3.292 \(\int \frac{1}{(a \cot (x)+b \csc (x))^5} \, dx\)

Optimal. Leaf size=100 \[ \frac{\left (a^2-b^2\right )^2}{4 a^5 (a \cos (x)+b)^4}+\frac{4 b \left (a^2-b^2\right )}{3 a^5 (a \cos (x)+b)^3}-\frac{a^2-3 b^2}{a^5 (a \cos (x)+b)^2}-\frac{4 b}{a^5 (a \cos (x)+b)}-\frac{\log (a \cos (x)+b)}{a^5} \]

[Out]

(a^2 - b^2)^2/(4*a^5*(b + a*Cos[x])^4) + (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cos[x])^3) - (a^2 - 3*b^2)/(a^5*(b +
a*Cos[x])^2) - (4*b)/(a^5*(b + a*Cos[x])) - Log[b + a*Cos[x]]/a^5

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Rubi [A]  time = 0.123181, antiderivative size = 100, 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} \[ \frac{\left (a^2-b^2\right )^2}{4 a^5 (a \cos (x)+b)^4}+\frac{4 b \left (a^2-b^2\right )}{3 a^5 (a \cos (x)+b)^3}-\frac{a^2-3 b^2}{a^5 (a \cos (x)+b)^2}-\frac{4 b}{a^5 (a \cos (x)+b)}-\frac{\log (a \cos (x)+b)}{a^5} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x] + b*Csc[x])^(-5),x]

[Out]

(a^2 - b^2)^2/(4*a^5*(b + a*Cos[x])^4) + (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cos[x])^3) - (a^2 - 3*b^2)/(a^5*(b +
a*Cos[x])^2) - (4*b)/(a^5*(b + a*Cos[x])) - Log[b + a*Cos[x]]/a^5

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))^5} \, dx &=\int \frac{\sin ^5(x)}{(b+a \cos (x))^5} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a^2-x^2\right )^2}{(b+x)^5} \, dx,x,a \cos (x)\right )}{a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a^2-b^2\right )^2}{(b+x)^5}-\frac{4 b \left (-a^2+b^2\right )}{(b+x)^4}-\frac{2 \left (a^2-3 b^2\right )}{(b+x)^3}-\frac{4 b}{(b+x)^2}+\frac{1}{b+x}\right ) \, dx,x,a \cos (x)\right )}{a^5}\\ &=\frac{\left (a^2-b^2\right )^2}{4 a^5 (b+a \cos (x))^4}+\frac{4 b \left (a^2-b^2\right )}{3 a^5 (b+a \cos (x))^3}-\frac{a^2-3 b^2}{a^5 (b+a \cos (x))^2}-\frac{4 b}{a^5 (b+a \cos (x))}-\frac{\log (b+a \cos (x))}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.3392, size = 138, normalized size = 1.38 \[ -\frac{12 a^2 \cos ^2(x) \left (a^2+6 b^2 \log (a \cos (x)+b)+9 b^2\right )+8 a b \cos (x) \left (a^2+6 b^2 \log (a \cos (x)+b)+11 b^2\right )+2 a^2 b^2+12 a^4 \cos ^4(x) \log (a \cos (x)+b)+48 a^3 b \cos ^3(x) (\log (a \cos (x)+b)+1)-3 a^4+12 b^4 \log (a \cos (x)+b)+25 b^4}{12 a^5 (a \cos (x)+b)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x] + b*Csc[x])^(-5),x]

[Out]

-(-3*a^4 + 2*a^2*b^2 + 25*b^4 + 12*b^4*Log[b + a*Cos[x]] + 12*a^4*Cos[x]^4*Log[b + a*Cos[x]] + 48*a^3*b*Cos[x]
^3*(1 + Log[b + a*Cos[x]]) + 12*a^2*Cos[x]^2*(a^2 + 9*b^2 + 6*b^2*Log[b + a*Cos[x]]) + 8*a*b*Cos[x]*(a^2 + 11*
b^2 + 6*b^2*Log[b + a*Cos[x]]))/(12*a^5*(b + a*Cos[x])^4)

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Maple [A]  time = 0.056, size = 132, normalized size = 1.3 \begin{align*}{\frac{1}{4\,a \left ( b+a\cos \left ( x \right ) \right ) ^{4}}}-{\frac{{b}^{2}}{2\,{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) ^{4}}}+{\frac{{b}^{4}}{4\,{a}^{5} \left ( b+a\cos \left ( x \right ) \right ) ^{4}}}-4\,{\frac{b}{{a}^{5} \left ( b+a\cos \left ( x \right ) \right ) }}-{\frac{\ln \left ( b+a\cos \left ( x \right ) \right ) }{{a}^{5}}}+{\frac{4\,b}{3\,{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) ^{3}}}-{\frac{4\,{b}^{3}}{3\,{a}^{5} \left ( b+a\cos \left ( x \right ) \right ) ^{3}}}-{\frac{1}{{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) ^{2}}}+3\,{\frac{{b}^{2}}{{a}^{5} \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))^5,x)

[Out]

