\(\int \frac {\log (\log (x) \sin (x))}{x^2} \, dx\) [309]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\operatorname {ExpIntegralEi}(-\log (x))-\frac {\log (\log (x) \sin (x))}{x}+\text {Int}\left (\frac {\cot (x)}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int \frac {\log (\log (x) \sin (x))}{x^2} \, dx \]

[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*} \text {integral}& = -\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+\text {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*}

Mathematica [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int \frac {\log (\log (x) \sin (x))}{x^2} \, dx \]

[In]

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

[Out]

Integrate[Log[Log[x]*Sin[x]]/x^2, x]

Maple [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (\ln \left (x \right ) \sin \left (x \right )\right )}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int { \frac {\log \left (\log \left (x\right ) \sin \left (x\right )\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(log(log(x)*sin(x))/x^2, x)

Sympy [N/A]

Not integrable

Time = 20.61 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int \frac {\log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(log(log(x)*sin(x))/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 121, normalized size of antiderivative = 12.10 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int { \frac {\log \left (\log \left (x\right ) \sin \left (x\right )\right )}{x^{2}} \,d x } \]

[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

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int { \frac {\log \left (\log \left (x\right ) \sin \left (x\right )\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(log(log(x)*sin(x))/x^2, x)

Mupad [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\log (\log (x) \sin (x))}{x^2} \, dx=\int \frac {\ln \left (\ln \left (x\right )\,\sin \left (x\right )\right )}{x^2} \,d x \]

[In]

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

[Out]

int(log(log(x)*sin(x))/x^2, x)