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

Optimal. Leaf size=83 \[ -\frac{\sqrt{x+\sqrt{x+1}}}{x}-\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right )+\frac{3}{4} \tanh ^{-1}\left (\frac{1-3 \sqrt{x+1}}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

[Out]

-(Sqrt[x + Sqrt[1 + x]]/x) - ArcTan[(3 + Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])]
/4 + (3*ArcTanh[(1 - 3*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/4

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Rubi [A]  time = 0.226901, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{\sqrt{x+\sqrt{x+1}}}{x}-\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right )+\frac{3}{4} \tanh ^{-1}\left (\frac{1-3 \sqrt{x+1}}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x + Sqrt[1 + x]]/x^2,x]

[Out]

-(Sqrt[x + Sqrt[1 + x]]/x) - ArcTan[(3 + Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])]
/4 + (3*ArcTanh[(1 - 3*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/4

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Rubi in Sympy [A]  time = 11.8989, size = 71, normalized size = 0.86 \[ - \frac{\operatorname{atan}{\left (- \frac{- \sqrt{x + 1} - 3}{2 \sqrt{x + \sqrt{x + 1}}} \right )}}{4} - \frac{3 \operatorname{atanh}{\left (\frac{3 \sqrt{x + 1} - 1}{2 \sqrt{x + \sqrt{x + 1}}} \right )}}{4} - \frac{\sqrt{x + \sqrt{x + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(-(-sqrt(x + 1) - 3)/(2*sqrt(x + sqrt(x + 1))))/4 - 3*atanh((3*sqrt(x + 1)
- 1)/(2*sqrt(x + sqrt(x + 1))))/4 - sqrt(x + sqrt(x + 1))/x

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Mathematica [A]  time = 0.0582356, size = 97, normalized size = 1.17 \[ -\frac{\sqrt{x+\sqrt{x+1}}}{x}+\frac{3}{4} \log \left (1-\sqrt{x+1}\right )-\frac{3}{4} \log \left (-3 \sqrt{x+1}-2 \sqrt{x+\sqrt{x+1}}+1\right )-\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x + Sqrt[1 + x]]/x^2,x]

[Out]

-(Sqrt[x + Sqrt[1 + x]]/x) - ArcTan[(3 + Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])]
/4 + (3*Log[1 - Sqrt[1 + x]])/4 - (3*Log[1 - 3*Sqrt[1 + x] - 2*Sqrt[x + Sqrt[1 +
x]]])/4

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Maple [B]  time = 0.022, size = 298, normalized size = 3.6 \[ -{\frac{1}{2} \left ( \left ( -1+\sqrt{1+x} \right ) ^{2}-2+3\,\sqrt{1+x} \right ) ^{{\frac{3}{2}}} \left ( -1+\sqrt{1+x} \right ) ^{-1}}+{\frac{3}{4}\sqrt{ \left ( -1+\sqrt{1+x} \right ) ^{2}-2+3\,\sqrt{1+x}}}+{\frac{1}{2}\ln \left ({\frac{1}{2}}+\sqrt{1+x}+\sqrt{ \left ( -1+\sqrt{1+x} \right ) ^{2}-2+3\,\sqrt{1+x}} \right ) }-{\frac{3}{4}{\it Artanh} \left ({\frac{1}{2} \left ( -1+3\,\sqrt{1+x} \right ){\frac{1}{\sqrt{ \left ( -1+\sqrt{1+x} \right ) ^{2}-2+3\,\sqrt{1+x}}}}} \right ) }+{\frac{1}{4} \left ( 2\,\sqrt{1+x}+1 \right ) \sqrt{ \left ( -1+\sqrt{1+x} \right ) ^{2}-2+3\,\sqrt{1+x}}}-{\frac{1}{2} \left ( \left ( 1+\sqrt{1+x} \right ) ^{2}-2-\sqrt{1+x} \right ) ^{{\frac{3}{2}}} \left ( 1+\sqrt{1+x} \right ) ^{-1}}-{\frac{1}{4}\sqrt{ \left ( 1+\sqrt{1+x} \right ) ^{2}-2-\sqrt{1+x}}}-{\frac{1}{2}\ln \left ({\frac{1}{2}}+\sqrt{1+x}+\sqrt{ \left ( 1+\sqrt{1+x} \right ) ^{2}-2-\sqrt{1+x}} \right ) }+{\frac{1}{4}\arctan \left ({\frac{1}{2} \left ( -3-\sqrt{1+x} \right ){\frac{1}{\sqrt{ \left ( 1+\sqrt{1+x} \right ) ^{2}-2-\sqrt{1+x}}}}} \right ) }+{\frac{1}{4} \left ( 2\,\sqrt{1+x}+1 \right ) \sqrt{ \left ( 1+\sqrt{1+x} \right ) ^{2}-2-\sqrt{1+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(-1+(1+x)^(1/2))*((-1+(1+x)^(1/2))^2-2+3*(1+x)^(1/2))^(3/2)+3/4*((-1+(1+x)^
(1/2))^2-2+3*(1+x)^(1/2))^(1/2)+1/2*ln(1/2+(1+x)^(1/2)+((-1+(1+x)^(1/2))^2-2+3*(
1+x)^(1/2))^(1/2))-3/4*arctanh(1/2*(-1+3*(1+x)^(1/2))/((-1+(1+x)^(1/2))^2-2+3*(1
+x)^(1/2))^(1/2))+1/4*(2*(1+x)^(1/2)+1)*((-1+(1+x)^(1/2))^2-2+3*(1+x)^(1/2))^(1/
2)-1/2/(1+(1+x)^(1/2))*((1+(1+x)^(1/2))^2-2-(1+x)^(1/2))^(3/2)-1/4*((1+(1+x)^(1/
2))^2-2-(1+x)^(1/2))^(1/2)-1/2*ln(1/2+(1+x)^(1/2)+((1+(1+x)^(1/2))^2-2-(1+x)^(1/
2))^(1/2))+1/4*arctan(1/2*(-3-(1+x)^(1/2))/((1+(1+x)^(1/2))^2-2-(1+x)^(1/2))^(1/
2))+1/4*(2*(1+x)^(1/2)+1)*((1+(1+x)^(1/2))^2-2-(1+x)^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x + sqrt(x + 1))/x**2, 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(sqrt(x + sqrt(x + 1))/x^2,x, algorithm="giac")

[Out]

Timed out