3.18 \(\int \frac{\sin (x)}{(1+\cos (x))^3} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{2 (\cos (x)+1)^2} \]

[Out]

1/(2*(1 + Cos[x])^2)

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Rubi [A]  time = 0.0189541, antiderivative size = 10, 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} \[ \frac{1}{2 (\cos (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(1 + Cos[x])^3,x]

[Out]

1/(2*(1 + Cos[x])^2)

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))^3} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(1+x)^3} \, dx,x,\cos (x)\right )\\ &=\frac{1}{2 (1+\cos (x))^2}\\ \end{align*}

Mathematica [A]  time = 0.0077746, size = 12, normalized size = 1.2 \[ \frac{1}{8} \sec ^4\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(1 + Cos[x])^3,x]

[Out]

Sec[x/2]^4/8

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Maple [A]  time = 0.029, size = 9, normalized size = 0.9 \begin{align*}{\frac{1}{2\, \left ( \cos \left ( x \right ) +1 \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x)+1)^3,x)

[Out]

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

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Maxima [A]  time = 1.25347, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{2 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))^3,x, algorithm="maxima")

[Out]

1/2/(cos(x) + 1)^2

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Fricas [A]  time = 1.5844, size = 42, normalized size = 4.2 \begin{align*} \frac{1}{2 \,{\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(sin(x)/(1+cos(x))^3,x, algorithm="fricas")

[Out]

1/2/(cos(x)^2 + 2*cos(x) + 1)

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Sympy [A]  time = 0.945813, size = 14, normalized size = 1.4 \begin{align*} \frac{1}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))**3,x)

[Out]

1/(2*cos(x)**2 + 4*cos(x) + 2)

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Giac [A]  time = 1.14107, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{2 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))^3,x, algorithm="giac")

[Out]

1/2/(cos(x) + 1)^2