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

Optimal. Leaf size=31 \[ \text {Ei}(-\log (x))+\log (x)-\frac {\log \left (e^x \log (x) \sin (x)\right )}{x}+\text {Int}\left (\frac {\cot (x)}{x},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx &=-\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*}

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Mathematica [A]
time = 1.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left ({\mathrm e}^{x} \ln \left (x \right ) \sin \left (x \right )\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (e^{x} \log {\left (x \right )} \sin {\left (x \right )} \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left ({\mathrm {e}}^x\,\ln \left (x\right )\,\sin \left (x\right )\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(log(exp(x)*log(x)*sin(x))/x^2, x)

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