3.765 \(\int \frac{\cos (x) \sin (x)}{1-\cos (x)} \, dx\)

Optimal. Leaf size=10 \[ \cos (x)+\log (1-\cos (x)) \]

[Out]

Cos[x] + Log[1 - Cos[x]]

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Rubi [A]  time = 0.0330559, antiderivative size = 10, 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 = {2833, 43} \[ \cos (x)+\log (1-\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]*Sin[x])/(1 - Cos[x]),x]

[Out]

Cos[x] + Log[1 - Cos[x]]

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, 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])

Rubi steps

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

Mathematica [A]  time = 0.0174454, size = 12, normalized size = 1.2 \[ \cos (x)+2 \log \left (\sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]*Sin[x])/(1 - Cos[x]),x]

[Out]

Cos[x] + 2*Log[Sin[x/2]]

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Maple [A]  time = 0.012, size = 9, normalized size = 0.9 \begin{align*} \cos \left ( x \right ) +\ln \left ( -1+\cos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(x)/(1-cos(x)),x)

[Out]

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

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Maxima [A]  time = 0.9537, size = 11, normalized size = 1.1 \begin{align*} \cos \left (x\right ) + \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x) + log(cos(x) - 1)

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Fricas [A]  time = 1.92776, size = 45, normalized size = 4.5 \begin{align*} \cos \left (x\right ) + \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x) + log(-1/2*cos(x) + 1/2)

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Sympy [A]  time = 0.19146, size = 8, normalized size = 0.8 \begin{align*} \log{\left (\cos{\left (x \right )} - 1 \right )} + \cos{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(cos(x) - 1) + cos(x)

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Giac [A]  time = 1.109, size = 14, normalized size = 1.4 \begin{align*} \cos \left (x\right ) + \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x) + log(-cos(x) + 1)