3.1.23 \(\int \frac {\sin ^3(x)}{(1-\cos (x))^3} \, dx\) [23]

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 = 0.02, 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 = {2746, 45} \begin {gather*} -\frac {2}{1-\cos (x)}-\log (1-\cos (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(1 - Cos[x])^3,x]

[Out]

-2/(1 - Cos[x]) - Log[1 - Cos[x]]

Rule 45

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 2746

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 &=\text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,-\cos (x)\right )\\ &=\text {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 \begin {gather*} -\cot ^2\left (\frac {x}{2}\right )-2 \log \left (\cos \left (\frac {x}{2}\right )\right )-2 \log \left (\tan \left (\frac {x}{2}\right )\right ) \end {gather*}

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 = 0.09, size = 17, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {2}{-1+\cos \left (x \right )}-\ln \left (-1+\cos \left (x \right )\right )\)	\(17\)
default	\(\frac {2}{-1+\cos \left (x \right )}-\ln \left (-1+\cos \left (x \right )\right )\)	\(17\)
risch	\(i x +\frac {4 \,{\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}-1\right )^{2}}-2 \ln \left ({\mathrm e}^{i x}-1\right )\)	\(32\)
norman	\(\frac {\tan ^{11}\left (\frac {x}{2}\right )+2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )-2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3} \tan \left (\frac {x}{2}\right )^{5}}-2 \ln \left (\tan \left (\frac {x}{2}\right )\right )+\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 16, normalized size = 0.80 \begin {gather*} \frac {2}{\cos \left (x\right ) - 1} - \log \left (\cos \left (x\right ) - 1\right ) \end {gather*}

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 = 0.36, size = 22, normalized size = 1.10 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (14) = 28\).
time = 0.25, size = 126, normalized size = 6.30 \begin {gather*} - \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 {gather*}

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 = 0.43, size = 18, normalized size = 0.90 \begin {gather*} \frac {2}{\cos \left (x\right ) - 1} - \log \left (-\cos \left (x\right ) + 1\right ) \end {gather*}

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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Mupad [B]
time = 0.04, size = 16, normalized size = 0.80 \begin {gather*} \frac {2}{\cos \left (x\right )-1}-\ln \left (\cos \left (x\right )-1\right ) \end {gather*}

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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