Integrand size = 11, antiderivative size = 100 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=\frac {\left (a^2-b^2\right )^2}{4 a^5 (b+a \cos (x))^4}+\frac {4 b \left (a^2-b^2\right )}{3 a^5 (b+a \cos (x))^3}-\frac {a^2-3 b^2}{a^5 (b+a \cos (x))^2}-\frac {4 b}{a^5 (b+a \cos (x))}-\frac {\log (b+a \cos (x))}{a^5} \] Output:
1/4*(a^2-b^2)^2/a^5/(b+a*cos(x))^4+4/3*b*(a^2-b^2)/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))-ln(b+a*cos(x))/a^5
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=-\frac {-3 a^4+2 a^2 b^2+25 b^4+12 b^4 \log (b+a \cos (x))+12 a^4 \cos ^4(x) \log (b+a \cos (x))+48 a^3 b \cos ^3(x) (1+\log (b+a \cos (x)))+12 a^2 \cos ^2(x) \left (a^2+9 b^2+6 b^2 \log (b+a \cos (x))\right )+8 a b \cos (x) \left (a^2+11 b^2+6 b^2 \log (b+a \cos (x))\right )}{12 a^5 (b+a \cos (x))^4} \] Input:
Integrate[(a*Cot[x] + b*Csc[x])^(-5),x]
Output:
-1/12*(-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]]))/(a^5*(b + a*Cos[x])^4)
Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4892, 3042, 3147, 476, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \cot (x)+b \csc (x))^5}dx\) |
| \(\Big \downarrow \) 4892 |
| \(\displaystyle \int \frac {\sin ^5(x)}{(a \cos (x)+b)^5}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (x-\frac {\pi }{2}\right )^5}{\left (b-a \sin \left (x-\frac {\pi }{2}\right )\right )^5}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle -\frac {\int \frac {\left (a^2-a^2 \cos ^2(x)\right )^2}{(b+a \cos (x))^5}d(a \cos (x))}{a^5}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle -\frac {\int \left (\frac {\left (a^2-b^2\right )^2}{(b+a \cos (x))^5}+\frac {1}{b+a \cos (x)}-\frac {4 b}{(b+a \cos (x))^2}-\frac {2 \left (a^2-3 b^2\right )}{(b+a \cos (x))^3}-\frac {4 b \left (b^2-a^2\right )}{(b+a \cos (x))^4}\right )d(a \cos (x))}{a^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {\left (a^2-b^2\right )^2}{4 (a \cos (x)+b)^4}-\frac {4 b \left (a^2-b^2\right )}{3 (a \cos (x)+b)^3}+\frac {a^2-3 b^2}{(a \cos (x)+b)^2}+\frac {4 b}{a \cos (x)+b}+\log (a \cos (x)+b)}{a^5}\) |
Input:
Int[(a*Cot[x] + b*Csc[x])^(-5),x]
Output:
-((-1/4*(a^2 - b^2)^2/(b + a*Cos[x])^4 - (4*b*(a^2 - b^2))/(3*(b + a*Cos[x ])^3) + (a^2 - 3*b^2)/(b + a*Cos[x])^2 + (4*b)/(b + a*Cos[x]) + Log[b + a* Cos[x]])/a^5)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[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] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b _.))^(p_)*(u_.), x_Symbol] :> Int[ActivateTrig[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]
