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

Optimal. Leaf size=6 \[ \frac{1}{\cos (x)+1} \]

[Out]

(1 + Cos[x])^(-1)

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Rubi [A]  time = 0.0185305, antiderivative size = 6, 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 = {2667, 32} \[ \frac{1}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(1 + Cos[x])^2,x]

[Out]

(1 + Cos[x])^(-1)

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{(1+\cos (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=\frac{1}{1+\cos (x)}\\ \end{align*}

Mathematica [A]  time = 0.0085246, size = 12, normalized size = 2. \[ \frac{1}{2} \sec ^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(1 + Cos[x])^2,x]

[Out]

Sec[x/2]^2/2

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Maple [A]  time = 0.031, size = 7, normalized size = 1.2 \begin{align*} \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(cos(x)+1)

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Maxima [A]  time = 1.14017, size = 8, normalized size = 1.33 \begin{align*} \frac{1}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(cos(x) + 1)

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Fricas [A]  time = 1.54871, size = 22, normalized size = 3.67 \begin{align*} \frac{1}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(cos(x) + 1)

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Sympy [A]  time = 0.439105, size = 8, normalized size = 1.33 \begin{align*} - \frac{\cos{\left (x \right )}}{\cos{\left (x \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)/(cos(x) + 1)

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Giac [A]  time = 1.15179, size = 8, normalized size = 1.33 \begin{align*} \frac{1}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(cos(x) + 1)