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)*ln(cos(x))

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 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]

Rule 2638

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

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \cos (x)-\cos (x) \log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x] - Cos[x]*Log[Cos[x]]

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fricas [A]  time = 0.47, size = 10, normalized size = 1.00 \[ -\cos \relax (x) \log \left (\cos \relax (x)\right ) + \cos \relax (x) \]

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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giac [A]  time = 0.17, size = 10, normalized size = 1.00 \[ -\cos \relax (x) \log \left (\cos \relax (x)\right ) + \cos \relax (x) \]

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)

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maple [A]  time = 0.18, size = 11, normalized size = 1.10 \[ -\cos \relax (x ) \ln \left (\cos \relax (x )\right )+\cos \relax (x ) \]

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 = 0.44, size = 10, normalized size = 1.00 \[ -\cos \relax (x) \log \left (\cos \relax (x)\right ) + \cos \relax (x) \]

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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mupad [B]  time = 0.38, size = 9, normalized size = 0.90 \[ -\cos \relax (x)\,\left (\ln \left (\cos \relax (x)\right )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)*(log(cos(x)) - 1)

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sympy [A]  time = 0.87, size = 10, normalized size = 1.00 \[ - \log {\left (\cos {\relax (x )} \right )} \cos {\relax (x )} + \cos {\relax (x )} \]

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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