3.21 \(\int \frac{1}{x \left (1+\log ^2(x)\right )} \, dx\)

Optimal. Leaf size=3 \[ \tan ^{-1}(\log (x)) \]

[Out]

ArcTan[Log[x]]

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Rubi [A]  time = 0.0316255, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \tan ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

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Rubi in Sympy [A]  time = 3.31803, size = 3, normalized size = 1. \[ \operatorname{atan}{\left (\log{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(log(x))

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Mathematica [A]  time = 0.00580001, size = 3, normalized size = 1. \[ \tan ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Log[x]]

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Maple [A]  time = 0.001, size = 4, normalized size = 1.3 \[ \arctan \left ( \ln \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(1+ln(x)^2),x)

[Out]

arctan(ln(x))

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Maxima [A]  time = 1.52938, size = 4, normalized size = 1.33 \[ \arctan \left (\log \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(log(x))

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Fricas [A]  time = 0.244666, size = 4, normalized size = 1.33 \[ \arctan \left (\log \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(log(x))

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Sympy [A]  time = 0.139617, size = 15, normalized size = 5. \[ \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (2 i + \log{\left (x \right )} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(1+ln(x)**2),x)

[Out]

RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + log(x))))

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GIAC/XCAS [A]  time = 0.200576, size = 4, normalized size = 1.33 \[ \arctan \left ({\rm ln}\left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(ln(x))