3.311 \(\int \frac{1}{\left (-2+b x^2\right ) \sqrt [4]{-1+b x^2}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.046275, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 57.1789, size = 170, normalized size = 2.21 \[ \frac{\sqrt{2} x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{b x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{b x^{2} - 1} + 1} \sqrt{i \sqrt{b x^{2} - 1} + 1}} - \frac{\sqrt{2} \sqrt{b x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt [4]{b x^{2} - 1}}{\sqrt{b x^{2}}} \right )}}{4 b x} - \frac{\sqrt{\frac{b x^{2}}{\left (\sqrt{b x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{b x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{b x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{4 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2-2)/(b*x**2-1)**(1/4),x)

[Out]

sqrt(2)*x*(1 - I)*elliptic_pi(I, asin(sqrt(2)*(1 + I)*(b*x**2 - 1)**(1/4)/2), -1
)/(2*sqrt(-I*sqrt(b*x**2 - 1) + 1)*sqrt(I*sqrt(b*x**2 - 1) + 1)) - sqrt(2)*sqrt(
b*x**2)*atanh(sqrt(2)*(b*x**2 - 1)**(1/4)/sqrt(b*x**2))/(4*b*x) - sqrt(b*x**2/(s
qrt(b*x**2 - 1) + 1)**2)*(sqrt(b*x**2 - 1) + 1)*elliptic_f(2*atan((b*x**2 - 1)**
(1/4)), 1/2)/(4*b*x)

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Mathematica [C]  time = 0.250127, size = 132, normalized size = 1.71 \[ \frac{6 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};b x^2,\frac{b x^2}{2}\right )}{\left (b x^2-2\right ) \sqrt [4]{b x^2-1} \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};b x^2,\frac{b x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};b x^2,\frac{b x^2}{2}\right )\right )+6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};b x^2,\frac{b x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{2}-2}{\frac{1}{\sqrt [4]{b{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2-2)/(b*x^2-1)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 9.33184, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}}}{\sqrt{b} x}\right ) + \log \left (\frac{4 \,{\left (b x^{2} - 1\right )}^{\frac{1}{4}} b^{2} x^{3} - 4 \, \sqrt{2} \sqrt{b x^{2} - 1} b^{\frac{3}{2}} x^{2} + 8 \,{\left (b x^{2} - 1\right )}^{\frac{3}{4}} b x - \sqrt{2}{\left (b^{2} x^{4} + 4 \, b x^{2} - 4\right )} \sqrt{b}}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )\right )}}{8 \, \sqrt{b}}, -\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b}}{b x}\right ) - \log \left (\frac{4 \,{\left (b x^{2} - 1\right )}^{\frac{1}{4}} b^{2} x^{3} + 4 \, \sqrt{2} \sqrt{b x^{2} - 1} \sqrt{-b} b x^{2} - 8 \,{\left (b x^{2} - 1\right )}^{\frac{3}{4}} b x - \sqrt{2}{\left (b^{2} x^{4} + 4 \, b x^{2} - 4\right )} \sqrt{-b}}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )\right )}}{8 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)),x, algorithm="fricas")

[Out]

[1/8*sqrt(2)*(2*arctan(sqrt(2)*(b*x^2 - 1)^(1/4)/(sqrt(b)*x)) + log((4*(b*x^2 -
1)^(1/4)*b^2*x^3 - 4*sqrt(2)*sqrt(b*x^2 - 1)*b^(3/2)*x^2 + 8*(b*x^2 - 1)^(3/4)*b
*x - sqrt(2)*(b^2*x^4 + 4*b*x^2 - 4)*sqrt(b))/(b^2*x^4 - 4*b*x^2 + 4)))/sqrt(b),
-1/8*sqrt(2)*(2*arctan(sqrt(2)*(b*x^2 - 1)^(1/4)*sqrt(-b)/(b*x)) - log((4*(b*x^
2 - 1)^(1/4)*b^2*x^3 + 4*sqrt(2)*sqrt(b*x^2 - 1)*sqrt(-b)*b*x^2 - 8*(b*x^2 - 1)^
(3/4)*b*x - sqrt(2)*(b^2*x^4 + 4*b*x^2 - 4)*sqrt(-b))/(b^2*x^4 - 4*b*x^2 + 4)))/
sqrt(-b)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{2} - 2\right ) \sqrt [4]{b x^{2} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2-2)/(b*x**2-1)**(1/4),x)

[Out]

Integral(1/((b*x**2 - 2)*(b*x**2 - 1)**(1/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - 1\right )}^{\frac{1}{4}}{\left (b x^{2} - 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)),x, algorithm="giac")

[Out]

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