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

Optimal. Leaf size=63 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

(-2*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]])/a^(3/2) - (2*ArcTanh[Sqrt[x + Sqr
t[a^2 + x^2]]/Sqrt[a]])/a^(3/2)

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Rubi [A]  time = 0.365434, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a]])/a^(3/2) - (2*ArcTanh[Sqrt[x + Sqr
t[a^2 + x^2]]/Sqrt[a]])/a^(3/2)

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Rubi in Sympy [A]  time = 9.55944, size = 58, normalized size = 0.92 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.250593, size = 85, normalized size = 1.35 \[ \frac{\log \left (\sqrt{a}-\sqrt{\sqrt{a^2+x^2}+x}\right )-\log \left (\sqrt{\sqrt{a^2+x^2}+x}+\sqrt{a}\right )-2 \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt{{a}^{2}+{x}^{2}}}}{\frac{1}{\sqrt{x+\sqrt{{a}^{2}+{x}^{2}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x/(a^2+x^2)^(1/2)/(x+(a^2+x^2)^(1/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243175, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{x + \sqrt{a^{2} + x^{2}}}}\right ) + \log \left (-\frac{{\left (a + x\right )} \sqrt{a} - 2 \, a \sqrt{x + \sqrt{a^{2} + x^{2}}} + \sqrt{a^{2} + x^{2}} \sqrt{a}}{a - x - \sqrt{a^{2} + x^{2}}}\right )}{a^{\frac{3}{2}}}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{-a} \sqrt{x + \sqrt{a^{2} + x^{2}}}}\right ) + \log \left (-\frac{\sqrt{-a}{\left (a - x\right )} + 2 \, a \sqrt{x + \sqrt{a^{2} + x^{2}}} - \sqrt{a^{2} + x^{2}} \sqrt{-a}}{a + x + \sqrt{a^{2} + x^{2}}}\right )}{\sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(2*arctan(sqrt(a)/sqrt(x + sqrt(a^2 + x^2))) + log(-((a + x)*sqrt(a) - 2*a*sqrt
(x + sqrt(a^2 + x^2)) + sqrt(a^2 + x^2)*sqrt(a))/(a - x - sqrt(a^2 + x^2))))/a^(
3/2), (2*arctan(a/(sqrt(-a)*sqrt(x + sqrt(a^2 + x^2)))) + log(-(sqrt(-a)*(a - x)
+ 2*a*sqrt(x + sqrt(a^2 + x^2)) - sqrt(a^2 + x^2)*sqrt(-a))/(a + x + sqrt(a^2 +
x^2))))/(sqrt(-a)*a)]

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Sympy [A]  time = 2.87045, size = 46, normalized size = 0.73 \[ - \frac{\Gamma ^{2}\left (\frac{3}{4}\right ) \Gamma \left (\frac{5}{4}\right ){{}_{3}F_{2}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4} \end{matrix}\middle |{\frac{a^{2} e^{i \pi }}{x^{2}}} \right )}}{\pi x^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(3/4)**2*gamma(5/4)*hyper((3/4, 3/4, 5/4), (3/2, 7/4), a**2*exp_polar(I*pi
)/x**2)/(pi*x**(3/2)*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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