### 3.188 $$\int \log (\cos (x)) \sin (x) \, dx$$

Optimal. Leaf size=10 $\cos (x)-\cos (x) \log (\cos (x))$

[Out]

Cos[x] - Cos[x]*Log[Cos[x]]

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Rubi [A]  time = 0.0078169, antiderivative size = 10, 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 = {2638, 2554} $\cos (x)-\cos (x) \log (\cos (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Cos[x] - Cos[x]*Log[Cos[x]]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[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 \log (\cos (x)) \sin (x) \, dx &=-\cos (x) \log (\cos (x))-\int \sin (x) \, dx\\ &=\cos (x)-\cos (x) \log (\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.007417, size = 10, normalized size = 1. $\cos (x)-\cos (x) \log (\cos (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Cos[x] - Cos[x]*Log[Cos[x]]

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Maple [A]  time = 0.004, size = 11, normalized size = 1.1 \begin{align*} \cos \left ( x \right ) -\cos \left ( x \right ) \ln \left ( \cos \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x)-cos(x)*ln(cos(x))

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Maxima [A]  time = 0.993473, size = 14, normalized size = 1.4 \begin{align*} -\cos \left (x\right ) \log \left (\cos \left (x\right )\right ) + \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)*log(cos(x)) + cos(x)

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Fricas [A]  time = 2.2622, size = 41, normalized size = 4.1 \begin{align*} -\cos \left (x\right ) \log \left (\cos \left (x\right )\right ) + \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)*log(cos(x)) + cos(x)

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Sympy [A]  time = 0.947031, size = 10, normalized size = 1. \begin{align*} - \log{\left (\cos{\left (x \right )} \right )} \cos{\left (x \right )} + \cos{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(cos(x))*cos(x) + cos(x)

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Giac [A]  time = 1.19456, size = 14, normalized size = 1.4 \begin{align*} -\cos \left (x\right ) \log \left (\cos \left (x\right )\right ) + \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)*log(cos(x)) + cos(x)