3.13.53 \(\int \frac {3-x^2}{(1-x^2) \sqrt [4]{1-6 x^2+x^4}} \, dx\)

Optimal. Leaf size=91 \[ \tan ^{-1}\left (\frac {x+i}{\sqrt [4]{x^4-6 x^2+1}}\right )-\tan ^{-1}\left (\frac {\sqrt [4]{x^4-6 x^2+1}}{x-i}\right )+\tanh ^{-1}\left (\frac {x+i}{\sqrt [4]{x^4-6 x^2+1}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{x^4-6 x^2+1}}{x-i}\right ) \]

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Rubi [F]  time = 0.05, 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 {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.17, size = 91, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\tan ^{-1}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )+\tanh ^{-1}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTan[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)] + ArcTanh[(I + x)/(1 - 6*x^
2 + x^4)^(1/4)] + ArcTanh[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 4.82, size = 233, normalized size = 2.56

method	result	size
trager	\(\frac {\ln \left (-\frac {\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x +\sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+x^{4}-\sqrt {x^{4}-6 x^{2}+1}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x -5 x^{2}}{\left (1+x \right ) \left (-1+x \right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x +\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (1+x \right ) \left (-1+x \right )}\right )}{2}\)	\(233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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