3.14.17 \(\int \frac {(1+x^2)^2}{(1-x^2) (1-6 x^2+x^4)^{3/4}} \, dx\) [1317]

Optimal. Leaf size=95 \[ \text {ArcTan}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {ArcTan}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )-\tanh ^{-1}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \]

[Out]

arctan((I+x)/(x^4-6*x^2+1)^(1/4))-arctan((x^4-6*x^2+1)^(1/4)/(-I+x))-arctanh((I+x)/(x^4-6*x^2+1)^(1/4))-arctan
h((x^4-6*x^2+1)^(1/4)/(-I+x))

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Rubi [F]
time = 0.04, 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 [A]
time = 6.13, size = 95, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {ArcTan}\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]

Integrate[(1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)),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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.06, size = 232, normalized size = 2.44

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (-1+x \right ) \left (1+x \right )}\right )}{2}+\frac {\ln \left (\frac {\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-x^{4}+\sqrt {x^{4}-6 x^{2}+1}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{\left (1+x \right ) \left (-1+x \right )}\right )}{2}\)	\(232\)

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)*(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))+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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {2 x^{2}}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx - \int \frac {x^{4}}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx - \int \frac {1}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \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(2*x**2/(x**2*(x**4 - 6*x**2 + 1)**(3/4) - (x**4 - 6*x**2 + 1)**(3/4)), x) - Integral(x**4/(x**2*(x**
4 - 6*x**2 + 1)**(3/4) - (x**4 - 6*x**2 + 1)**(3/4)), x) - Integral(1/(x**2*(x**4 - 6*x**2 + 1)**(3/4) - (x**4
- 6*x**2 + 1)**(3/4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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)*(x^4 - 6*x^2 + 1)^(3/4)),x)

[Out]

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

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