3.309 \(\int \frac {\log (\log (x) \sin (x))}{x^2} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\cot (x)}{x},x\right )+\text {Ei}(-\log (x))-\frac {\log (\log (x) \sin (x))}{x} \]

[Out]

Ei(-ln(x))-ln(ln(x)*sin(x))/x+Unintegrable(cot(x)/x,x)

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Rubi [A]  time = 0.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[Log[x]*Sin[x]]/x^2,x]

[Out]

ExpIntegralEi[-Log[x]] - Log[Log[x]*Sin[x]]/x + Defer[Int][Cot[x]/x, x]

Rubi steps

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

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Mathematica [A]  time = 2.01, size = 0, normalized size = 0.00 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[Log[x]*Sin[x]]/x^2,x]

[Out]

Integrate[Log[Log[x]*Sin[x]]/x^2, x]

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\log \relax (x) \sin \relax (x)\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x)*sin(x))/x^2,x, algorithm="fricas")

[Out]

integral(log(log(x)*sin(x))/x^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\log \relax (x) \sin \relax (x)\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x)*sin(x))/x^2,x, algorithm="giac")

[Out]

integrate(log(log(x)*sin(x))/x^2, x)

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maple [A]  time = 3.26, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\ln \relax (x ) \sin \relax (x )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x)*sin(x))/x^2,x)

[Out]

int(ln(ln(x)*sin(x))/x^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x {\left ({\rm Ei}\left (-\log \relax (x)\right ) + \overline {{\rm Ei}\left (-\log \relax (x)\right )}\right )} - 2 \, x \int \frac {\sin \relax (x)}{{\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )} x}\,{d x} + 2 \, x \int \frac {\sin \relax (x)}{{\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )} x}\,{d x} + 2 \, \log \relax (2) - \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 2 \, \log \left (\log \relax (x)\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x)*sin(x))/x^2,x, algorithm="maxima")

[Out]

1/2*(x*(Ei(-log(x)) + conjugate(Ei(-log(x)))) - 2*x*integrate(sin(x)/(x*cos(x)^2 + x*sin(x)^2 + 2*x*cos(x) + x
), x) + 2*x*integrate(sin(x)/(x*cos(x)^2 + x*sin(x)^2 - 2*x*cos(x) + x), x) + 2*log(2) - log(cos(x)^2 + sin(x)
^2 + 2*cos(x) + 1) - log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) - 2*log(log(x)))/x

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (\ln \relax (x)\,\sin \relax (x)\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(log(x)*sin(x))/x^2,x)

[Out]

int(log(log(x)*sin(x))/x^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\log {\relax (x )} \sin {\relax (x )} \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(ln(x)*sin(x))/x**2,x)

[Out]

Integral(log(log(x)*sin(x))/x**2, x)

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