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 = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.01, size = 23, normalized size = 2.30 \[ \frac {2 \sin \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )} \]

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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fricas [A]  time = 0.41, size = 18, normalized size = 1.80 \[ -\frac {\cos \relax (x) - \sin \relax (x) + 1}{\cos \relax (x) + \sin \relax (x) + 1} \]

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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giac [A]  time = 1.04, size = 10, normalized size = 1.00 \[ -\frac {2}{\tan \left (\frac {1}{2} \, x\right ) + 1} \]

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)

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maple [A]  time = 0.02, size = 11, normalized size = 1.10 \[ -\frac {2}{\tan \left (\frac {x}{2}\right )+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 15, normalized size = 1.50 \[ -\frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1} \]

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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mupad [B]  time = 0.02, size = 10, normalized size = 1.00 \[ -\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x) + 1),x)

[Out]

-2/(tan(x/2) + 1)

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sympy [A]  time = 0.36, size = 8, normalized size = 0.80 \[ - \frac {2}{\tan {\left (\frac {x}{2} \right )} + 1} \]

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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