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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]