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

Optimal. Leaf size=79 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}} \]

[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])

_______________________________________________________________________________________

Rubi [A]  time = 0.0500597, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.5565, size = 187, normalized size = 2.37 \[ \frac{\sqrt{2} x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{- b x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{- b x^{2} - 1} + 1} \sqrt{i \sqrt{- b x^{2} - 1} + 1}} + \frac{\sqrt{2} \sqrt{- b x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt [4]{- b x^{2} - 1}}{\sqrt{- b x^{2}}} \right )}}{4 b x} + \frac{\sqrt{- \frac{b x^{2}}{\left (\sqrt{- b x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{- b x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{- b x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{4 b x} \]

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)*sq
rt(-b*x**2)*atanh(sqrt(2)*(-b*x**2 - 1)**(1/4)/sqrt(-b*x**2))/(4*b*x) + sqrt(-b*
x**2/(sqrt(-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)

_______________________________________________________________________________________

Mathematica [C]  time = 0.238794, size = 137, normalized size = 1.73 \[ \frac{6 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-b x^2,-\frac{b x^2}{2}\right )}{\sqrt [4]{-b x^2-1} \left (b x^2+2\right ) \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )\right )-6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-b x^2,-\frac{b x^2}{2}\right )\right )} \]

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])/((-1 - b*x^2)^(1/4)*(2 +
b*x^2)*(-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])))

_______________________________________________________________________________________

Maple [F]  time = 0.054, 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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 9.12209, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}}}{\sqrt{b} x}\right ) + \log \left (\frac{4 \,{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} b^{2} x^{3} + 8 \,{\left (-b x^{2} - 1\right )}^{\frac{3}{4}} b x - \sqrt{2}{\left (b^{2} x^{4} + 4 \, \sqrt{-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 4\right )} \sqrt{b}}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )\right )}}{8 \, \sqrt{b}}, -\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b}}{b x}\right ) - \log \left (\frac{4 \,{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} b^{2} x^{3} - 8 \,{\left (-b x^{2} - 1\right )}^{\frac{3}{4}} b x - \sqrt{2}{\left (b^{2} x^{4} - 4 \, \sqrt{-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 4\right )} \sqrt{-b}}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )\right )}}{8 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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