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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {203} \[ \tan ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\log (x)\right )\\ &=\tan ^{-1}(\log (x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 3, normalized size = 1.00 \[ \arctan \left (\log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x)^2),x, algorithm="fricas")

[Out]

arctan(log(x))

________________________________________________________________________________________

giac [A]  time = 0.93, size = 3, normalized size = 1.00 \[ \arctan \left (\log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x)^2),x, algorithm="giac")

[Out]

arctan(log(x))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 4, normalized size = 1.33

method	result	size
derivativedivides	\(\arctan \left (\ln \relax (x )\right )\)	\(4\)
default	\(\arctan \left (\ln \relax (x )\right )\)	\(4\)
risch	\(\frac {i \ln \left (\ln \relax (x )+i\right )}{2}-\frac {i \ln \left (\ln \relax (x )-i\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(1+ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

arctan(ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.96, size = 3, normalized size = 1.00 \[ \arctan \left (\log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x)^2),x, algorithm="maxima")

[Out]

arctan(log(x))

________________________________________________________________________________________

mupad [B]  time = 0.34, size = 3, normalized size = 1.00 \[ \mathrm {atan}\left (\ln \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(log(x)^2 + 1)),x)

[Out]

atan(log(x))

________________________________________________________________________________________

sympy [B]  time = 0.14, size = 15, normalized size = 5.00 \[ \operatorname {RootSum} {\left (4 z^{2} + 1, \left (i \mapsto i \log {\left (2 i + \log {\relax (x )} \right )} \right )\right )} \]

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))))

________________________________________________________________________________________