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

Optimal. Leaf size=129 \[ -\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{b x^2+2}+2\ 2^{3/4}}{2 \sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{b x^2+2}}{2 \sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt{b}} \]

[Out]

-ArcTan[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + b*x^2])/(2*Sqrt[b]*x*(2 + b*x^2)^(1/4))]
/(2*2^(3/4)*Sqrt[b]) - ArcTanh[(2*2^(3/4) - 2*2^(1/4)*Sqrt[2 + b*x^2])/(2*Sqrt[b
]*x*(2 + b*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.0689154, antiderivative size = 129, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{b x^2+2}+2^{3/4}}{\sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{b x^2+2}}{\sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(2^(3/4) + 2^(1/4)*Sqrt[2 + b*x^2])/(Sqrt[b]*x*(2 + b*x^2)^(1/4))]/(2*2^
(3/4)*Sqrt[b]) - ArcTanh[(2^(3/4) - 2^(1/4)*Sqrt[2 + b*x^2])/(Sqrt[b]*x*(2 + b*x
^2)^(1/4))]/(2*2^(3/4)*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 73.7195, size = 88, normalized size = 0.68 \[ \frac{\sqrt [4]{2} i \sqrt{- b x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{b x^{2} + 2}}{2} \right )}\middle | -1\right )}{2 b x} - \frac{\sqrt [4]{2} i \sqrt{- b x^{2}} \Pi \left (i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{b x^{2} + 2}}{2} \right )}\middle | -1\right )}{2 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*I*sqrt(-b*x**2)*elliptic_pi(-I, asin(2**(3/4)*(b*x**2 + 2)**(1/4)/2), -
1)/(2*b*x) - 2**(1/4)*I*sqrt(-b*x**2)*elliptic_pi(I, asin(2**(3/4)*(b*x**2 + 2)*
*(1/4)/2), -1)/(2*b*x)

_______________________________________________________________________________________

Mathematica [C]  time = 0.23179, size = 144, normalized size = 1.12 \[ -\frac{12 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )}{\sqrt [4]{b x^2+2} \left (b x^2+4\right ) \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )\right )-12 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-12*x*AppellF1[1/2, 1/4, 1, 3/2, -(b*x^2)/2, -(b*x^2)/4])/((2 + b*x^2)^(1/4)*(4
+ b*x^2)*(-12*AppellF1[1/2, 1/4, 1, 3/2, -(b*x^2)/2, -(b*x^2)/4] + b*x^2*(2*App
ellF1[3/2, 1/4, 2, 5/2, -(b*x^2)/2, -(b*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, -(b
*x^2)/2, -(b*x^2)/4])))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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