3.1.17 \(\int \frac {\sin ^3(x)}{(1-\cos (x))^2} \, dx\) [17]

Optimal. Leaf size=12 \[ \cos (x)+2 \log (1-\cos (x)) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 12, 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*} \cos (x)+2 \log (1-\cos (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(1 - Cos[x])^2,x]

[Out]

Cos[x] + 2*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))^2} \, dx &=\text {Subst}\left (\int \frac {1-x}{1+x} \, dx,x,-\cos (x)\right )\\ &=\text {Subst}\left (\int \left (-1+\frac {2}{1+x}\right ) \, dx,x,-\cos (x)\right )\\ &=\cos (x)+2 \log (1-\cos (x))\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 13, normalized size = 1.08 \begin {gather*} -1+\cos (x)+4 \log \left (\sin \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(1 - Cos[x])^2,x]

[Out]

-1 + Cos[x] + 4*Log[Sin[x/2]]

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Maple [A]
time = 0.08, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(\cos \left (x \right )+2 \ln \left (-1+\cos \left (x \right )\right )\)	\(11\)
default	\(\cos \left (x \right )+2 \ln \left (-1+\cos \left (x \right )\right )\)	\(11\)
risch	\(-2 i x +\frac {{\mathrm e}^{i x}}{2}+\frac {{\mathrm e}^{-i x}}{2}+4 \ln \left ({\mathrm e}^{i x}-1\right )\)	\(30\)
norman	\(\frac {-2 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )-\frac {4 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3} \tan \left (\frac {x}{2}\right )^{3}}+4 \ln \left (\tan \left (\frac {x}{2}\right )\right )-2 \ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(1-cos(x))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 10, normalized size = 0.83 \begin {gather*} \cos \left (x\right ) + 2 \, \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))^2,x, algorithm="maxima")

[Out]

cos(x) + 2*log(cos(x) - 1)

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Fricas [A]
time = 0.37, size = 12, normalized size = 1.00 \begin {gather*} \cos \left (x\right ) + 2 \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(1-cos(x))^2,x, algorithm="fricas")

[Out]

cos(x) + 2*log(-1/2*cos(x) + 1/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (10) = 20\).
time = 0.20, size = 58, normalized size = 4.83 \begin {gather*} \frac {2 \log {\left (\cos {\left (x \right )} - 1 \right )} \cos {\left (x \right )}}{\cos {\left (x \right )} - 1} - \frac {2 \log {\left (\cos {\left (x \right )} - 1 \right )}}{\cos {\left (x \right )} - 1} + \frac {\sin ^{2}{\left (x \right )}}{\cos {\left (x \right )} - 1} + \frac {2 \cos ^{2}{\left (x \right )}}{\cos {\left (x \right )} - 1} - \frac {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))**2,x)

[Out]

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

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Giac [A]
time = 0.39, size = 12, normalized size = 1.00 \begin {gather*} \cos \left (x\right ) + 2 \, \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))^2,x, algorithm="giac")

[Out]

cos(x) + 2*log(-cos(x) + 1)

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Mupad [B]
time = 0.04, size = 10, normalized size = 0.83 \begin {gather*} 2\,\ln \left (\cos \left (x\right )-1\right )+\cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(cos(x) - 1)^2,x)

[Out]

2*log(cos(x) - 1) + cos(x)

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