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

Optimal. Leaf size=11 \[ x+\frac{\cos (x)}{\sin (x)+1} \]

[Out]

x + Cos[x]/(1 + Sin[x])

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Rubi [A]  time = 0.0218378, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2735, 2648} \[ x+\frac{\cos (x)}{\sin (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(1 + Sin[x]),x]

[Out]

x + Cos[x]/(1 + Sin[x])

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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]

Rubi steps

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

Mathematica [B]  time = 0.0394707, size = 25, normalized size = 2.27 \[ x-\frac{2 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(1 + Sin[x]),x]

[Out]

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

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Maple [A]  time = 0.011, size = 13, normalized size = 1.2 \begin{align*} 2\, \left ( 1+\tan \left ( x/2 \right ) \right ) ^{-1}+x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.44419, size = 38, normalized size = 3.45 \begin{align*} \frac{2}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sin(x)/(cos(x) + 1) + 1) + 2*arctan(sin(x)/(cos(x) + 1))

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Fricas [B]  time = 1.96256, size = 88, normalized size = 8. \begin{align*} \frac{{\left (x + 1\right )} \cos \left (x\right ) +{\left (x - 1\right )} \sin \left (x\right ) + x + 1}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x + 1)*cos(x) + (x - 1)*sin(x) + x + 1)/(cos(x) + sin(x) + 1)

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Sympy [B]  time = 0.53787, size = 34, normalized size = 3.09 \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{2 \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(sin(x)/(1+sin(x)),x)

[Out]

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

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Giac [A]  time = 1.06276, size = 16, normalized size = 1.45 \begin{align*} x + \frac{2}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2/(tan(1/2*x) + 1)