x} \, dx\right )+\frac {1}{2} \int \frac {\left (1+4 x-3 x^2+4 x^3-5 x^4+6 x^5+x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+\int \frac {\left (-1-3 x-4 x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1+x} \, dx\right )+\frac {1}{2} \int \left (\frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {4 x \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}-\frac {3 x^2 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {4 x^3 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}-\frac {5 x^4 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {6 x^5 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}\right ) \, dx+\int \left (\frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{-1-x^2-2 x^3-x^8}-\frac {3 x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8}-\frac {4 x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1+x} \, dx\right )+\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+\frac {1}{2} \int \frac {x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx-\frac {3}{2} \int \frac {x^2 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+2 \int \frac {x \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+2 \int \frac {x^3 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx-\frac {5}{2} \int \frac {x^4 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+3 \int \frac {x^5 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx-3 \int \frac {x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8} \, dx-4 \int \frac {x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8} \, dx+\int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{-1-x^2-2 x^3-x^8} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8} \left (-1+x^3+3 x^8\right )}{\left (1+2 x^3+x^8\right ) \left (1+x^2+2 x^3+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]*(-1 + x^3 + 3*x^8))/((1 + 2*x^3 + x^8)*(1 + x^2 + 2*x^3 + x^8)),x]

[Out]

Integrate[(Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]*(-1 + x^3 + 3*x^8))/((1 + 2*x^3 + x^8)*(1 + x^2 + 2*x^3 + x^8)), x]

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IntegrateAlgebraic [A]  time = 0.58, size = 95, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]*(-1 + x^3 + 3*x^8))/((1 + 2*x^3 + x^8)*(1 + x^2 + 2*x^3 + x
^8)),x]

[Out]

ArcTan[(x*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 + 2*x^2 + 2*x^3 + x^8)] - Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[-1 - 2*x
^2 - 2*x^3 - x^8])/(1 + 2*x^2 + 2*x^3 + x^8)]

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fricas [C]  time = 1.47, size = 241, normalized size = 2.54 \begin {gather*} -\frac {1}{4} \, \sqrt {-2} \log \left (-\frac {2 \, {\left (\sqrt {-2} {\left (x^{8} + 2 \, x^{3} + 4 \, x^{2} + 1\right )} + 4 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x\right )}}{x^{8} + 2 \, x^{3} + 1}\right ) + \frac {1}{4} \, \sqrt {-2} \log \left (\frac {2 \, {\left (\sqrt {-2} {\left (x^{8} + 2 \, x^{3} + 4 \, x^{2} + 1\right )} - 4 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x\right )}}{x^{8} + 2 \, x^{3} + 1}\right ) - \frac {1}{4} i \, \log \left (\frac {i \, x^{8} + 2 i \, x^{3} + 3 i \, x^{2} - 2 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x + i}{x^{8} + 2 \, x^{3} + x^{2} + 1}\right ) + \frac {1}{4} i \, \log \left (\frac {-i \, x^{8} - 2 i \, x^{3} - 3 i \, x^{2} - 2 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x - i}{x^{8} + 2 \, x^{3} + x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^8-2*x^3-2*x^2-1)^(1/2)*(3*x^8+x^3-1)/(x^8+2*x^3+1)/(x^8+2*x^3+x^2+1),x, algorithm="fricas")

[Out]

-1/4*sqrt(-2)*log(-2*(sqrt(-2)*(x^8 + 2*x^3 + 4*x^2 + 1) + 4*sqrt(-x^8 - 2*x^3 - 2*x^2 - 1)*x)/(x^8 + 2*x^3 +
1)) + 1/4*sqrt(-2)*log(2*(sqrt(-2)*(x^8 + 2*x^3 + 4*x^2 + 1) - 4*sqrt(-x^8 - 2*x^3 - 2*x^2 - 1)*x)/(x^8 + 2*x^
3 + 1)) - 1/4*I*log((I*x^8 + 2*I*x^3 + 3*I*x^2 - 2*sqrt(-x^8 - 2*x^3 - 2*x^2 - 1)*x + I)/(x^8 + 2*x^3 + x^2 +
1)) + 1/4*I*log((-I*x^8 - 2*I*x^3 - 3*I*x^2 - 2*sqrt(-x^8 - 2*x^3 - 2*x^2 - 1)*x - I)/(x^8 + 2*x^3 + x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{8} + x^{3} - 1\right )} \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1}}{{\left (x^{8} + 2 \, x^{3} + x^{2} + 1\right )} {\left (x^{8} + 2 \, x^{3} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^8-2*x^3-2*x^2-1)^(1/2)*(3*x^8+x^3-1)/(x^8+2*x^3+1)/(x^8+2*x^3+x^2+1),x, algorithm="giac")

