3.7 \(\int \frac{\cot ^3(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\cot ^4(x)}{4 a}+\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) \csc (x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a} \]

[Out]

(3*ArcTanh[Cos[x]])/(8*a) - Cot[x]^4/(4*a) - (3*Cot[x]*Csc[x])/(8*a) + (Cot[x]^3*Csc[x])/(4*a)

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Rubi [A]  time = 0.0962216, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2611, 3770} \[ -\frac{\cot ^4(x)}{4 a}+\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) \csc (x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3/(a + a*Cos[x]),x]

[Out]

(3*ArcTanh[Cos[x]])/(8*a) - Cot[x]^4/(4*a) - (3*Cot[x]*Csc[x])/(8*a) + (Cot[x]^3*Csc[x])/(4*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 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 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 ^3(x)}{a+a \cos (x)} \, dx &=-\frac{\int \cot ^4(x) \csc (x) \, dx}{a}+\frac{\int \cot ^3(x) \csc ^2(x) \, dx}{a}\\ &=\frac{\cot ^3(x) \csc (x)}{4 a}+\frac{3 \int \cot ^2(x) \csc (x) \, dx}{4 a}-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,-\cot (x)\right )}{a}\\ &=-\frac{\cot ^4(x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a}+\frac{\cot ^3(x) \csc (x)}{4 a}-\frac{3 \int \csc (x) \, dx}{8 a}\\ &=\frac{3 \tanh ^{-1}(\cos (x))}{8 a}-\frac{\cot ^4(x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a}+\frac{\cot ^3(x) \csc (x)}{4 a}\\ \end{align*}

Mathematica [A]  time = 0.144443, size = 60, normalized size = 1.3 \[ -\frac{2 \cot ^2\left (\frac{x}{2}\right )+\sec ^2\left (\frac{x}{2}\right )-12 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )-8}{16 a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3/(a + a*Cos[x]),x]

[Out]

-(-8 + 2*Cot[x/2]^2 - 12*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]) + Sec[x/2]^2)/(16*a*(1 + Cos[x]))

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Maple [A]  time = 0.052, size = 55, normalized size = 1.2 \begin{align*}{\frac{1}{8\,a \left ( -1+\cos \left ( x \right ) \right ) }}-{\frac{3\,\ln \left ( -1+\cos \left ( x \right ) \right ) }{16\,a}}-{\frac{1}{8\,a \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}+{\frac{1}{2\,a \left ( \cos \left ( x \right ) +1 \right ) }}+{\frac{3\,\ln \left ( \cos \left ( x \right ) +1 \right ) }{16\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^3/(a+a*cos(x)),x)

[Out]

1/8/a/(-1+cos(x))-3/16/a*ln(-1+cos(x))-1/8/a/(cos(x)+1)^2+1/2/a/(cos(x)+1)+3/16*ln(cos(x)+1)/a

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Maxima [A]  time = 1.15431, size = 76, normalized size = 1.65 \begin{align*} \frac{5 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2}{8 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} + \frac{3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} - \frac{3 \, \log \left (\cos \left (x\right ) - 1\right )}{16 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3/(a+a*cos(x)),x, algorithm="maxima")

[Out]

1/8*(5*cos(x)^2 + cos(x) - 2)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a) + 3/16*log(cos(x) + 1)/a - 3/16*log(cos
(x) - 1)/a

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Fricas [B]  time = 1.44269, size = 269, normalized size = 5.85 \begin{align*} \frac{10 \, \cos \left (x\right )^{2} + 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \, \cos \left (x\right ) - 4}{16 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3/(a+a*cos(x)),x, algorithm="fricas")

[Out]

1/16*(10*cos(x)^2 + 3*(cos(x)^3 + cos(x)^2 - cos(x) - 1)*log(1/2*cos(x) + 1/2) - 3*(cos(x)^3 + cos(x)^2 - cos(
x) - 1)*log(-1/2*cos(x) + 1/2) + 2*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{3}{\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)**3/(a+a*cos(x)),x)

[Out]

Integral(cot(x)**3/(cos(x) + 1), x)/a

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Giac [A]  time = 1.29375, size = 68, normalized size = 1.48 \begin{align*} \frac{3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} - \frac{3 \, \log \left (-\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{5 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2}{8 \, a{\left (\cos \left (x\right ) + 1\right )}^{2}{\left (\cos \left (x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3/(a+a*cos(x)),x, algorithm="giac")

[Out]

3/16*log(cos(x) + 1)/a - 3/16*log(-cos(x) + 1)/a + 1/8*(5*cos(x)^2 + cos(x) - 2)/(a*(cos(x) + 1)^2*(cos(x) - 1
))