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]

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

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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]

-1/3*Cot[x/2]^3

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fricas [A]  time = 0.79, size = 22, normalized size = 1.38 \[ \frac {\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1}{3 \, {\left (\cos \relax (x) - 1\right )} \sin \relax (x)} \]

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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giac [A]  time = 0.98, size = 8, normalized size = 0.50 \[ -\frac {1}{3 \, \tan \left (\frac {1}{2} \, x\right )^{3}} \]

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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maple [A]  time = 0.07, size = 9, normalized size = 0.56 \[ -\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}} \]

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 = 0.30, size = 12, normalized size = 0.75 \[ -\frac {{\left (\cos \relax (x) + 1\right )}^{3}}{3 \, \sin \relax (x)^{3}} \]

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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mupad [B]  time = 0.35, size = 8, normalized size = 0.50 \[ -\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-sin(x)^2/(cos(x) - 1)^3,x)

[Out]

-cot(x/2)^3/3

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sympy [A]  time = 1.33, size = 10, normalized size = 0.62 \[ - \frac {1}{3 \tan ^{3}{\left (\frac {x}{2} \right )}} \]

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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