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

Optimal. Leaf size=20 \[ -\frac{2}{1-\cos (x)}-\log (1-\cos (x)) \]

[Out]

-2/(1 - Cos[x]) - Log[1 - Cos[x]]

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Rubi [A]  time = 0.0372703, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ -\frac{2}{1-\cos (x)}-\log (1-\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(1 - Cos[x])^3,x]

[Out]

-2/(1 - Cos[x]) - Log[1 - Cos[x]]

Rule 2667

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])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0110897, size = 29, normalized size = 1.45 \[ -\cot ^2\left (\frac{x}{2}\right )-2 \log \left (\tan \left (\frac{x}{2}\right )\right )-2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(1 - Cos[x])^3,x]

[Out]

-Cot[x/2]^2 - 2*Log[Cos[x/2]] - 2*Log[Tan[x/2]]

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Maple [A]  time = 0.064, size = 17, normalized size = 0.9 \begin{align*} -\ln \left ( -1+\cos \left ( x \right ) \right ) +2\, \left ( -1+\cos \left ( x \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(1-cos(x))^3,x)

[Out]

-ln(-1+cos(x))+2/(-1+cos(x))

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Maxima [A]  time = 1.51156, size = 22, normalized size = 1.1 \begin{align*} \frac{2}{\cos \left (x\right ) - 1} - \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(cos(x) - 1) - log(cos(x) - 1)

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Fricas [A]  time = 1.61262, size = 77, normalized size = 3.85 \begin{align*} -\frac{{\left (\cos \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2}{\cos \left (x\right ) - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(1-cos(x))^3,x, algorithm="fricas")

[Out]

-((cos(x) - 1)*log(-1/2*cos(x) + 1/2) - 2)/(cos(x) - 1)

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Sympy [B]  time = 0.89091, size = 126, normalized size = 6.3 \begin{align*} - \frac{2 \log{\left (\cos{\left (x \right )} - 1 \right )} \cos ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} + \frac{4 \log{\left (\cos{\left (x \right )} - 1 \right )} \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} - \frac{2 \log{\left (\cos{\left (x \right )} - 1 \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} - \frac{\sin ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} + \frac{2 \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} - \frac{2}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(cos(x) - 1)*cos(x)**2/(2*cos(x)**2 - 4*cos(x) + 2) + 4*log(cos(x) - 1)*cos(x)/(2*cos(x)**2 - 4*cos(x) +
2) - 2*log(cos(x) - 1)/(2*cos(x)**2 - 4*cos(x) + 2) - sin(x)**2/(2*cos(x)**2 - 4*cos(x) + 2) + 2*cos(x)/(2*co
s(x)**2 - 4*cos(x) + 2) - 2/(2*cos(x)**2 - 4*cos(x) + 2)

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Giac [A]  time = 1.14637, size = 24, normalized size = 1.2 \begin{align*} \frac{2}{\cos \left (x\right ) - 1} - \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(cos(x) - 1) - log(-cos(x) + 1)