[Out]

integrate((3*x^8 + x^3 - 1)*sqrt(-x^8 - 2*x^3 - 2*x^2 - 1)/((x^8 + 2*x^3 + x^2 + 1)*(x^8 + 2*x^3 + 1)), x)

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maple [C]  time = 1.26, size = 195, normalized size = 2.05

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{8}-2 x^{3}-2 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{8}+2 x^{3}+x^{2}+1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-x^{8}-2 x^{3}-2 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (1+x \right ) \left (x^{7}-x^{6}+x^{5}-x^{4}+x^{3}+x^{2}-x +1\right )}\right )}{2}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^8-2*x^3-2*x^2-1)^(1/2)*(3*x^8+x^3-1)/(x^8+2*x^3+1)/(x^8+2*x^3+x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^8+2*RootOf(_Z^2+1)*x^3+3*RootOf(_Z^2+1)*x^2+2*(-x^8-2*x^3-2*x^2-1)^(1
/2)*x+RootOf(_Z^2+1))/(x^8+2*x^3+x^2+1))-1/2*RootOf(_Z^2+2)*ln(-(RootOf(_Z^2+2)*x^8+2*RootOf(_Z^2+2)*x^3+4*Roo
tOf(_Z^2+2)*x^2+4*(-x^8-2*x^3-2*x^2-1)^(1/2)*x+RootOf(_Z^2+2))/(1+x)/(x^7-x^6+x^5-x^4+x^3+x^2-x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{8} + x^{3} - 1\right )} \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1}}{{\left (x^{8} + 2 \, x^{3} + x^{2} + 1\right )} {\left (x^{8} + 2 \, x^{3} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^8-2*x^3-2*x^2-1)^(1/2)*(3*x^8+x^3-1)/(x^8+2*x^3+1)/(x^8+2*x^3+x^2+1),x, algorithm="maxima")

[Out]

integrate((3*x^8 + x^3 - 1)*sqrt(-x^8 - 2*x^3 - 2*x^2 - 1)/((x^8 + 2*x^3 + x^2 + 1)*(x^8 + 2*x^3 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (3\,x^8+x^3-1\right )\,\sqrt {-x^8-2\,x^3-2\,x^2-1}}{\left (x^8+2\,x^3+1\right )\,\left (x^8+2\,x^3+x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + 3*x^8 - 1)*(- 2*x^2 - 2*x^3 - x^8 - 1)^(1/2))/((2*x^3 + x^8 + 1)*(x^2 + 2*x^3 + x^8 + 1)),x)

[Out]

int(((x^3 + 3*x^8 - 1)*(- 2*x^2 - 2*x^3 - x^8 - 1)^(1/2))/((2*x^3 + x^8 + 1)*(x^2 + 2*x^3 + x^8 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x^{8} + x^{3} - 1\right ) \sqrt {- x^{8} - 2 x^{3} - 2 x^{2} - 1}}{\left (x + 1\right ) \left (x^{8} + 2 x^{3} + x^{2} + 1\right ) \left (x^{7} - x^{6} + x^{5} - 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**8-2*x**3-2*x**2-1)**(1/2)*(3*x**8+x**3-1)/(x**8+2*x**3+1)/(x**8+2*x**3+x**2+1),x)

[Out]

Integral((3*x**8 + x**3 - 1)*sqrt(-x**8 - 2*x**3 - 2*x**2 - 1)/((x + 1)*(x**8 + 2*x**3 + x**2 + 1)*(x**7 - x**
6 + x**5 - x**4 + x**3 + x**2 - x + 1)), x)

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