### 3.314 $$\int \frac{\log (e^x \log (x) \sin (x))}{x^2} \, dx$$

Optimal. Leaf size=30 $\text{Unintegrable}\left (\frac{\cot (x)}{x},x\right )+\text{Ei}(-\log (x))+\log (x)-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x}$

[Out]

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

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Rubi [A]  time = 0.0688721, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx$

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+\operatorname{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 [A]  time = 2.26533, size = 0, normalized size = 0. $\int \frac{\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx$

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 = 0.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ({{\rm e}^{x}}\ln \left ( x \right ) \sin \left ( x \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \frac{x{\left ({\rm Ei}\left (-\log \left (x\right )\right ) + \overline{{\rm Ei}\left (-\log \left (x\right )\right )}\right )} - 2 \, x \int \frac{\sin \left (x\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )} x}\,{d x} + 2 \, x \int \frac{\sin \left (x\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )} x}\,{d x} + 2 \, x \log \left (x\right ) + 2 \, \log \left (2\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 \, \log \left (\log \left (x\right )\right )}{2 \, x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (e^{x} \log \left (x\right ) \sin \left (x\right )\right )}{x^{2}}, x\right ) \end{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError