3.177 \(\int \frac{1}{\left (a+b x^2\right ) \sqrt{4+d x^4}} \, dx\)

Optimal. Leaf size=287 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{a d}{b}+\frac{4 b}{a}}}{\sqrt{d x^4+4}}\right )}{2 a \sqrt{\frac{a d}{b}+\frac{4 b}{a}}}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^2+2\right ) \sqrt{\frac{d x^4+4}{\left (\sqrt{d} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{d x^4+4} \left (2 b-a \sqrt{d}\right )}+\frac{\left (\sqrt{d} x^2+2\right ) \sqrt{\frac{d x^4+4}{\left (\sqrt{d} x^2+2\right )^2}} \left (a \sqrt{d}+2 b\right ) \Pi \left (-\frac{\left (2 b-a \sqrt{d}\right )^2}{8 a b \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )|\frac{1}{2}\right )}{4 \sqrt{2} a \sqrt [4]{d} \sqrt{d x^4+4} \left (2 b-a \sqrt{d}\right )} \]

[Out]

ArcTan[(Sqrt[(4*b)/a + (a*d)/b]*x)/Sqrt[4 + d*x^4]]/(2*a*Sqrt[(4*b)/a + (a*d)/b]
) - (d^(1/4)*(2 + Sqrt[d]*x^2)*Sqrt[(4 + d*x^4)/(2 + Sqrt[d]*x^2)^2]*EllipticF[2
*ArcTan[(d^(1/4)*x)/Sqrt[2]], 1/2])/(2*Sqrt[2]*(2*b - a*Sqrt[d])*Sqrt[4 + d*x^4]
) + ((2*b + a*Sqrt[d])*(2 + Sqrt[d]*x^2)*Sqrt[(4 + d*x^4)/(2 + Sqrt[d]*x^2)^2]*E
llipticPi[-(2*b - a*Sqrt[d])^2/(8*a*b*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/Sqrt[2]], 1
/2])/(4*Sqrt[2]*a*(2*b - a*Sqrt[d])*d^(1/4)*Sqrt[4 + d*x^4])

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Rubi [A]  time = 0.289906, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{a d}{b}+\frac{4 b}{a}}}{\sqrt{d x^4+4}}\right )}{2 a \sqrt{\frac{a d}{b}+\frac{4 b}{a}}}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^2+2\right ) \sqrt{\frac{d x^4+4}{\left (\sqrt{d} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{d x^4+4} \left (2 b-a \sqrt{d}\right )}+\frac{\left (\sqrt{d} x^2+2\right ) \sqrt{\frac{d x^4+4}{\left (\sqrt{d} x^2+2\right )^2}} \left (a \sqrt{d}+2 b\right ) \Pi \left (-\frac{\left (2 b-a \sqrt{d}\right )^2}{8 a b \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )|\frac{1}{2}\right )}{4 \sqrt{2} a \sqrt [4]{d} \sqrt{d x^4+4} \left (2 b-a \sqrt{d}\right )} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(Sqrt[(4*b)/a + (a*d)/b]*x)/Sqrt[4 + d*x^4]]/(2*a*Sqrt[(4*b)/a + (a*d)/b]
) - (d^(1/4)*(2 + Sqrt[d]*x^2)*Sqrt[(4 + d*x^4)/(2 + Sqrt[d]*x^2)^2]*EllipticF[2
*ArcTan[(d^(1/4)*x)/Sqrt[2]], 1/2])/(2*Sqrt[2]*(2*b - a*Sqrt[d])*Sqrt[4 + d*x^4]
) + ((2*b + a*Sqrt[d])*(2 + Sqrt[d]*x^2)*Sqrt[(4 + d*x^4)/(2 + Sqrt[d]*x^2)^2]*E
llipticPi[-(2*b - a*Sqrt[d])^2/(8*a*b*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/Sqrt[2]], 1
/2])/(4*Sqrt[2]*a*(2*b - a*Sqrt[d])*d^(1/4)*Sqrt[4 + d*x^4])

