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

Optimal. Leaf size=10 \[ -\frac{\cos (x)}{\sin (x)+1} \]

[Out]

-(Cos[x]/(1 + Sin[x]))

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Rubi [A]  time = 0.005755, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2648} \[ -\frac{\cos (x)}{\sin (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sin[x])^(-1),x]

[Out]

-(Cos[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]

Rubi steps

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

Mathematica [B]  time = 0.009546, size = 23, normalized size = 2.3 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sin[x])^(-1),x]

[Out]

(2*Sin[x/2])/(Cos[x/2] + Sin[x/2])

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Maple [A]  time = 0.006, size = 11, normalized size = 1.1 \begin{align*} -2\, \left ( 1+\tan \left ( x/2 \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.938928, size = 20, normalized size = 2. \begin{align*} -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sin(x)/(cos(x) + 1) + 1)

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Fricas [A]  time = 1.66611, size = 62, normalized size = 6.2 \begin{align*} -\frac{\cos \left (x\right ) - \sin \left (x\right ) + 1}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.376199, size = 8, normalized size = 0.8 \begin{align*} - \frac{2}{\tan{\left (\frac{x}{2} \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(tan(x/2) + 1)

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Giac [A]  time = 1.07258, size = 14, normalized size = 1.4 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(tan(1/2*x) + 1)