3.3 \(\int \frac{1}{(1+\cos (x))^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{\sin (x)}{3 (\cos (x)+1)}+\frac{\sin (x)}{3 (\cos (x)+1)^2} \]

[Out]

Sin[x]/(3*(1 + Cos[x])^2) + Sin[x]/(3*(1 + Cos[x]))

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Rubi [A]  time = 0.013607, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2650, 2648} \[ \frac{\sin (x)}{3 (\cos (x)+1)}+\frac{\sin (x)}{3 (\cos (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x])^(-2),x]

[Out]

Sin[x]/(3*(1 + Cos[x])^2) + Sin[x]/(3*(1 + Cos[x]))

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

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))^2} \, dx &=\frac{\sin (x)}{3 (1+\cos (x))^2}+\frac{1}{3} \int \frac{1}{1+\cos (x)} \, dx\\ &=\frac{\sin (x)}{3 (1+\cos (x))^2}+\frac{\sin (x)}{3 (1+\cos (x))}\\ \end{align*}

Mathematica [A]  time = 0.0088695, size = 16, normalized size = 0.64 \[ \frac{\sin (x) (\cos (x)+2)}{3 (\cos (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x])^(-2),x]

[Out]

((2 + Cos[x])*Sin[x])/(3*(1 + Cos[x])^2)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.6 \begin{align*}{\frac{1}{6} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{1}{2}\tan \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x))^2,x)

[Out]

1/6*tan(1/2*x)^3+1/2*tan(1/2*x)

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Maxima [A]  time = 0.964996, size = 31, normalized size = 1.24 \begin{align*} \frac{\sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{\sin \left (x\right )^{3}}{6 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x))^2,x, algorithm="maxima")

[Out]

1/2*sin(x)/(cos(x) + 1) + 1/6*sin(x)^3/(cos(x) + 1)^3

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Fricas [A]  time = 1.11919, size = 69, normalized size = 2.76 \begin{align*} \frac{{\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right )}{3 \,{\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x))^2,x, algorithm="fricas")

[Out]

1/3*(cos(x) + 2)*sin(x)/(cos(x)^2 + 2*cos(x) + 1)

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Sympy [A]  time = 0.408357, size = 14, normalized size = 0.56 \begin{align*} \frac{\tan ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{\tan{\left (\frac{x}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x))**2,x)

[Out]

tan(x/2)**3/6 + tan(x/2)/2

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Giac [A]  time = 1.10102, size = 20, normalized size = 0.8 \begin{align*} \frac{1}{6} \, \tan \left (\frac{1}{2} \, x\right )^{3} + \frac{1}{2} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x))^2,x, algorithm="giac")

[Out]

1/6*tan(1/2*x)^3 + 1/2*tan(1/2*x)