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Rubi in Sympy [A]  time = 18.6951, size = 253, normalized size = 0.88 \[ - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{\frac{d x^{4} + 4}{\left (\frac{\sqrt{d} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{d} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{d} x}{2} \right )}\middle | \frac{1}{2}\right )}{4 \left (- a \sqrt{d} + 2 b\right ) \sqrt{d x^{4} + 4}} + \frac{\operatorname{atan}{\left (\frac{x \sqrt{\frac{a d}{b} + \frac{4 b}{a}}}{\sqrt{d x^{4} + 4}} \right )}}{2 a \sqrt{\frac{a d}{b} + \frac{4 b}{a}}} + \frac{\sqrt{2} \sqrt{\frac{d x^{4} + 4}{\left (\frac{\sqrt{d} x^{2}}{2} + 1\right )^{2}}} \left (a \sqrt{d} + 2 b\right ) \left (\frac{\sqrt{d} x^{2}}{2} + 1\right ) \Pi \left (- \frac{\left (- \frac{a \sqrt{d}}{2} + b\right )^{2}}{2 a b \sqrt{d}}; 2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{d} x}{2} \right )}\middle | \frac{1}{2}\right )}{8 a \sqrt [4]{d} \left (- a \sqrt{d} + 2 b\right ) \sqrt{d x^{4} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*d**(1/4)*sqrt((d*x**4 + 4)/(sqrt(d)*x**2/2 + 1)**2)*(sqrt(d)*x**2/2 + 1
)*elliptic_f(2*atan(sqrt(2)*d**(1/4)*x/2), 1/2)/(4*(-a*sqrt(d) + 2*b)*sqrt(d*x**
4 + 4)) + atan(x*sqrt(a*d/b + 4*b/a)/sqrt(d*x**4 + 4))/(2*a*sqrt(a*d/b + 4*b/a))
+ sqrt(2)*sqrt((d*x**4 + 4)/(sqrt(d)*x**2/2 + 1)**2)*(a*sqrt(d) + 2*b)*(sqrt(d)
*x**2/2 + 1)*elliptic_pi(-(-a*sqrt(d)/2 + b)**2/(2*a*b*sqrt(d)), 2*atan(sqrt(2)*
d**(1/4)*x/2), 1/2)/(8*a*d**(1/4)*(-a*sqrt(d) + 2*b)*sqrt(d*x**4 + 4))

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Mathematica [C]  time = 0.0634357, size = 65, normalized size = 0.23 \[ -\frac{i \Pi \left (-\frac{2 i b}{a \sqrt{d}};\left .i \sinh ^{-1}\left (\frac{\sqrt{i \sqrt{d}} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt{i \sqrt{d}}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*EllipticPi[((-2*I)*b)/(a*Sqrt[d]), I*ArcSinh[(Sqrt[I*Sqrt[d]]*x)/Sqrt[2]],
-1])/(Sqrt[2]*a*Sqrt[I*Sqrt[d]])

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Maple [C]  time = 0.033, size = 86, normalized size = 0.3 \[{\frac{1}{a}\sqrt{1-{\frac{i}{2}}\sqrt{d}{x}^{2}}\sqrt{1+{\frac{i}{2}}\sqrt{d}{x}^{2}}{\it EllipticPi} \left ( \sqrt{{\frac{i}{2}}\sqrt{d}}x,{\frac{2\,ib}{a}{\frac{1}{\sqrt{d}}}},{1\sqrt{-{\frac{i}{2}}\sqrt{d}}{\frac{1}{\sqrt{{\frac{i}{2}}\sqrt{d}}}}} \right ){\frac{1}{\sqrt{{\frac{i}{2}}\sqrt{d}}}}{\frac{1}{\sqrt{d{x}^{4}+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a/(1/2*I*d^(1/2))^(1/2)*(1-1/2*I*d^(1/2)*x^2)^(1/2)*(1+1/2*I*d^(1/2)*x^2)^(1/2
)/(d*x^4+4)^(1/2)*EllipticPi((1/2*I*d^(1/2))^(1/2)*x,2*I/d^(1/2)*b/a,(-1/2*I*d^(
1/2))^(1/2)/(1/2*I*d^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{d x^{4} + 4}{\left (b x^{2} + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(d*x^4 + 4)*(b*x^2 + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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