\(\int \frac {\log (e^x \log (x) \sin (x))}{x^2} \, dx\) [314]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\operatorname {ExpIntegralEi}(-\log (x))+\log (x)-\frac {\log \left (e^x \log (x) \sin (x)\right )}{x}+\text {Int}\left (\frac {\cot (x)}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 13, 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 = {} \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.60 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int { \frac {\log \left (e^{x} \log \left (x\right ) \sin \left (x\right )\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(log(e^x*log(x)*sin(x))/x^2, x)

Sympy [N/A]

Not integrable

Time = 59.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int \frac {\log {\left (e^{x} \log {\left (x \right )} \sin {\left (x \right )} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(log(exp(x)*log(x)*sin(x))/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 126, normalized size of antiderivative = 9.69 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int { \frac {\log \left (e^{x} \log \left (x\right ) \sin \left (x\right )\right )}{x^{2}} \,d x } \]

[In]

integrate(log(exp(x)*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*x*log(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 = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int { \frac {\log \left (e^{x} \log \left (x\right ) \sin \left (x\right )\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(log(e^x*log(x)*sin(x))/x^2, x)

Mupad [N/A]

Not integrable

Time = 1.61 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx=\int \frac {\ln \left ({\mathrm {e}}^x\,\ln \left (x\right )\,\sin \left (x\right )\right )}{x^2} \,d x \]

[In]

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

[Out]

int(log(exp(x)*log(x)*sin(x))/x^2, x)