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

Optimal. Leaf size=56 \[ \frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left (\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right )-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

Sqrt[2 + x^2]/4 - ArcSinh[x/Sqrt[2]]/16 - (Sqrt[33]*ArcTanh[(8 - x)/(Sqrt[33]*Sq
rt[2 + x^2])])/16

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Rubi [A]  time = 0.109809, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left (\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right )-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + x^2]/(1 + 4*x),x]

[Out]

Sqrt[2 + x^2]/4 - ArcSinh[x/Sqrt[2]]/16 - (Sqrt[33]*ArcTanh[(8 - x)/(Sqrt[33]*Sq
rt[2 + x^2])])/16

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Rubi in Sympy [A]  time = 8.33541, size = 48, normalized size = 0.86 \[ \frac{\sqrt{x^{2} + 2}}{4} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} - \frac{\sqrt{33} \operatorname{atanh}{\left (\frac{\sqrt{33} \left (- x + 8\right )}{33 \sqrt{x^{2} + 2}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x**2 + 2)/4 - asinh(sqrt(2)*x/2)/16 - sqrt(33)*atanh(sqrt(33)*(-x + 8)/(33*
sqrt(x**2 + 2)))/16

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Mathematica [A]  time = 0.0661654, size = 68, normalized size = 1.21 \[ \frac{1}{16} \left (4 \sqrt{x^2+2}-\sqrt{33} \log \left (4 \left (\sqrt{33} \sqrt{x^2+2}-x+8\right )\right )+\sqrt{33} \log (4 x+1)-\sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[2 + x^2] - ArcSinh[x/Sqrt[2]] + Sqrt[33]*Log[1 + 4*x] - Sqrt[33]*Log[4*(
8 - x + Sqrt[33]*Sqrt[2 + x^2])])/16

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Maple [A]  time = 0.009, size = 57, normalized size = 1. \[{\frac{1}{16}\sqrt{16\, \left ( x+1/4 \right ) ^{2}-8\,x+31}}-{\frac{1}{16}{\it Arcsinh} \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{\sqrt{33}}{16}{\it Artanh} \left ({\frac{8\,\sqrt{33}}{33} \left ( 4-{\frac{x}{2}} \right ){\frac{1}{\sqrt{16\, \left ( x+1/4 \right ) ^{2}-8\,x+31}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2)^(1/2)/(1+4*x),x)

[Out]

1/16*(16*(x+1/4)^2-8*x+31)^(1/2)-1/16*arcsinh(1/2*x*2^(1/2))-1/16*33^(1/2)*arcta
nh(8/33*(4-1/2*x)*33^(1/2)/(16*(x+1/4)^2-8*x+31)^(1/2))

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Maxima [A]  time = 0.794295, size = 72, normalized size = 1.29 \[ \frac{1}{16} \, \sqrt{33} \operatorname{arsinh}\left (\frac{\sqrt{2} x}{2 \,{\left | 4 \, x + 1 \right |}} - \frac{4 \, \sqrt{2}}{{\left | 4 \, x + 1 \right |}}\right ) + \frac{1}{4} \, \sqrt{x^{2} + 2} - \frac{1}{16} \, \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(33)*arcsinh(1/2*sqrt(2)*x/abs(4*x + 1) - 4*sqrt(2)/abs(4*x + 1)) + 1/4
*sqrt(x^2 + 2) - 1/16*arcsinh(1/2*sqrt(2)*x)

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Fricas [A]  time = 0.232636, size = 186, normalized size = 3.32 \[ -\frac{4 \, x^{2} -{\left (x - \sqrt{x^{2} + 2}\right )} \log \left (-x + \sqrt{x^{2} + 2}\right ) -{\left (\sqrt{33} x - \sqrt{33} \sqrt{x^{2} + 2}\right )} \log \left (\frac{16 \, x^{2} - 4 \, \sqrt{x^{2} + 2}{\left (4 \, x + \sqrt{33} + 1\right )} + \sqrt{33}{\left (4 \, x + 1\right )} + 4 \, x + 33}{4 \, x^{2} - \sqrt{x^{2} + 2}{\left (4 \, x + 1\right )} + x}\right ) - 4 \, \sqrt{x^{2} + 2} x + 8}{16 \,{\left (x - \sqrt{x^{2} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(4*x^2 - (x - sqrt(x^2 + 2))*log(-x + sqrt(x^2 + 2)) - (sqrt(33)*x - sqrt(
33)*sqrt(x^2 + 2))*log((16*x^2 - 4*sqrt(x^2 + 2)*(4*x + sqrt(33) + 1) + sqrt(33)
*(4*x + 1) + 4*x + 33)/(4*x^2 - sqrt(x^2 + 2)*(4*x + 1) + x)) - 4*sqrt(x^2 + 2)*
x + 8)/(x - sqrt(x^2 + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**2 + 2)/(4*x + 1), x)

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GIAC/XCAS [A]  time = 0.226753, size = 96, normalized size = 1.71 \[ \frac{1}{16} \, \sqrt{33}{\rm ln}\left (\frac{{\left | -4 \, x - \sqrt{33} + 4 \, \sqrt{x^{2} + 2} - 1 \right |}}{{\left | -4 \, x + \sqrt{33} + 4 \, \sqrt{x^{2} + 2} - 1 \right |}}\right ) + \frac{1}{4} \, \sqrt{x^{2} + 2} + \frac{1}{16} \,{\rm ln}\left (-x + \sqrt{x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(33)*ln(abs(-4*x - sqrt(33) + 4*sqrt(x^2 + 2) - 1)/abs(-4*x + sqrt(33)
+ 4*sqrt(x^2 + 2) - 1)) + 1/4*sqrt(x^2 + 2) + 1/16*ln(-x + sqrt(x^2 + 2))