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

Optimal. Leaf size=93 \[ \frac{a^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}-\frac{\csc ^2(x) (a-b \cos (x))}{2 \left (a^2-b^2\right )}-\frac{(2 a+b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(2 a-b) \log (\cos (x)+1)}{4 (a-b)^2} \]

[Out]

-((a - b*Cos[x])*Csc[x]^2)/(2*(a^2 - b^2)) - ((2*a + b)*Log[1 - Cos[x]])/(4*(a + b)^2) - ((2*a - b)*Log[1 + Co
s[x]])/(4*(a - b)^2) + (a^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2

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Rubi [A]  time = 0.182853, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2721, 1647, 801} \[ \frac{a^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}-\frac{\csc ^2(x) (a-b \cos (x))}{2 \left (a^2-b^2\right )}-\frac{(2 a+b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(2 a-b) \log (\cos (x)+1)}{4 (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3/(a + b*Cos[x]),x]

[Out]

-((a - b*Cos[x])*Csc[x]^2)/(2*(a^2 - b^2)) - ((2*a + b)*Log[1 - Cos[x]])/(4*(a + b)^2) - ((2*a - b)*Log[1 + Co
s[x]])/(4*(a - b)^2) + (a^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

\begin{align*} \int \frac{\cot ^3(x)}{a+b \cos (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \cos (x)\right )\\ &=-\frac{(a-b \cos (x)) \csc ^2(x)}{2 \left (a^2-b^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a b^4}{a^2-b^2}-\frac{b^2 \left (2 a^2-b^2\right ) x}{a^2-b^2}}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cos (x)\right )}{2 b^2}\\ &=-\frac{(a-b \cos (x)) \csc ^2(x)}{2 \left (a^2-b^2\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{b^2 (2 a+b)}{2 (a+b)^2 (b-x)}-\frac{2 a^3 b^2}{(a-b)^2 (a+b)^2 (a+x)}+\frac{(2 a-b) b^2}{2 (a-b)^2 (b+x)}\right ) \, dx,x,b \cos (x)\right )}{2 b^2}\\ &=-\frac{(a-b \cos (x)) \csc ^2(x)}{2 \left (a^2-b^2\right )}-\frac{(2 a+b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(2 a-b) \log (1+\cos (x))}{4 (a-b)^2}+\frac{a^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.547953, size = 100, normalized size = 1.08 \[ \frac{1}{8} \left (\frac{8 a^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}-\frac{\csc ^2\left (\frac{x}{2}\right )}{a+b}-\frac{\sec ^2\left (\frac{x}{2}\right )}{a-b}-\frac{4 (2 a+b) \log \left (\sin \left (\frac{x}{2}\right )\right )}{(a+b)^2}+\frac{4 (b-2 a) \log \left (\cos \left (\frac{x}{2}\right )\right )}{(a-b)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3/(a + b*Cos[x]),x]

[Out]

(-(Csc[x/2]^2/(a + b)) + (4*(-2*a + b)*Log[Cos[x/2]])/(a - b)^2 + (8*a^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2 - (4
*(2*a + b)*Log[Sin[x/2]])/(a + b)^2 - Sec[x/2]^2/(a - b))/8

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Maple [A]  time = 0.056, size = 114, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}\ln \left ( a+b\cos \left ( x \right ) \right ) }{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}}}+{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( -1+\cos \left ( x \right ) \right ) }}-{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) a}{2\, \left ( a+b \right ) ^{2}}}-{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) b}{4\, \left ( a+b \right ) ^{2}}}-{\frac{1}{ \left ( 4\,a-4\,b \right ) \left ( \cos \left ( x \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) a}{2\, \left ( a-b \right ) ^{2}}}+{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) b}{4\, \left ( a-b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3/(a+b)^2/(a-b)^2*ln(a+b*cos(x))+1/(4*a+4*b)/(-1+cos(x))-1/2/(a+b)^2*ln(-1+cos(x))*a-1/4/(a+b)^2*ln(-1+cos(x
))*b-1/(4*a-4*b)/(cos(x)+1)-1/2/(a-b)^2*ln(cos(x)+1)*a+1/4/(a-b)^2*ln(cos(x)+1)*b

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Maxima [A]  time = 1.09561, size = 157, normalized size = 1.69 \begin{align*} \frac{a^{3} \log \left (b \cos \left (x\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (2 \, a - b\right )} \log \left (\cos \left (x\right ) + 1\right )}{4 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac{{\left (2 \, a + b\right )} \log \left (\cos \left (x\right ) - 1\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac{b \cos \left (x\right ) - a}{2 \,{\left ({\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} + b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.76721, size = 431, normalized size = 4.63 \begin{align*} -\frac{2 \, a^{3} - 2 \, a b^{2} - 2 \,{\left (a^{2} b - b^{3}\right )} \cos \left (x\right ) + 4 \,{\left (a^{3} \cos \left (x\right )^{2} - a^{3}\right )} \log \left (-b \cos \left (x\right ) - a\right ) +{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3} -{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (2 \, a^{3} - 3 \, a^{2} b + b^{3} -{\left (2 \, a^{3} - 3 \, a^{2} b + b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**3/(a+b*cos(x)),x)

[Out]

Integral(cot(x)**3/(a + b*cos(x)), x)

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Giac [A]  time = 1.34972, size = 186, normalized size = 2. \begin{align*} \frac{a^{3} b \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac{{\left (2 \, a - b\right )} \log \left (\cos \left (x\right ) + 1\right )}{4 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac{{\left (2 \, a + b\right )} \log \left (-\cos \left (x\right ) + 1\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac{a^{3} - a b^{2} -{\left (a^{2} b - b^{3}\right )} \cos \left (x\right )}{2 \,{\left (a + b\right )}^{2}{\left (a - b\right )}^{2}{\left (\cos \left (x\right ) + 1\right )}{\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+b*cos(x)),x, algorithm="giac")

[Out]

a^3*b*log(abs(b*cos(x) + a))/(a^4*b - 2*a^2*b^3 + b^5) - 1/4*(2*a - b)*log(cos(x) + 1)/(a^2 - 2*a*b + b^2) - 1
/4*(2*a + b)*log(-cos(x) + 1)/(a^2 + 2*a*b + b^2) + 1/2*(a^3 - a*b^2 - (a^2*b - b^3)*cos(x))/((a + b)^2*(a - b
)^2*(cos(x) + 1)*(cos(x) - 1))