3.572 \(\int \frac{1+2 \sqrt{1+x}}{x \sqrt{1+x} \sqrt{x+\sqrt{1+x}}} \, dx\)

Optimal. Leaf size=61 \[ 3 \tanh ^{-1}\left (\frac{1-3 \sqrt{x+1}}{2 \sqrt{x+\sqrt{x+1}}}\right )-\tan ^{-1}\left (\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.889224, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ 3 \tanh ^{-1}\left (\frac{1-3 \sqrt{x+1}}{2 \sqrt{x+\sqrt{x+1}}}\right )-\tan ^{-1}\left (\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 71.7827, size = 78, normalized size = 1.28 \[ - 2 \operatorname{atan}{\left (- \frac{- \sqrt{x + 1} - 3}{2 \sqrt{x + \sqrt{x + 1}}} \right )} + \operatorname{atan}{\left (\frac{\sqrt{x + 1} + 3}{2 \sqrt{x + \sqrt{x + 1}}} \right )} - 3 \operatorname{atanh}{\left (\frac{3 \sqrt{x + 1} - 1}{2 \sqrt{x + \sqrt{x + 1}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atan(-(-sqrt(x + 1) - 3)/(2*sqrt(x + sqrt(x + 1)))) + atan((sqrt(x + 1) + 3)/
(2*sqrt(x + sqrt(x + 1)))) - 3*atanh((3*sqrt(x + 1) - 1)/(2*sqrt(x + sqrt(x + 1)
)))

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Mathematica [A]  time = 0.029641, size = 73, normalized size = 1.2 \[ 3 \log \left (1-\sqrt{x+1}\right )-3 \log \left (-3 \sqrt{x+1}-2 \sqrt{x+\sqrt{x+1}}+1\right )-\tan ^{-1}\left (\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(3 + Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])] + 3*Log[1 - Sqrt[1 + x]] -
3*Log[1 - 3*Sqrt[1 + x] - 2*Sqrt[x + Sqrt[1 + x]]]

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Maple [A]  time = 0.021, size = 68, normalized size = 1.1 \[ \arctan \left ({\frac{1}{2} \left ( -3-\sqrt{1+x} \right ){\frac{1}{\sqrt{ \left ( 1+\sqrt{1+x} \right ) ^{2}-2-\sqrt{1+x}}}}} \right ) -3\,{\it Artanh} \left ( 1/2\,{\frac{-1+3\,\sqrt{1+x}}{\sqrt{ \left ( \sqrt{1+x}-1 \right ) ^{2}+3\,\sqrt{1+x}-2}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(1/2*(-3-(1+x)^(1/2))/((1+(1+x)^(1/2))^2-2-(1+x)^(1/2))^(1/2))-3*arctanh(1
/2*(-1+3*(1+x)^(1/2))/(((1+x)^(1/2)-1)^2+3*(1+x)^(1/2)-2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, \sqrt{x + 1} + 1}{\sqrt{x + \sqrt{x + 1}} \sqrt{x + 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((2*sqrt(x + 1) + 1)/(sqrt(x + sqrt(x + 1))*sqrt(x + 1)*x),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out