Time = 84.64 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(-\frac {2 a^{2}-6 b^{2}}{2 a^{5} \left (b +a \cos \left (x \right )\right )^{2}}+\frac {4 b \left (a^{2}-b^{2}\right )}{3 a^{5} \left (b +a \cos \left (x \right )\right )^{3}}-\frac {\ln \left (b +a \cos \left (x \right )\right )}{a^{5}}-\frac {4 b}{a^{5} \left (b +a \cos \left (x \right )\right )}-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{4 a^{5} \left (b +a \cos \left (x \right )\right )^{4}}\) | \(107\) |
| risch | \(\frac {i x}{a^{5}}-\frac {4 \left (6 a^{3} b \,{\mathrm e}^{7 i x}+3 a^{4} {\mathrm e}^{6 i x}+27 a^{2} b^{2} {\mathrm e}^{6 i x}+22 a^{3} b \,{\mathrm e}^{5 i x}+44 a \,b^{3} {\mathrm e}^{5 i x}+3 a^{4} {\mathrm e}^{4 i x}+56 a^{2} b^{2} {\mathrm e}^{4 i x}+25 b^{4} {\mathrm e}^{4 i x}+22 a^{3} b \,{\mathrm e}^{3 i x}+44 a \,b^{3} {\mathrm e}^{3 i x}+3 a^{4} {\mathrm e}^{2 i x}+27 a^{2} b^{2} {\mathrm e}^{2 i x}+6 a^{3} b \,{\mathrm e}^{i x}\right )}{3 a^{5} \left (a \,{\mathrm e}^{2 i x}+2 b \,{\mathrm e}^{i x}+a \right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 b \,{\mathrm e}^{i x}}{a}+1\right )}{a^{5}}\) | \(203\) |
Input:
int(1/(a*cot(x)+b*csc(x))^5,x,method=_RETURNVERBOSE)
Output:
-1/2*(2*a^2-6*b^2)/a^5/(b+a*cos(x))^2+4/3*b*(a^2-b^2)/a^5/(b+a*cos(x))^3-l n(b+a*cos(x))/a^5-4*b/a^5/(b+a*cos(x))-1/4*(-a^4+2*a^2*b^2-b^4)/a^5/(b+a*c os(x))^4
Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=-\frac {48 \, a^{3} b \cos \left (x\right )^{3} - 3 \, a^{4} + 2 \, a^{2} b^{2} + 25 \, b^{4} + 12 \, {\left (a^{4} + 9 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2} + 8 \, {\left (a^{3} b + 11 \, a b^{3}\right )} \cos \left (x\right ) + 12 \, {\left (a^{4} \cos \left (x\right )^{4} + 4 \, a^{3} b \cos \left (x\right )^{3} + 6 \, a^{2} b^{2} \cos \left (x\right )^{2} + 4 \, a b^{3} \cos \left (x\right ) + b^{4}\right )} \log \left (a \cos \left (x\right ) + b\right )}{12 \, {\left (a^{9} \cos \left (x\right )^{4} + 4 \, a^{8} b \cos \left (x\right )^{3} + 6 \, a^{7} b^{2} \cos \left (x\right )^{2} + 4 \, a^{6} b^{3} \cos \left (x\right ) + a^{5} b^{4}\right )}} \] Input:
integrate(1/(a*cot(x)+b*csc(x))^5,x, algorithm="fricas")
Output:
-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)*cos(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)
Timed out. \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=\text {Timed out} \] Input:
integrate(1/(a*cot(x)+b*csc(x))**5,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (96) = 192\).
