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

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

[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])

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Rubi [A]  time = 0.0081616, antiderivative size = 53, normalized size of antiderivative = 1., 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 = {398} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt{2}} \]

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 398

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b^2/a), 4]}, Simp[(b*Ar
cTan[(q*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(2*Sqrt[2]*a*d*q), x] + Simp[(b*ArcTanh[(q*x)/(Sqrt[2]*(a + b*x^2)^(1
/4))])/(2*Sqrt[2]*a*d*q), 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*}

Mathematica [C]  time = 0.137361, size = 115, normalized size = 2.17 \[ -\frac{6 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};x^2,\frac{x^2}{2}\right )}{\left (x^2-2\right ) \sqrt [4]{x^2-1} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};x^2,\frac{x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};x^2,\frac{x^2}{2}\right )\right )+6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};x^2,\frac{x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-6*x*AppellF1[1/2, 1/4, 1, 3/2, x^2, x^2/2])/((-2 + x^2)*(-1 + x^2)^(1/4)*(6*AppellF1[1/2, 1/4, 1, 3/2, x^2,
x^2/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, x^2, x^2/2] + AppellF1[3/2, 5/4, 1, 5/2, x^2, x^2/2])))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{x}^{2}+2}{\frac{1}{\sqrt [4]{{x}^{2}-1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (x^{2} - 1\right )}^{\frac{1}{4}}{\left (x^{2} - 2\right )}}\,{d x} \end{align*}

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]  time = 23.3346, size = 254, normalized size = 4.79 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (x^{2} - 1\right )}^{\frac{1}{4}}{\left (x^{2} - 2\right )}}\,{d x} \end{align*}

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)