3.40 \(\int \frac{\log (\sin (x))}{1+\sin (x)} \, dx\)

Optimal. Leaf size=22 \[ -x-\tanh ^{-1}(\cos (x))-\frac{\cos (x) \log (\sin (x))}{\sin (x)+1} \]

[Out]

-x - ArcTanh[Cos[x]] - (Cos[x]*Log[Sin[x]])/(1 + Sin[x])

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Rubi [A]  time = 0.0713951, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2648, 2554, 2839, 3770, 8} \[ -x-\tanh ^{-1}(\cos (x))-\frac{\cos (x) \log (\sin (x))}{\sin (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Log[Sin[x]]/(1 + Sin[x]),x]

[Out]

-x - ArcTanh[Cos[x]] - (Cos[x]*Log[Sin[x]])/(1 + Sin[x])

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

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 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3770

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\log (\sin (x))}{1+\sin (x)} \, dx &=-\frac{\cos (x) \log (\sin (x))}{1+\sin (x)}+\int \frac{\cos (x) \cot (x)}{1+\sin (x)} \, dx\\ &=-\frac{\cos (x) \log (\sin (x))}{1+\sin (x)}-\int 1 \, dx+\int \csc (x) \, dx\\ &=-x-\tanh ^{-1}(\cos (x))-\frac{\cos (x) \log (\sin (x))}{1+\sin (x)}\\ \end{align*}

Mathematica [A]  time = 0.0421438, size = 39, normalized size = 1.77 \[ -x-2 \log \left (\cos \left (\frac{x}{2}\right )\right )+\frac{2 \sin \left (\frac{x}{2}\right ) \log (\sin (x))}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[Sin[x]]/(1 + Sin[x]),x]

[Out]

-x - 2*Log[Cos[x/2]] + (2*Log[Sin[x]]*Sin[x/2])/(Cos[x/2] + Sin[x/2])

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Maple [B]  time = 0.093, size = 54, normalized size = 2.5 \begin{align*}{ \left ( -x-x\tan \left ({\frac{x}{2}} \right ) +2\,\tan \left ( x/2 \right ) \ln \left ( 2\,{\frac{\tan \left ( x/2 \right ) }{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1}} \right ) \right ) \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(sin(x))/(1+sin(x)),x)

[Out]

(-x-x*tan(1/2*x)+2*tan(1/2*x)*ln(2*tan(1/2*x)/(tan(1/2*x)^2+1)))/(1+tan(1/2*x))+ln(tan(1/2*x)^2+1)

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Maxima [B]  time = 1.43296, size = 111, normalized size = 5.05 \begin{align*} -\frac{2 \, \log \left (\frac{2 \, \sin \left (x\right )}{{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}}\right )}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) + 2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) - \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(sin(x))/(1+sin(x)),x, algorithm="maxima")

[Out]

-2*log(2*sin(x)/((sin(x)^2/(cos(x) + 1)^2 + 1)*(cos(x) + 1)))/(sin(x)/(cos(x) + 1) + 1) - 2*arctan(sin(x)/(cos
(x) + 1)) + 2*log(sin(x)/(cos(x) + 1)) - log(sin(x)^2/(cos(x) + 1)^2 + 1)

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Fricas [B]  time = 2.03369, size = 365, normalized size = 16.59 \begin{align*} -\frac{4 \,{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \arctan \left (-\frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1}\right ) + 4 \, x \cos \left (x\right ) +{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \,{\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )} \log \left (\sin \left (x\right )\right ) + 4 \, x \sin \left (x\right ) + 4 \, x}{2 \,{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(sin(x))/(1+sin(x)),x, algorithm="fricas")

[Out]

-1/2*(4*(cos(x) + sin(x) + 1)*arctan(-(cos(x) + sin(x) + 1)/(cos(x) - sin(x) + 1)) + 4*x*cos(x) + (cos(x) + si
n(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + sin(x) + 1)*log(-1/2*cos(x) + 1/2) + 2*(cos(x) - sin(x) + 1)*log(s
in(x)) + 4*x*sin(x) + 4*x)/(cos(x) + sin(x) + 1)

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Sympy [B]  time = 1.67584, size = 97, normalized size = 4.41 \begin{align*} - \frac{x \tan{\left (\frac{x}{2} \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} - \frac{x}{\tan{\left (\frac{x}{2} \right )} + 1} - \frac{\log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} \tan{\left (\frac{x}{2} \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} + \frac{\log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} + \frac{2 \log{\left (\tan{\left (\frac{x}{2} \right )} \right )} \tan{\left (\frac{x}{2} \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} + \frac{2 \log{\left (2 \right )} \tan{\left (\frac{x}{2} \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(sin(x))/(1+sin(x)),x)

[Out]

-x*tan(x/2)/(tan(x/2) + 1) - x/(tan(x/2) + 1) - log(tan(x/2)**2 + 1)*tan(x/2)/(tan(x/2) + 1) + log(tan(x/2)**2
+ 1)/(tan(x/2) + 1) + 2*log(tan(x/2))*tan(x/2)/(tan(x/2) + 1) + 2*log(2)*tan(x/2)/(tan(x/2) + 1)

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Giac [A]  time = 1.14278, size = 49, normalized size = 2.23 \begin{align*} -x - \frac{2 \, \log \left (\sin \left (x\right )\right )}{\tan \left (\frac{1}{2} \, x\right ) + 1} - 2 \, \log \left (\tan \left (\frac{1}{4} \, x\right )^{2} + 1\right ) + 2 \, \log \left ({\left | \tan \left (\frac{1}{4} \, x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(sin(x))/(1+sin(x)),x, algorithm="giac")

[Out]

-x - 2*log(sin(x))/(tan(1/2*x) + 1) - 2*log(tan(1/4*x)^2 + 1) + 2*log(abs(tan(1/4*x)))