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

Optimal. Leaf size=16 \[ -\frac{\sin ^3(x)}{3 (1-\cos (x))^3} \]

[Out]

-Sin[x]^3/(3*(1 - Cos[x])^3)

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Rubi [A]  time = 0.0308387, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2671} \[ -\frac{\sin ^3(x)}{3 (1-\cos (x))^3} \]

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 = 0.0293909, size = 12, normalized size = 0.75 \[ -\frac{1}{3} \cot ^3\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(1 - Cos[x])^3,x]

[Out]

-Cot[x/2]^3/3

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Maple [A]  time = 0.06, 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/(1-cos(x))^3,x)

[Out]

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

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Maxima [A]  time = 1.3989, size = 16, normalized size = 1. \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )}^{3}}{3 \, \sin \left (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="maxima")

[Out]

-1/3*(cos(x) + 1)^3/sin(x)^3

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Fricas [A]  time = 1.56474, size = 72, normalized size = 4.5 \begin{align*} \frac{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1}{3 \,{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\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)^2 + 2*cos(x) + 1)/((cos(x) - 1)*sin(x))

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Sympy [A]  time = 2.27375, size = 10, normalized size = 0.62 \begin{align*} - \frac{1}{3 \tan ^{3}{\left (\frac{x}{2} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15603, size = 11, normalized size = 0.69 \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