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))-ln(1-cos(x))

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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])

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])

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.67, size = 22, normalized size = 1.10 \[ -\frac {{\left (\cos \relax (x) - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2}{\cos \relax (x) - 1} \]

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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giac [A]  time = 0.41, size = 18, normalized size = 0.90 \[ \frac {2}{\cos \relax (x) - 1} - \log \left (-\cos \relax (x) + 1\right ) \]

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)

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maple [A]  time = 0.07, size = 17, normalized size = 0.85 \[ \frac {2}{-1+\cos \relax (x )}-\ln \left (-1+\cos \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.76, size = 16, normalized size = 0.80 \[ \frac {2}{\cos \relax (x) - 1} - \log \left (\cos \relax (x) - 1\right ) \]

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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mupad [B]  time = 0.04, size = 16, normalized size = 0.80 \[ \frac {2}{\cos \relax (x)-1}-\ln \left (\cos \relax (x)-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-sin(x)^3/(cos(x) - 1)^3,x)

[Out]

2/(cos(x) - 1) - log(cos(x) - 1)

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sympy [B]  time = 0.58, size = 126, normalized size = 6.30 \[ - \frac {2 \log {\left (\cos {\relax (x )} - 1 \right )} \cos ^{2}{\relax (x )}}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} + \frac {4 \log {\left (\cos {\relax (x )} - 1 \right )} \cos {\relax (x )}}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} - \frac {2 \log {\left (\cos {\relax (x )} - 1 \right )}}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} - \frac {\sin ^{2}{\relax (x )}}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} + \frac {2 \cos {\relax (x )}}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} - \frac {2}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} \]

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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