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

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Maxima [B]  time = 1.69827, size = 671, normalized size = 6.71 \begin{align*} -\frac{2 \,{\left (5 \, a^{4} b + 10 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 6 \, a b^{4} - 3 \, b^{5} + \frac{{\left (3 \, a^{5} - 17 \, a^{4} b - 6 \, a^{3} b^{2} + 26 \, a^{2} b^{3} + 3 \, a b^{4} - 9 \, b^{5}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{3 \,{\left (4 \, a^{5} - 13 \, a^{4} b + 12 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 8 \, a b^{4} + 3 \, b^{5}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{3 \,{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}}{3 \,{\left (a^{10} + 2 \, a^{9} b - a^{8} b^{2} - 4 \, a^{7} b^{3} - a^{6} b^{4} + 2 \, a^{5} b^{5} + a^{4} b^{6} - \frac{4 \,{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{6 \,{\left (a^{10} - 2 \, a^{9} b - a^{8} b^{2} + 4 \, a^{7} b^{3} - a^{6} b^{4} - 2 \, a^{5} b^{5} + a^{4} b^{6}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{4 \,{\left (a^{10} - 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} + 4 \, a^{5} b^{5} - a^{4} b^{6}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{{\left (a^{10} - 6 \, a^{9} b + 15 \, a^{8} b^{2} - 20 \, a^{7} b^{3} + 15 \, a^{6} b^{4} - 6 \, a^{5} b^{5} + a^{4} b^{6}\right )} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} - \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^{5}} + \frac{\log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))^5,x, algorithm="maxima")

[Out]

-2/3*(5*a^4*b + 10*a^3*b^2 + 2*a^2*b^3 - 6*a*b^4 - 3*b^5 + (3*a^5 - 17*a^4*b - 6*a^3*b^2 + 26*a^2*b^3 + 3*a*b^
4 - 9*b^5)*sin(x)^2/(cos(x) + 1)^2 - 3*(4*a^5 - 13*a^4*b + 12*a^3*b^2 + 2*a^2*b^3 - 8*a*b^4 + 3*b^5)*sin(x)^4/
(cos(x) + 1)^4 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*sin(x)^6/(cos(x) + 1)^6)/(a^10 +
2*a^9*b - a^8*b^2 - 4*a^7*b^3 - a^6*b^4 + 2*a^5*b^5 + a^4*b^6 - 4*(a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sin
(x)^2/(cos(x) + 1)^2 + 6*(a^10 - 2*a^9*b - a^8*b^2 + 4*a^7*b^3 - a^6*b^4 - 2*a^5*b^5 + a^4*b^6)*sin(x)^4/(cos(
x) + 1)^4 - 4*(a^10 - 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 + 4*a^5*b^5 - a^4*b^6)*sin(x)^6/(cos(x) + 1)^6 + (a^10 -
6*a^9*b + 15*a^8*b^2 - 20*a^7*b^3 + 15*a^6*b^4 - 6*a^5*b^5 + a^4*b^6)*sin(x)^8/(cos(x) + 1)^8) - log(a + b -
(a - b)*sin(x)^2/(cos(x) + 1)^2)/a^5 + log(sin(x)^2/(cos(x) + 1)^2 + 1)/a^5

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Fricas [A]  time = 2.53773, size = 409, normalized size = 4.09 \begin{align*} -\frac{48 \, a^{3} b \cos \left (x\right )^{3} - 3 \, a^{4} + 2 \, a^{2} b^{2} + 25 \, b^{4} + 12 \,{\left (a^{4} + 9 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2} + 8 \,{\left (a^{3} b + 11 \, a b^{3}\right )} \cos \left (x\right ) + 12 \,{\left (a^{4} \cos \left (x\right )^{4} + 4 \, a^{3} b \cos \left (x\right )^{3} + 6 \, a^{2} b^{2} \cos \left (x\right )^{2} + 4 \, a b^{3} \cos \left (x\right ) + b^{4}\right )} \log \left (a \cos \left (x\right ) + b\right )}{12 \,{\left (a^{9} \cos \left (x\right )^{4} + 4 \, a^{8} b \cos \left (x\right )^{3} + 6 \, a^{7} b^{2} \cos \left (x\right )^{2} + 4 \, a^{6} b^{3} \cos \left (x\right ) + a^{5} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))^5,x, algorithm="fricas")

[Out]

-1/12*(48*a^3*b*cos(x)^3 - 3*a^4 + 2*a^2*b^2 + 25*b^4 + 12*(a^4 + 9*a^2*b^2)*cos(x)^2 + 8*(a^3*b + 11*a*b^3)*c
os(x) + 12*(a^4*cos(x)^4 + 4*a^3*b*cos(x)^3 + 6*a^2*b^2*cos(x)^2 + 4*a*b^3*cos(x) + b^4)*log(a*cos(x) + b))/(a
^9*cos(x)^4 + 4*a^8*b*cos(x)^3 + 6*a^7*b^2*cos(x)^2 + 4*a^6*b^3*cos(x) + a^5*b^4)

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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))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.14898, size = 126, normalized size = 1.26 \begin{align*} -\frac{\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{a^{5}} - \frac{48 \, a^{2} b \cos \left (x\right )^{3} + 12 \,{\left (a^{3} + 9 \, a b^{2}\right )} \cos \left (x\right )^{2} + 8 \,{\left (a^{2} b + 11 \, b^{3}\right )} \cos \left (x\right ) - \frac{3 \, a^{4} - 2 \, a^{2} b^{2} - 25 \, b^{4}}{a}}{12 \,{\left (a \cos \left (x\right ) + b\right )}^{4} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x))^5,x, algorithm="giac")

[Out]

-log(abs(a*cos(x) + b))/a^5 - 1/12*(48*a^2*b*cos(x)^3 + 12*(a^3 + 9*a*b^2)*cos(x)^2 + 8*(a^2*b + 11*b^3)*cos(x
) - (3*a^4 - 2*a^2*b^2 - 25*b^4)/a)/((a*cos(x) + b)^4*a^4)