3.11.79 \(\int \frac {(1-x^2)^2}{(1+x^2) (1+6 x^2+x^4)^{3/4}} \, dx\)

Optimal. Leaf size=81 \[ -\tan ^{-1}\left (\frac {x-1}{\sqrt [4]{x^4+6 x^2+1}}\right )-\tan ^{-1}\left (\frac {x+1}{\sqrt [4]{x^4+6 x^2+1}}\right )+\tanh ^{-1}\left (\frac {x-1}{\sqrt [4]{x^4+6 x^2+1}}\right )+\tanh ^{-1}\left (\frac {x+1}{\sqrt [4]{x^4+6 x^2+1}}\right ) \]

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Rubi [F]  time = 0.08, 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^2\right )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x^2\right )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.29, size = 81, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )-\tan ^{-1}\left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.34, size = 380, normalized size = 4.69

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )^{3} \sqrt {x^{4}+6 x^{2}+1}\, x^{4}+\RootOf \left (\textit {\_Z}^{2}+1\right )^{3} x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{5}-\RootOf \left (\textit {\_Z}^{2}+1\right )^{3} \sqrt {x^{4}+6 x^{2}+1}\, x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+1\right )^{3} x^{4}+\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+6 x^{2}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x -\left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+6 x^{2}+1}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-3 \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (\RootOf \left (\textit {\_Z}^{2}+1\right ) x -1\right )^{2} \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) x +1\right )^{2}}\right )}{2}+\frac {\ln \left (\frac {\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x +x^{2} \sqrt {x^{4}+6 x^{2}+1}+\left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+x^{4}+\sqrt {x^{4}+6 x^{2}+1}+3 \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{x^{2}+1}\right )}{2}\)	\(380\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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