3.5 \(\int \frac{\cot (x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a} \]

[Out]

-ArcTanh[Cos[x]]/(2*a) + (Cot[x]*Csc[x])/(2*a) - Csc[x]^2/(2*a)

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Rubi [A]  time = 0.0548422, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {2706, 2606, 30, 2611, 3770} \[ -\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(a + a*Cos[x]),x]

[Out]

-ArcTanh[Cos[x]]/(2*a) + (Cot[x]*Csc[x])/(2*a) - Csc[x]^2/(2*a)

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\cot (x)}{a+a \cos (x)} \, dx &=-\frac{\int \cot ^2(x) \csc (x) \, dx}{a}+\frac{\int \cot (x) \csc ^2(x) \, dx}{a}\\ &=\frac{\cot (x) \csc (x)}{2 a}+\frac{\int \csc (x) \, dx}{2 a}-\frac{\operatorname{Subst}(\int x \, dx,x,\csc (x))}{a}\\ &=-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a}-\frac{\csc ^2(x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0401624, size = 42, normalized size = 1.27 \[ -\frac{2 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )+1}{2 a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(a + a*Cos[x]),x]

[Out]

-(1 + 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(2*a*(1 + Cos[x]))

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Maple [A]  time = 0.053, size = 33, normalized size = 1. \begin{align*}{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{4\,a}}-{\frac{1}{2\,a \left ( \cos \left ( x \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a+a*cos(x)),x)

[Out]

1/4/a*ln(-1+cos(x))-1/2/a/(cos(x)+1)-1/4*ln(cos(x)+1)/a

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Maxima [A]  time = 1.14122, size = 42, normalized size = 1.27 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{\log \left (\cos \left (x\right ) - 1\right )}{4 \, a} - \frac{1}{2 \,{\left (a \cos \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*cos(x)),x, algorithm="maxima")

[Out]

-1/4*log(cos(x) + 1)/a + 1/4*log(cos(x) - 1)/a - 1/2/(a*cos(x) + a)

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Fricas [A]  time = 1.3867, size = 135, normalized size = 4.09 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2}{4 \,{\left (a \cos \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*cos(x)),x, algorithm="fricas")

[Out]

-1/4*((cos(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + 1)*log(-1/2*cos(x) + 1/2) + 2)/(a*cos(x) + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot{\left (x \right )}}{\cos{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*cos(x)),x)

[Out]

Integral(cot(x)/(cos(x) + 1), x)/a

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Giac [A]  time = 1.2206, size = 46, normalized size = 1.39 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{\log \left (-\cos \left (x\right ) + 1\right )}{4 \, a} - \frac{1}{2 \, a{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*cos(x)),x, algorithm="giac")

[Out]

-1/4*log(cos(x) + 1)/a + 1/4*log(-cos(x) + 1)/a - 1/2/(a*(cos(x) + 1))