3.13.92 \(\int \frac {(1+x^4) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx\)

Optimal. Leaf size=93 \[ \frac {1}{10} \left (-5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {x^4-x^2-1}}\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {x^4-x^2-1}}\right ) \]

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Rubi [F]  time = 0.59, 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^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.48, size = 5470, normalized size = 58.82 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 + x^4)*Sqrt[-1 - x^2 + x^4])/(1 - x^2 - 3*x^4 + x^6 + x^8),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.40, size = 81, normalized size = 0.87 \begin {gather*} \frac {1}{10} \left (5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )+\frac {1}{10} \left (-5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^4)*Sqrt[-1 - x^2 + x^4])/(1 - x^2 - 3*x^4 + x^6 + x^8),x]

[Out]

((5 - Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^4])])/10 + ((-5 - Sqrt[5])*ArcTan[((1 + Sqrt[5])
*x)/(2*Sqrt[-1 - x^2 + x^4])])/10

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fricas [B]  time = 0.60, size = 313, normalized size = 3.37 \begin {gather*} \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 3} \arctan \left (-\frac {2 \, \sqrt {10} \sqrt {x^{4} - x^{2} - 1} {\left (5 \, x^{3} + \sqrt {5} {\left (2 \, x^{5} - 5 \, x^{3} - 2 \, x\right )}\right )} \sqrt {\sqrt {5} + 3} + \sqrt {10} {\left (15 \, x^{8} - 65 \, x^{6} + 5 \, x^{4} + 65 \, x^{2} - \sqrt {5} {\left (7 \, x^{8} - 29 \, x^{6} + x^{4} + 29 \, x^{2} + 7\right )} + 15\right )} \sqrt {4 \, \sqrt {5} + 9} \sqrt {\sqrt {5} + 3}}{20 \, {\left (x^{8} - 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + 1\right )}}\right ) + \frac {1}{10} \, \sqrt {10} \sqrt {-\sqrt {5} + 3} \arctan \left (-\frac {40 \, \sqrt {10} \sqrt {x^{4} - x^{2} - 1} {\left (5 \, x^{3} - \sqrt {5} {\left (2 \, x^{5} - 5 \, x^{3} - 2 \, x\right )}\right )} \sqrt {-\sqrt {5} + 3} + \sqrt {10} {\left (15 \, x^{8} - 65 \, x^{6} + 5 \, x^{4} + 65 \, x^{2} + \sqrt {5} {\left (7 \, x^{8} - 29 \, x^{6} + x^{4} + 29 \, x^{2} + 7\right )} + 15\right )} \sqrt {-\sqrt {5} + 3} \sqrt {-1600 \, \sqrt {5} + 3600}}{400 \, {\left (x^{8} - 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + 1\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x, algorithm="fricas")

[Out]

1/10*sqrt(10)*sqrt(sqrt(5) + 3)*arctan(-1/20*(2*sqrt(10)*sqrt(x^4 - x^2 - 1)*(5*x^3 + sqrt(5)*(2*x^5 - 5*x^3 -
2*x))*sqrt(sqrt(5) + 3) + sqrt(10)*(15*x^8 - 65*x^6 + 5*x^4 + 65*x^2 - sqrt(5)*(7*x^8 - 29*x^6 + x^4 + 29*x^2
+ 7) + 15)*sqrt(4*sqrt(5) + 9)*sqrt(sqrt(5) + 3))/(x^8 - 5*x^6 + 3*x^4 + 5*x^2 + 1)) + 1/10*sqrt(10)*sqrt(-sq
rt(5) + 3)*arctan(-1/400*(40*sqrt(10)*sqrt(x^4 - x^2 - 1)*(5*x^3 - sqrt(5)*(2*x^5 - 5*x^3 - 2*x))*sqrt(-sqrt(5
) + 3) + sqrt(10)*(15*x^8 - 65*x^6 + 5*x^4 + 65*x^2 + sqrt(5)*(7*x^8 - 29*x^6 + x^4 + 29*x^2 + 7) + 15)*sqrt(-
sqrt(5) + 3)*sqrt(-1600*sqrt(5) + 3600))/(x^8 - 5*x^6 + 3*x^4 + 5*x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^4 - x^2 - 1)*(x^4 + 1)/(x^8 + x^6 - 3*x^4 - x^2 + 1), x)

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maple [B]  time = 1.85, size = 209, normalized size = 2.25

method	result	size
default	\(\frac {\left (\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{2 \sqrt {10}+2 \sqrt {2}}+\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}-\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{2 \sqrt {10}-2 \sqrt {2}}\right ) \sqrt {2}}{2}\)	\(209\)
elliptic	\(\frac {\left (\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{2 \sqrt {10}+2 \sqrt {2}}+\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}-\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{2 \sqrt {10}-2 \sqrt {2}}\right ) \sqrt {2}}{2}\)	\(209\)
trager	\(-20 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {-1400 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}+70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}-460 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}+9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+30 \sqrt {x^{4}-x^{2}-1}\, \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} x -70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}-36 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+4 x \sqrt {x^{4}-x^{2}-1}-9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )-\RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {1200 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}+60 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}-20 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}+2 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+60 \sqrt {x^{4}-x^{2}-1}\, \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} x -60 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}-2 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+x \sqrt {x^{4}-x^{2}-1}-2 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}-x^{4}+x^{2}+1}\right )-3 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-1400 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}+70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}-460 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}+9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+30 \sqrt {x^{4}-x^{2}-1}\, \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} x -70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}-36 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+4 x \sqrt {x^{4}-x^{2}-1}-9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )\)	\(673\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(4/(2*10^(1/2)+2*2^(1/2))*arctan(4*(x^4-x^2-1)^(1/2)*2^(1/2)/x/(2*10^(1/2)+2*2^(1/2)))+12/5*5^(1/2)/(2*10^
(1/2)+2*2^(1/2))*arctan(4*(x^4-x^2-1)^(1/2)*2^(1/2)/x/(2*10^(1/2)+2*2^(1/2)))-12/5*5^(1/2)/(2*10^(1/2)-2*2^(1/
2))*arctan(4*(x^4-x^2-1)^(1/2)*2^(1/2)/x/(2*10^(1/2)-2*2^(1/2)))+4/(2*10^(1/2)-2*2^(1/2))*arctan(4*(x^4-x^2-1)
^(1/2)*2^(1/2)/x/(2*10^(1/2)-2*2^(1/2))))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - x^2 - 1)*(x^4 + 1)/(x^8 + x^6 - 3*x^4 - 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^4+1\right )\,\sqrt {x^4-x^2-1}}{x^8+x^6-3\,x^4-x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 1)*(x^4 - x^2 - 1)^(1/2))/(x^6 - 3*x^4 - x^2 + x^8 + 1),x)

[Out]

int(((x^4 + 1)*(x^4 - x^2 - 1)^(1/2))/(x^6 - 3*x^4 - x^2 + x^8 + 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)*(x**4-x**2-1)**(1/2)/(x**8+x**6-3*x**4-x**2+1),x)

[Out]

Timed out

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