3.4.20 \(\int \frac {1}{(2-x^2) \sqrt [4]{-1+x^2}} \, dx\) [320]

Optimal. Leaf size=53 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {407} \begin {gather*} \frac {\text {ArcTan}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((2 - x^2)*(-1 + x^2)^(1/4)),x]

[Out]

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

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 48, normalized size = 0.91 \begin {gather*} \frac {-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{x}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.11, size = 121, normalized size = 2.28

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {3}{4}}-\sqrt {x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {3}{4}}+\sqrt {x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{4}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1/(-x^2+2)/(x^2-1)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (39) = 78\).
time = 3.42, size = 91, normalized size = 1.72 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{2} - 1} x^{2} + 4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} x + 4 \, x^{2} - 4}{x^{4} - 4 \, x^{2} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*arctan(sqrt(2)*(x^2 - 1)^(1/4)/x) + 1/8*sqrt(2)*log(-(x^4 + 2*sqrt(2)*(x^2 - 1)^(1/4)*x^3 + 4*sqr
t(x^2 - 1)*x^2 + 4*sqrt(2)*(x^2 - 1)^(3/4)*x + 4*x^2 - 4)/(x^4 - 4*x^2 + 4))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {1}{{\left (x^2-1\right )}^{1/4}\,\left (x^2-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((x^2 - 1)^(1/4)*(x^2 - 2)),x)

[Out]

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

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