3.31 \(\int \frac{\tan ^{-1}(x)}{x^2 \sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

-((x-I)*(x+I))^(1/2)*arctan(x)/x+ln((1+I*x)/(x^2+1)^(1/2)-1)-ln(1+(1+I*x)/(x^2+1
)^(1/2))

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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]

-sqrt(x^2 + 1)*arctan(x)/x - arcsinh(1/abs(x))

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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]

-(x*log(-x + sqrt(x^2 + 1) + 1) - x*log(-x + sqrt(x^2 + 1) - 1) + sqrt(x^2 + 1)*
arctan(x))/x

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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]

-asinh(1/x) - sqrt(x**2 + 1)*atan(x)/x

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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]

2*arctan(x)/((x - sqrt(x^2 + 1))^2 - 1) + arctan(x) - ln(abs(-x + sqrt(x^2 + 1)
+ 1)) + ln(abs(-x + sqrt(x^2 + 1) - 1))