### 3.190 $$\int \cos (x) \log (\sin (x)) \, dx$$

Optimal. Leaf size=11 $\sin (x) \log (\sin (x))-\sin (x)$

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0093442, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {2637, 2554} $\sin (x) \log (\sin (x))-\sin (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

\begin{align*} \int \cos (x) \log (\sin (x)) \, dx &=\log (\sin (x)) \sin (x)-\int \cos (x) \, dx\\ &=-\sin (x)+\log (\sin (x)) \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0025112, size = 11, normalized size = 1. $\sin (x) \log (\sin (x))-\sin (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 12, normalized size = 1.1 \begin{align*} -\sin \left ( x \right ) +\ln \left ( \sin \left ( x \right ) \right ) \sin \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(x)+ln(sin(x))*sin(x)

________________________________________________________________________________________

Maxima [A]  time = 1.00768, size = 15, normalized size = 1.36 \begin{align*} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

________________________________________________________________________________________

Fricas [A]  time = 2.12528, size = 39, normalized size = 3.55 \begin{align*} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

________________________________________________________________________________________

Sympy [A]  time = 0.946213, size = 10, normalized size = 0.91 \begin{align*} \log{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )} - \sin{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x))*sin(x) - sin(x)

________________________________________________________________________________________

Giac [A]  time = 1.17447, size = 15, normalized size = 1.36 \begin{align*} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*log(sin(x)),x, algorithm="giac")

[Out]

log(sin(x))*sin(x) - sin(x)