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

Optimal. Leaf size=9 \[ \frac{\sin (x)}{\cos (x)+1} \]

[Out]

Sin[x]/(1 + Cos[x])

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Rubi [A]  time = 0.0075139, antiderivative size = 9, 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} \[ \frac{\sin (x)}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x])^(-1),x]

[Out]

Sin[x]/(1 + Cos[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+\cos (x)} \, dx &=\frac{\sin (x)}{1+\cos (x)}\\ \end{align*}

Mathematica [A]  time = 0.0036081, size = 6, normalized size = 0.67 \[ \tan \left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x])^(-1),x]

[Out]

Tan[x/2]

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Maple [A]  time = 0., size = 5, normalized size = 0.6 \begin{align*} \tan \left ({\frac{x}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(1/2*x)

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Maxima [A]  time = 0.918008, size = 12, normalized size = 1.33 \begin{align*} \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(cos(x) + 1)

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Fricas [A]  time = 1.94506, size = 28, normalized size = 3.11 \begin{align*} \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(cos(x) + 1)

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Sympy [A]  time = 0.179115, size = 3, normalized size = 0.33 \begin{align*} \tan{\left (\frac{x}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x/2)

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Giac [B]  time = 1.10891, size = 41, normalized size = 4.56 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{{\left (x^{2} + 1\right )}{\left (\frac{x^{2} - 1}{x^{2} + 1} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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