Optimal. Leaf size=29 \[ -\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]
[Out]
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Rubi [A] time = 0.0824241, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[ArcTan[x]/(x^2*Sqrt[1 + x^2]),x]
[Out]
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Rubi in Sympy [A] time = 5.63885, size = 24, normalized size = 0.83 \[ - \operatorname{atanh}{\left (\sqrt{x^{2} + 1} \right )} - \frac{\sqrt{x^{2} + 1} \operatorname{atan}{\left (x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(atan(x)/x**2/(x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0291508, size = 33, normalized size = 1.14 \[ -\log \left (\sqrt{x^2+1}+1\right )-\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{x}+\log (x) \]
Antiderivative was successfully verified.
[In] Integrate[ArcTan[x]/(x^2*Sqrt[1 + x^2]),x]
[Out]
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Maple [C] time = 0.129, size = 56, normalized size = 1.9 \[ -{\frac{\arctan \left ( x \right ) }{x}\sqrt{ \left ( x-i \right ) \left ( x+i \right ) }}+\ln \left ({(1+ix){\frac{1}{\sqrt{{x}^{2}+1}}}}-1 \right ) -\ln \left ( 1+{(1+ix){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(arctan(x)/x^2/(x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.64557, size = 30, normalized size = 1.03 \[ -\frac{\sqrt{x^{2} + 1} \arctan \left (x\right )}{x} - \operatorname{arsinh}\left (\frac{1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(arctan(x)/(sqrt(x^2 + 1)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246935, size = 63, normalized size = 2.17 \[ -\frac{x \log \left (-x + \sqrt{x^{2} + 1} + 1\right ) - x \log \left (-x + \sqrt{x^{2} + 1} - 1\right ) + \sqrt{x^{2} + 1} \arctan \left (x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(arctan(x)/(sqrt(x^2 + 1)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 50.9079, size = 19, normalized size = 0.66 \[ - \operatorname{asinh}{\left (\frac{1}{x} \right )} - \frac{\sqrt{x^{2} + 1} \operatorname{atan}{\left (x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(atan(x)/x**2/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.202946, size = 73, normalized size = 2.52 \[ \frac{2 \, \arctan \left (x\right )}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} + \arctan \left (x\right ) -{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} + 1 \right |}\right ) +{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(arctan(x)/(sqrt(x^2 + 1)*x^2),x, algorithm="giac")
[Out]