3.203 \(\int \frac{\sqrt{x+\sqrt{a^2+x^2}}}{x} \, dx\)

Optimal. Leaf size=82 \[ 2 \sqrt{\sqrt{a^2+x^2}+x}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right ) \]

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]] - 2*Sqrt[a]*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]
] - 2*Sqrt[a]*ArcTanh[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]]

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Rubi [A]  time = 0.137614, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ 2 \sqrt{\sqrt{a^2+x^2}+x}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]] - 2*Sqrt[a]*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]
] - 2*Sqrt[a]*ArcTanh[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]]

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Rubi in Sympy [A]  time = 8.18798, size = 73, normalized size = 0.89 \[ - 2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a}} \right )} - 2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a}} \right )} + 2 \sqrt{x + \sqrt{a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*atan(sqrt(x + sqrt(a**2 + x**2))/sqrt(a)) - 2*sqrt(a)*atanh(sqrt(x +
sqrt(a**2 + x**2))/sqrt(a)) + 2*sqrt(x + sqrt(a**2 + x**2))

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Mathematica [A]  time = 0.274323, size = 161, normalized size = 1.96 \[ -\frac{\sqrt{a^2+x^2} \left (\sqrt{a^2+x^2}+x\right ) \left (-2 \sqrt{\sqrt{a^2+x^2}+x}-\sqrt{a} \log \left (\sqrt{a}-\sqrt{\sqrt{a^2+x^2}+x}\right )+\sqrt{a} \log \left (\sqrt{\sqrt{a^2+x^2}+x}+\sqrt{a}\right )+2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )\right )}{x \left (\sqrt{a^2+x^2}+x\right )+a^2} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[a^2 + x^2]*(x + Sqrt[a^2 + x^2])*(-2*Sqrt[x + Sqrt[a^2 + x^2]] + 2*Sqrt[
a]*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]] - Sqrt[a]*Log[Sqrt[a] - Sqrt[x + Sq
rt[a^2 + x^2]]] + Sqrt[a]*Log[Sqrt[a] + Sqrt[x + Sqrt[a^2 + x^2]]]))/(a^2 + x*(x
+ Sqrt[a^2 + x^2])))

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Maple [C]  time = 0.049, size = 25, normalized size = 0.3 \[ 2\,\sqrt{2}\sqrt{x}{\mbox{$_3$F$_2$}(-1/4,-1/4,1/4;\,1/2,3/4;\,-{\frac{{a}^{2}}{{x}^{2}}})} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*2^(1/2)*x^(1/2)*hypergeom([-1/4,-1/4,1/4],[1/2,3/4],-1/x^2*a^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + sqrt(a^2 + x^2))/x, x)

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Fricas [A]  time = 0.240301, size = 1, normalized size = 0.01 \[ \left [-2 \, \sqrt{a} \arctan \left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a}}\right ) + \sqrt{a} \log \left (-\frac{a - 2 \, \sqrt{a} \sqrt{x + \sqrt{a^{2} + x^{2}}} + x + \sqrt{a^{2} + x^{2}}}{a - x - \sqrt{a^{2} + x^{2}}}\right ) + 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}}, -2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{-a}}\right ) + \sqrt{-a} \log \left (-\frac{a + 2 \, \sqrt{-a} \sqrt{x + \sqrt{a^{2} + x^{2}}} - x - \sqrt{a^{2} + x^{2}}}{a + x + \sqrt{a^{2} + x^{2}}}\right ) + 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-2*sqrt(a)*arctan(sqrt(x + sqrt(a^2 + x^2))/sqrt(a)) + sqrt(a)*log(-(a - 2*sqrt
(a)*sqrt(x + sqrt(a^2 + x^2)) + x + sqrt(a^2 + x^2))/(a - x - sqrt(a^2 + x^2)))
+ 2*sqrt(x + sqrt(a^2 + x^2)), -2*sqrt(-a)*arctan(sqrt(x + sqrt(a^2 + x^2))/sqrt
(-a)) + sqrt(-a)*log(-(a + 2*sqrt(-a)*sqrt(x + sqrt(a^2 + x^2)) - x - sqrt(a^2 +
x^2))/(a + x + sqrt(a^2 + x^2))) + 2*sqrt(x + sqrt(a^2 + x^2))]

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Sympy [A]  time = 2.27418, size = 51, normalized size = 0.62 \[ \frac{\sqrt{x} \Gamma ^{2}\left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right ){{}_{3}F_{2}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4}, \frac{1}{4} \\ \frac{1}{2}, \frac{3}{4} \end{matrix}\middle |{\frac{a^{2} e^{i \pi }}{x^{2}}} \right )}}{8 \pi \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*gamma(-1/4)**2*gamma(1/4)*hyper((-1/4, -1/4, 1/4), (1/2, 3/4), a**2*exp_
polar(I*pi)/x**2)/(8*pi*gamma(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + sqrt(a^2 + x^2))/x, x)