3.307 \(\int \log (\log (x) \sin (x)) \, dx\)

Optimal. Leaf size=52 \[ \frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )-\text{li}(x)+\frac{i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]

[Out]

(I/2)*x^2 - x*Log[1 - E^((2*I)*x)] + x*Log[Log[x]*Sin[x]] - LogIntegral[x] + (I/2)*PolyLog[2, E^((2*I)*x)]

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Rubi [A]  time = 0.0615273, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2549, 3717, 2190, 2279, 2391, 2298} \[ \frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )-\text{li}(x)+\frac{i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Log[Log[x]*Sin[x]],x]

[Out]

(I/2)*x^2 - x*Log[1 - E^((2*I)*x)] + x*Log[Log[x]*Sin[x]] - LogIntegral[x] + (I/2)*PolyLog[2, E^((2*I)*x)]

Rule 2549

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[
u]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin{align*} \int \log (\log (x) \sin (x)) \, dx &=x \log (\log (x) \sin (x))-\int \left (x \cot (x)+\frac{1}{\log (x)}\right ) \, dx\\ &=x \log (\log (x) \sin (x))-\int x \cot (x) \, dx-\int \frac{1}{\log (x)} \, dx\\ &=\frac{i x^2}{2}+x \log (\log (x) \sin (x))-\text{li}(x)+2 i \int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx\\ &=\frac{i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text{li}(x)+\int \log \left (1-e^{2 i x}\right ) \, dx\\ &=\frac{i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text{li}(x)-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac{i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text{li}(x)+\frac{1}{2} i \text{Li}_2\left (e^{2 i x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0307747, size = 47, normalized size = 0.9 \[ \frac{1}{2} i \left (x^2+\text{PolyLog}\left (2,e^{2 i x}\right )\right )-\text{li}(x)-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[Log[x]*Sin[x]],x]

[Out]

-(x*Log[1 - E^((2*I)*x)]) + x*Log[Log[x]*Sin[x]] - LogIntegral[x] + (I/2)*(x^2 + PolyLog[2, E^((2*I)*x)])

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Maple [C]  time = 0.167, size = 368, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x)*sin(x)),x)

[Out]

-x*ln(exp(I*x))-1/2*I*Pi*x+1/2*I*Pi*csgn(I*(exp(2*I*x)-1))*csgn(I*ln(x)*(exp(2*I*x)-1))^2*x-1/2*I*Pi*csgn(I*ln
(x)*(exp(2*I*x)-1))^3*x+1/2*I*x^2+I*ln(exp(I*x))*ln(exp(I*x)+1)-I*dilog(exp(I*x))-ln(2)*x+1/2*I*Pi*csgn(I*exp(
-I*x))*csgn(I*ln(x)*(exp(2*I*x)-1))*csgn(ln(x)*sin(x))*x+1/2*I*Pi*csgn(I*exp(-I*x))*csgn(ln(x)*sin(x))^2*x+1/2
*I*Pi*csgn(I*ln(x)*sin(x))^2*x+1/2*I*Pi*csgn(I*ln(x)*(exp(2*I*x)-1))*csgn(ln(x)*sin(x))^2*x-I*ln(exp(I*x))*ln(
exp(2*I*x)-1)-1/2*I*Pi*csgn(I*(exp(2*I*x)-1))*csgn(I*ln(x))*csgn(I*ln(x)*(exp(2*I*x)-1))*x+I*dilog(exp(I*x)+1)
+1/2*I*Pi*csgn(ln(x)*sin(x))^3*x+1/2*I*Pi*csgn(I*ln(x))*csgn(I*ln(x)*(exp(2*I*x)-1))^2*x-1/2*I*Pi*csgn(I*ln(x)
*sin(x))^3*x+1/2*I*Pi*csgn(ln(x)*sin(x))*csgn(I*ln(x)*sin(x))*x-1/2*I*Pi*csgn(ln(x)*sin(x))*csgn(I*ln(x)*sin(x
))^2*x+x*ln(ln(x))+Ei(1,-ln(x))

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Maxima [A]  time = 1.94638, size = 58, normalized size = 1.12 \begin{align*} \frac{1}{2} \,{\left (i \, \pi - 2 \, \log \left (2\right )\right )} x - \frac{1}{2} i \, x^{2} + x \log \left (\log \left (x\right )\right ) -{\rm Ei}\left (\log \left (x\right )\right ) + i \,{\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \,{\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(I*pi - 2*log(2))*x - 1/2*I*x^2 + x*log(log(x)) - Ei(log(x)) + I*dilog(-e^(I*x)) + I*dilog(e^(I*x))

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Fricas [B]  time = 2.3886, size = 427, normalized size = 8.21 \begin{align*} x \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac{1}{2} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \logintegral \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(log(x)*sin(x)) - 1/2*x*log(cos(x) + I*sin(x) + 1) - 1/2*x*log(cos(x) - I*sin(x) + 1) - 1/2*x*log(-cos(x)
+ I*sin(x) + 1) - 1/2*x*log(-cos(x) - I*sin(x) + 1) + 1/2*I*dilog(cos(x) + I*sin(x)) - 1/2*I*dilog(cos(x) - I
*sin(x)) - 1/2*I*dilog(-cos(x) + I*sin(x)) + 1/2*I*dilog(-cos(x) - I*sin(x)) - log_integral(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\log{\left (x \right )} \sin{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(log(x)*sin(x)), x)

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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(log(x)*sin(x)),x, algorithm="giac")

[Out]

Exception raised: TypeError