3.47 \(\int \tan ^{-1}\left (x \sqrt{1+x^2}\right ) \, dx\)

Optimal. Leaf size=120 \[ -\frac{1}{4} \sqrt{3} \log \left (x^2-\sqrt{3} \sqrt{x^2+1}+2\right )+\frac{1}{4} \sqrt{3} \log \left (x^2+\sqrt{3} \sqrt{x^2+1}+2\right )+x \tan ^{-1}\left (x \sqrt{x^2+1}\right )+\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 \sqrt{x^2+1}\right )-\frac{1}{2} \tan ^{-1}\left (2 \sqrt{x^2+1}+\sqrt{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.497335, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{1}{4} \sqrt{3} \log \left (x^2-\sqrt{3} \sqrt{x^2+1}+2\right )+\frac{1}{4} \sqrt{3} \log \left (x^2+\sqrt{3} \sqrt{x^2+1}+2\right )+x \tan ^{-1}\left (x \sqrt{x^2+1}\right )+\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 \sqrt{x^2+1}\right )-\frac{1}{2} \tan ^{-1}\left (2 \sqrt{x^2+1}+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[x*Sqrt[1 + x^2]],x]

[Out]

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

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Rubi in Sympy [A]  time = 43.8189, size = 104, normalized size = 0.87 \[ x \operatorname{atan}{\left (x \sqrt{x^{2} + 1} \right )} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} \sqrt{x^{2} + 1} + 2 \right )}}{4} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} \sqrt{x^{2} + 1} + 2 \right )}}{4} - \frac{\operatorname{atan}{\left (2 \sqrt{x^{2} + 1} - \sqrt{3} \right )}}{2} - \frac{\operatorname{atan}{\left (2 \sqrt{x^{2} + 1} + \sqrt{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*atan(x*sqrt(x**2 + 1)) - sqrt(3)*log(x**2 - sqrt(3)*sqrt(x**2 + 1) + 2)/4 + sq
rt(3)*log(x**2 + sqrt(3)*sqrt(x**2 + 1) + 2)/4 - atan(2*sqrt(x**2 + 1) - sqrt(3)
)/2 - atan(2*sqrt(x**2 + 1) + sqrt(3))/2

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Mathematica [C]  time = 0.249549, size = 116, normalized size = 0.97 \[ \frac{1}{2} \left (-\sqrt{-2+2 i \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{x^2+1}}{\sqrt{-1-i \sqrt{3}}}\right )-\sqrt{-2-2 i \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{x^2+1}}{\sqrt{-1+i \sqrt{3}}}\right )+2 x \tan ^{-1}\left (x \sqrt{x^2+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[x*Sqrt[1 + x^2]],x]

[Out]

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

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Maple [B]  time = 0.068, size = 510, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x*(x^2+1)^(1/2)),x)

[Out]

x*arctan(x*(x^2+1)^(1/2))+1/3*2^(1/2)/(((-1+x)^2/(-1-x)^2+1)/(1+(-1+x)/(-1-x))^2
)^(1/2)/(1+(-1+x)/(-1-x))*(2*(-1+x)^2/(-1-x)^2+2)^(1/2)*3^(1/2)*arctanh(1/2*(2*(
-1+x)^2/(-1-x)^2+2)^(1/2)*3^(1/2))+1/3*2^(1/2)/(((1+x)^2/(1-x)^2+1)/(1+(1+x)/(1-
x))^2)^(1/2)/(1+(1+x)/(1-x))*(2*(1+x)^2/(1-x)^2+2)^(1/2)*3^(1/2)*arctanh(1/2*(2*
(1+x)^2/(1-x)^2+2)^(1/2)*3^(1/2))+1/12*2^(1/2)*(2*(-1+x)^2/(-1-x)^2+2)^(1/2)*(-3
^(1/2)*arctanh(1/2*(2*(-1+x)^2/(-1-x)^2+2)^(1/2)*3^(1/2))+3*arctan(1/((-1+x)^2/(
-1-x)^2+1)*(2*(-1+x)^2/(-1-x)^2+2)^(1/2)*(-1+x)/(-1-x)))/(((-1+x)^2/(-1-x)^2+1)/
(1+(-1+x)/(-1-x))^2)^(1/2)/(1+(-1+x)/(-1-x))+1/12*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^
(1/2)*(-3^(1/2)*arctanh(1/2*(2*(1+x)^2/(1-x)^2+2)^(1/2)*3^(1/2))+3*arctan(1/((1+
x)^2/(1-x)^2+1)*(2*(1+x)^2/(1-x)^2+2)^(1/2)*(1+x)/(1-x)))/(((1+x)^2/(1-x)^2+1)/(
1+(1+x)/(1-x))^2)^(1/2)/(1+(1+x)/(1-x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ x \arctan \left (\sqrt{x^{2} + 1} x\right ) - \int \frac{{\left (2 \, x^{3} + x\right )} \sqrt{x^{2} + 1}}{{\left (x^{4} + x^{2}\right )}{\left (x^{2} + 1\right )} + x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arctan(sqrt(x^2 + 1)*x) - integrate((2*x^3 + x)*sqrt(x^2 + 1)/((x^4 + x^2)*(x^
2 + 1) + x^2 + 1), x)

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Fricas [A]  time = 0.24219, size = 505, normalized size = 4.21 \[ x \arctan \left (\sqrt{x^{2} + 1} x\right ) - \frac{1}{4} \, \sqrt{3} \log \left (16 \, x^{4} + 40 \, x^{2} + 16 \, \sqrt{3}{\left (x^{3} + x\right )} - 8 \,{\left (2 \, x^{3} + \sqrt{3}{\left (2 \, x^{2} + 1\right )} + 4 \, x\right )} \sqrt{x^{2} + 1} + 16\right ) + \frac{1}{4} \, \sqrt{3} \log \left (16 \, x^{4} + 40 \, x^{2} - 16 \, \sqrt{3}{\left (x^{3} + x\right )} - 8 \,{\left (2 \, x^{3} - \sqrt{3}{\left (2 \, x^{2} + 1\right )} + 4 \, x\right )} \sqrt{x^{2} + 1} + 16\right ) + \arctan \left (-\frac{\sqrt{2} x - \sqrt{2} \sqrt{x^{2} + 1}}{\sqrt{3} \sqrt{2} x + 2 \, \sqrt{2}{\left (x^{2} + 1\right )} - \sqrt{x^{2} + 1}{\left (2 \, \sqrt{2} x + \sqrt{3} \sqrt{2}\right )} + 2 \, \sqrt{2} \sqrt{2 \, x^{4} + 5 \, x^{2} + 2 \, \sqrt{3}{\left (x^{3} + x\right )} -{\left (2 \, x^{3} + \sqrt{3}{\left (2 \, x^{2} + 1\right )} + 4 \, x\right )} \sqrt{x^{2} + 1} + 2}}\right ) + \arctan \left (\frac{\sqrt{2} x - \sqrt{2} \sqrt{x^{2} + 1}}{\sqrt{3} \sqrt{2} x - 2 \, \sqrt{2}{\left (x^{2} + 1\right )} + \sqrt{x^{2} + 1}{\left (2 \, \sqrt{2} x - \sqrt{3} \sqrt{2}\right )} - 2 \, \sqrt{2} \sqrt{2 \, x^{4} + 5 \, x^{2} - 2 \, \sqrt{3}{\left (x^{3} + x\right )} -{\left (2 \, x^{3} - \sqrt{3}{\left (2 \, x^{2} + 1\right )} + 4 \, x\right )} \sqrt{x^{2} + 1} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arctan(sqrt(x^2 + 1)*x) - 1/4*sqrt(3)*log(16*x^4 + 40*x^2 + 16*sqrt(3)*(x^3 +
x) - 8*(2*x^3 + sqrt(3)*(2*x^2 + 1) + 4*x)*sqrt(x^2 + 1) + 16) + 1/4*sqrt(3)*log
(16*x^4 + 40*x^2 - 16*sqrt(3)*(x^3 + x) - 8*(2*x^3 - sqrt(3)*(2*x^2 + 1) + 4*x)*
sqrt(x^2 + 1) + 16) + arctan(-(sqrt(2)*x - sqrt(2)*sqrt(x^2 + 1))/(sqrt(3)*sqrt(
2)*x + 2*sqrt(2)*(x^2 + 1) - sqrt(x^2 + 1)*(2*sqrt(2)*x + sqrt(3)*sqrt(2)) + 2*s
qrt(2)*sqrt(2*x^4 + 5*x^2 + 2*sqrt(3)*(x^3 + x) - (2*x^3 + sqrt(3)*(2*x^2 + 1) +
4*x)*sqrt(x^2 + 1) + 2))) + arctan((sqrt(2)*x - sqrt(2)*sqrt(x^2 + 1))/(sqrt(3)
*sqrt(2)*x - 2*sqrt(2)*(x^2 + 1) + sqrt(x^2 + 1)*(2*sqrt(2)*x - sqrt(3)*sqrt(2))
- 2*sqrt(2)*sqrt(2*x^4 + 5*x^2 - 2*sqrt(3)*(x^3 + x) - (2*x^3 - sqrt(3)*(2*x^2
+ 1) + 4*x)*sqrt(x^2 + 1) + 2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \arctan \left (\sqrt{x^{2} + 1} x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(arctan(sqrt(x^2 + 1)*x), x)