3.239 \(\int \frac{1}{\left (4+x^2\right ) \sqrt{1+4 x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{15} x}{2 \sqrt{4 x^2+1}}\right )}{2 \sqrt{15}} \]

[Out]

ArcTanh[(Sqrt[15]*x)/(2*Sqrt[1 + 4*x^2])]/(2*Sqrt[15])

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Rubi [A]  time = 0.0298029, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{15} x}{2 \sqrt{4 x^2+1}}\right )}{2 \sqrt{15}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(Sqrt[15]*x)/(2*Sqrt[1 + 4*x^2])]/(2*Sqrt[15])

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Rubi in Sympy [A]  time = 2.99983, size = 26, normalized size = 0.84 \[ \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} x}{2 \sqrt{4 x^{2} + 1}} \right )}}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(15)*atanh(sqrt(15)*x/(2*sqrt(4*x**2 + 1)))/30

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Mathematica [A]  time = 0.0717559, size = 56, normalized size = 1.81 \[ \frac{\log \left (31 x^2+4 \sqrt{60 x^2+15} x+4\right )-\log \left (31 x^2-4 \sqrt{60 x^2+15} x+4\right )}{8 \sqrt{15}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[4 + 31*x^2 - 4*x*Sqrt[15 + 60*x^2]] + Log[4 + 31*x^2 + 4*x*Sqrt[15 + 60*x^
2]])/(8*Sqrt[15])

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Maple [A]  time = 0.013, size = 22, normalized size = 0.7 \[{\frac{\sqrt{15}}{30}{\it Artanh} \left ({\frac{x\sqrt{15}}{2}{\frac{1}{\sqrt{4\,{x}^{2}+1}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*arctanh(1/2*x*15^(1/2)/(4*x^2+1)^(1/2))*15^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244482, size = 119, normalized size = 3.84 \[ \frac{1}{60} \, \sqrt{15} \log \left (\frac{120 \, x^{2} + \sqrt{15}{\left (8 \, x^{4} + 33 \, x^{2} + 124\right )} - 4 \, \sqrt{4 \, x^{2} + 1}{\left (\sqrt{15}{\left (x^{3} + 4 \, x\right )} + 15 \, x\right )} + 480}{8 \, x^{4} + 33 \, x^{2} - 4 \,{\left (x^{3} + 4 \, x\right )} \sqrt{4 \, x^{2} + 1} + 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*sqrt(15)*log((120*x^2 + sqrt(15)*(8*x^4 + 33*x^2 + 124) - 4*sqrt(4*x^2 + 1)
*(sqrt(15)*(x^3 + 4*x) + 15*x) + 480)/(8*x^4 + 33*x^2 - 4*(x^3 + 4*x)*sqrt(4*x^2
+ 1) + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217359, size = 77, normalized size = 2.48 \[ -\frac{1}{60} \, \sqrt{15}{\rm ln}\left (\frac{{\left (2 \, x - \sqrt{4 \, x^{2} + 1}\right )}^{2} - 8 \, \sqrt{15} + 31}{{\left (2 \, x - \sqrt{4 \, x^{2} + 1}\right )}^{2} + 8 \, \sqrt{15} + 31}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*sqrt(15)*ln(((2*x - sqrt(4*x^2 + 1))^2 - 8*sqrt(15) + 31)/((2*x - sqrt(4*x
^2 + 1))^2 + 8*sqrt(15) + 31))