3.290 \(\int \frac{1}{(a \cot (x)+b \csc (x))^3} \, dx\)

Optimal. Leaf size=50 \[ \frac{a^2-b^2}{2 a^3 (a \cos (x)+b)^2}+\frac{2 b}{a^3 (a \cos (x)+b)}+\frac{\log (a \cos (x)+b)}{a^3} \]

[Out]

(a^2 - b^2)/(2*a^3*(b + a*Cos[x])^2) + (2*b)/(a^3*(b + a*Cos[x])) + Log[b + a*Cos[x]]/a^3

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Rubi [A]  time = 0.0787822, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4392, 2668, 697} \[ \frac{a^2-b^2}{2 a^3 (a \cos (x)+b)^2}+\frac{2 b}{a^3 (a \cos (x)+b)}+\frac{\log (a \cos (x)+b)}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x] + b*Csc[x])^(-3),x]

[Out]

(a^2 - b^2)/(2*a^3*(b + a*Cos[x])^2) + (2*b)/(a^3*(b + a*Cos[x])) + Log[b + a*Cos[x]]/a^3

Rule 4392

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

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(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] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.111372, size = 77, normalized size = 1.54 \[ \frac{a^2 \cos (2 x) \log (a \cos (x)+b)+a^2 \log (a \cos (x)+b)+a^2+2 b^2 \log (a \cos (x)+b)+4 a b \cos (x) (\log (a \cos (x)+b)+1)+3 b^2}{2 a^3 (a \cos (x)+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x] + b*Csc[x])^(-3),x]

[Out]

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

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Maple [A]  time = 0.048, size = 56, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( b+a\cos \left ( x \right ) \right ) }{{a}^{3}}}+2\,{\frac{b}{{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) }}+{\frac{1}{2\,a \left ( b+a\cos \left ( x \right ) \right ) ^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cot(x)+b*csc(x))^3,x)

[Out]

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

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Maxima [B]  time = 1.51703, size = 239, normalized size = 4.78 \begin{align*} \frac{2 \,{\left (a b + b^{2} + \frac{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{a^{5} + a^{4} b - a^{3} b^{2} - a^{2} b^{3} - \frac{2 \,{\left (a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} + \frac{\log \left (a + b - \frac{{\left (a - b\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{3}} - \frac{\log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.31057, size = 181, normalized size = 3.62 \begin{align*} \frac{4 \, a b \cos \left (x\right ) + a^{2} + 3 \, b^{2} + 2 \,{\left (a^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + b^{2}\right )} \log \left (a \cos \left (x\right ) + b\right )}{2 \,{\left (a^{5} \cos \left (x\right )^{2} + 2 \, a^{4} b \cos \left (x\right ) + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.11597, size = 61, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{a^{3}} + \frac{4 \, b \cos \left (x\right ) + \frac{a^{2} + 3 \, b^{2}}{a}}{2 \,{\left (a \cos \left (x\right ) + b\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))^3,x, algorithm="giac")

[Out]

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