3.1.18 \(\int \frac {\sin (x)}{(1+\cos (x))^3} \, dx\) [18]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 = {2746, 32} \begin {gather*} \frac {1}{2 (\cos (x)+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(1 + Cos[x])^3,x]

[Out]

1/(2*(1 + Cos[x])^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]

Rule 2746

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])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.20 \begin {gather*} \frac {1}{8} \sec ^4\left (\frac {x}{2}\right ) \end {gather*}

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 = 0.04, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {1}{2 \left (\cos \left (x \right )+1\right )^{2}}\)	\(9\)
default	\(\frac {1}{2 \left (\cos \left (x \right )+1\right )^{2}}\)	\(9\)
risch	\(\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{i x}+1\right )^{4}}\)	\(17\)
norman	\(\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4}+\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} \end {gather*}

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 = 0.36, size = 14, normalized size = 1.40 \begin {gather*} \frac {1}{2 \, {\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \end {gather*}

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 = 0.26, size = 14, normalized size = 1.40 \begin {gather*} \frac {1}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} \end {gather*}

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 = 0.49, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} \end {gather*}

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

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Mupad [B]
time = 0.04, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2\,{\left (\cos \left (x\right )+1\right )}^2} \end {gather*}

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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