3.199 \(\int \frac{\cot (x)}{\sec (x)-\tan (x)} \, dx\)

Optimal. Leaf size=7 \[ x-\tanh ^{-1}(\cos (x)) \]

[Out]

x - ArcTanh[Cos[x]]

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Rubi [A]  time = 0.0888162, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4391, 2839, 3770, 8} \[ x-\tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(Sec[x] - Tan[x]),x]

[Out]

x - ArcTanh[Cos[x]]

Rule 4391

Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x_)]^(n_.))^(p_), x_Symbol] :> Int[A
ctivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cot (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac{\cos (x) \cot (x)}{1-\sin (x)} \, dx\\ &=\int 1 \, dx+\int \csc (x) \, dx\\ &=x-\tanh ^{-1}(\cos (x))\\ \end{align*}

Mathematica [B]  time = 0.0221246, size = 18, normalized size = 2.57 \[ x+\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(Sec[x] - Tan[x]),x]

[Out]

x - Log[Cos[x/2]] + Log[Sin[x/2]]

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Maple [A]  time = 0.105, size = 8, normalized size = 1.1 \begin{align*} \ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) +x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(sec(x)-tan(x)),x)

[Out]

ln(tan(1/2*x))+x

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Maxima [B]  time = 1.50547, size = 31, normalized size = 4.43 \begin{align*} 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(sec(x)-tan(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.490267, size = 81, normalized size = 11.57 \begin{align*} x - \frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \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(cot(x)/(sec(x)-tan(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{- \tan{\left (x \right )} + \sec{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(sec(x)-tan(x)),x)

[Out]

Integral(cot(x)/(-tan(x) + sec(x)), x)

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Giac [A]  time = 1.1556, size = 11, normalized size = 1.57 \begin{align*} x + \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(sec(x)-tan(x)),x, algorithm="giac")

[Out]

x + log(abs(tan(1/2*x)))