3.147 \(\int \frac {\sin (x)}{1+\cos (x)+\sin (x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {x}{2}-\frac {1}{2} \log \left (\tan \left (\frac {x}{2}\right )+1\right )-\frac {1}{2} \log (\sin (x)+\cos (x)+1) \]

[Out]

1/2*x-1/2*ln(1+cos(x)+sin(x))-1/2*ln(1+tan(1/2*x))

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3137, 3124, 31} \[ \frac {x}{2}-\frac {1}{2} \log \left (\tan \left (\frac {x}{2}\right )+1\right )-\frac {1}{2} \log (\sin (x)+\cos (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(1 + Cos[x] + Sin[x]),x]

[Out]

x/2 - Log[1 + Cos[x] + Sin[x]]/2 - Log[1 + Tan[x/2]]/2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rule 3137

Int[((A_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(
x_)]), x_Symbol] :> Simp[(c*C*(d + e*x))/(e*(b^2 + c^2)), x] + (Dist[(A*(b^2 + c^2) - a*c*C)/(b^2 + c^2), Int[
1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] - Simp[(b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2
+ c^2)), x]) /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*c*C, 0]

Rubi steps

\begin {align*} \int \frac {\sin (x)}{1+\cos (x)+\sin (x)} \, dx &=\frac {x}{2}-\frac {1}{2} \log (1+\cos (x)+\sin (x))-\frac {1}{2} \int \frac {1}{1+\cos (x)+\sin (x)} \, dx\\ &=\frac {x}{2}-\frac {1}{2} \log (1+\cos (x)+\sin (x))-\operatorname {Subst}\left (\int \frac {1}{2+2 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {x}{2}-\frac {1}{2} \log (1+\cos (x)+\sin (x))-\frac {1}{2} \log \left (1+\tan \left (\frac {x}{2}\right )\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 22, normalized size = 0.73 \[ \frac {x}{2}-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(1 + Cos[x] + Sin[x]),x]

[Out]

x/2 - Log[Cos[x/2] + Sin[x/2]]

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 11, normalized size = 0.37 \[ \frac {1}{2} \, x - \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/2*log(sin(x) + 1)

________________________________________________________________________________________

giac [A]  time = 0.02, size = 25, normalized size = 0.83 \[ \frac {1}{2} \, x + \frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 25, normalized size = 0.83 \[ \frac {x}{2}-\ln \left (\tan \left (\frac {x}{2}\right )+1\right )+\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.15, size = 41, normalized size = 1.37 \[ \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) - \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + \frac {1}{2} \, \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.31, size = 34, normalized size = 1.13 \[ -\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x) + sin(x) + 1),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.32, size = 22, normalized size = 0.73 \[ \frac {x}{2} - \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________