3.20 \(\int \frac{\sin ^2(x)}{(1+\cos (x))^3} \, dx\)

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

[Out]

Sin[x]^3/(3*(1 + Cos[x])^3)

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Rubi [A]  time = 0.0308714, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {2671} \[ \frac{\sin ^3(x)}{3 (\cos (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(1 + Cos[x])^3,x]

[Out]

Sin[x]^3/(3*(1 + Cos[x])^3)

Rule 2671

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[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*}

Mathematica [A]  time = 0.0280643, size = 12, normalized size = 0.86 \[ \frac{1}{3} \tan ^3\left (\frac{x}{2}\right ) \]

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 = 0.046, size = 9, normalized size = 0.6 \begin{align*}{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.60098, size = 16, normalized size = 1.14 \begin{align*} \frac{\sin \left (x\right )^{3}}{3 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} \end{align*}

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 = 1.53684, size = 70, normalized size = 5. \begin{align*} -\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{align*}

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 = 1.60673, size = 7, normalized size = 0.5 \begin{align*} \frac{\tan ^{3}{\left (\frac{x}{2} \right )}}{3} \end{align*}

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 = 1.16325, size = 11, normalized size = 0.79 \begin{align*} \frac{1}{3} \, \tan \left (\frac{1}{2} \, x\right )^{3} \end{align*}

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