3.1.20 \(\int \frac {\sin ^2(x)}{(1+\cos (x))^3} \, dx\) [20]

Optimal. Leaf size=14 \[ \frac {\sin ^3(x)}{3 (1+\cos (x))^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2750} \begin {gather*} \frac {\sin ^3(x)}{3 (\cos (x)+1)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(1 + Cos[x])^3,x]

[Out]

Sin[x]^3/(3*(1 + Cos[x])^3)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{3} \tan ^3\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(1 + Cos[x])^3,x]

[Out]

Tan[x/2]^3/3

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Maple [A]
time = 0.06, size = 9, normalized size = 0.64

method	result	size
default	\(\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}\)	\(9\)
risch	\(-\frac {2 i \left (3 \,{\mathrm e}^{2 i x}+1\right )}{3 \left ({\mathrm e}^{i x}+1\right )^{3}}\)	\(22\)
norman	\(\frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3}+\frac {\left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(cos(x)+1)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 12, normalized size = 0.86 \begin {gather*} \frac {\sin \left (x\right )^{3}}{3 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(1+cos(x))^3,x, algorithm="maxima")

[Out]

1/3*sin(x)^3/(cos(x) + 1)^3

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Fricas [A]
time = 0.35, size = 20, normalized size = 1.43 \begin {gather*} -\frac {{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}{3 \, {\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)^2/(1+cos(x))^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.34, size = 7, normalized size = 0.50 \begin {gather*} \frac {\tan ^{3}{\left (\frac {x}{2} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(1+cos(x))**3,x)

[Out]

tan(x/2)**3/3

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Giac [A]
time = 0.42, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{3} \, \tan \left (\frac {1}{2} \, x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(1+cos(x))^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.26, size = 8, normalized size = 0.57 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(cos(x) + 1)^3,x)

[Out]

tan(x/2)^3/3

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