3.35 \(\int \frac{1}{x (1+\sin (\log (x)))} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\cos (\log (x))}{\sin (\log (x))+1} \]

[Out]

-(Cos[Log[x]]/(1 + Sin[Log[x]]))

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Rubi [A]  time = 0.0289448, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2648} \[ -\frac{\cos (\log (x))}{\sin (\log (x))+1} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + Sin[Log[x]])),x]

[Out]

-(Cos[Log[x]]/(1 + Sin[Log[x]]))

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [B]  time = 0.0196309, size = 26, normalized size = 2.17 \[ \frac{2 \sin \left (\frac{\log (x)}{2}\right )}{\sin \left (\frac{\log (x)}{2}\right )+\cos \left (\frac{\log (x)}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + Sin[Log[x]])),x]

[Out]

(2*Sin[Log[x]/2])/(Cos[Log[x]/2] + Sin[Log[x]/2])

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Maple [A]  time = 0.017, size = 12, normalized size = 1. \begin{align*} -2\, \left ( 1+\tan \left ( 1/2\,\ln \left ( x \right ) \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(1+sin(ln(x))),x)

[Out]

-2/(1+tan(1/2*ln(x)))

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Maxima [A]  time = 1.09909, size = 23, normalized size = 1.92 \begin{align*} -\frac{2}{\frac{\sin \left (\log \left (x\right )\right )}{\cos \left (\log \left (x\right )\right ) + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sin(log(x))/(cos(log(x)) + 1) + 1)

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Fricas [A]  time = 2.02112, size = 89, normalized size = 7.42 \begin{align*} -\frac{\cos \left (\log \left (x\right )\right ) - \sin \left (\log \left (x\right )\right ) + 1}{\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(log(x)) - sin(log(x)) + 1)/(cos(log(x)) + sin(log(x)) + 1)

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Sympy [A]  time = 2.35437, size = 15, normalized size = 1.25 \begin{align*} \frac{2 \tan{\left (\frac{\log{\left (x \right )}}{2} \right )}}{\tan{\left (\frac{\log{\left (x \right )}}{2} \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+sin(ln(x))),x)

[Out]

2*tan(log(x)/2)/(tan(log(x)/2) + 1)

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Giac [A]  time = 1.11301, size = 15, normalized size = 1.25 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, \log \left (x\right )\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(tan(1/2*log(x)) + 1)