Time = 0.15 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.97 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=-\frac {2 \, {\left (5 \, a^{4} b + 10 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 6 \, a b^{4} - 3 \, b^{5} + \frac {{\left (3 \, a^{5} - 17 \, a^{4} b - 6 \, a^{3} b^{2} + 26 \, a^{2} b^{3} + 3 \, a b^{4} - 9 \, b^{5}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {3 \, {\left (4 \, a^{5} - 13 \, a^{4} b + 12 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 8 \, a b^{4} + 3 \, b^{5}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}}{3 \, {\left (a^{10} + 2 \, a^{9} b - a^{8} b^{2} - 4 \, a^{7} b^{3} - a^{6} b^{4} + 2 \, a^{5} b^{5} + a^{4} b^{6} - \frac {4 \, {\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{10} - 2 \, a^{9} b - a^{8} b^{2} + 4 \, a^{7} b^{3} - a^{6} b^{4} - 2 \, a^{5} b^{5} + a^{4} b^{6}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {4 \, {\left (a^{10} - 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} + 4 \, a^{5} b^{5} - a^{4} b^{6}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {{\left (a^{10} - 6 \, a^{9} b + 15 \, a^{8} b^{2} - 20 \, a^{7} b^{3} + 15 \, a^{6} b^{4} - 6 \, a^{5} b^{5} + a^{4} b^{6}\right )} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} - \frac {\log \left (a + b - \frac {{\left (a - b\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{5}} + \frac {\log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{5}} \] Input:
integrate(1/(a*cot(x)+b*csc(x))^5,x, algorithm="maxima")
Output:
-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/(c os(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(s in(x)^2/(cos(x) + 1)^2 + 1)/a^5
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=-\frac {\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{a^{5}} - \frac {48 \, a^{2} b \cos \left (x\right )^{3} + 12 \, {\left (a^{3} + 9 \, a b^{2}\right )} \cos \left (x\right )^{2} + 8 \, {\left (a^{2} b + 11 \, b^{3}\right )} \cos \left (x\right ) - \frac {3 \, a^{4} - 2 \, a^{2} b^{2} - 25 \, b^{4}}{a}}{12 \, {\left (a \cos \left (x\right ) + b\right )}^{4} a^{4}} \] Input:
integrate(1/(a*cot(x)+b*csc(x))^5,x, algorithm="giac")
Output:
-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)
Time = 17.13 (sec) , antiderivative size = 538, normalized size of antiderivative = 5.38 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {32\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{\frac {32\,b^3}{a^3}-\frac {32\,b^2}{a^2}-\frac {32\,b}{a}+\frac {32\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a}-\frac {64\,b^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a^2}+\frac {32\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a^3}+32}-\frac {64\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{32\,a-32\,b+32\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-\frac {32\,b^2}{a}+\frac {32\,b^3}{a^2}-\frac {64\,b^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a}+\frac {32\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a^2}}+\frac {32\,b^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{32\,a^2-32\,a\,b-32\,b^2-64\,b^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {32\,b^3}{a}+\frac {32\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a}+32\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}\right )}{a^5}-\frac {\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{a^4}+\frac {2\,\left (5\,a^4\,b+10\,a^3\,b^2+2\,a^2\,b^3-6\,a\,b^4-3\,b^5\right )}{3\,a^4\,{\left (a-b\right )}^2}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (-4\,a^3+5\,a^2\,b+2\,a\,b^2-3\,b^3\right )}{a^4}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (3\,a^4-14\,a^3\,b-20\,a^2\,b^2+6\,a\,b^3+9\,b^4\right )}{3\,a^4\,\left (a-b\right )}}{4\,a\,b^3+4\,a^3\,b+{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (6\,a^4-12\,a^2\,b^2+6\,b^4\right )+{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (-4\,a^4-8\,a^3\,b+8\,a\,b^3+4\,b^4\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^6\,\left (4\,a^4-8\,a^3\,b+8\,a\,b^3-4\,b^4\right )+a^4+b^4+{\mathrm {tan}\left (\frac {x}{2}\right )}^8\,\left (a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4\right )+6\,a^2\,b^2} \] Input:
int(1/(b/sin(x) + a*cot(x))^5,x)
Output:
(2*atanh((32*tan(x/2)^2)/((32*b^3)/a^3 - (32*b^2)/a^2 - (32*b)/a + (32*b*t an(x/2)^2)/a - (64*b^2*tan(x/2)^2)/a^2 + (32*b^3*tan(x/2)^2)/a^3 + 32) - ( 64*b*tan(x/2)^2)/(32*a - 32*b + 32*b*tan(x/2)^2 - (32*b^2)/a + (32*b^3)/a^ 2 - (64*b^2*tan(x/2)^2)/a + (32*b^3*tan(x/2)^2)/a^2) + (32*b^2*tan(x/2)^2) /(32*a^2 - 32*a*b - 32*b^2 - 64*b^2*tan(x/2)^2 + (32*b^3)/a + (32*b^3*tan( x/2)^2)/a + 32*a*b*tan(x/2)^2)))/a^5 - ((2*tan(x/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))/a^4 + (2*(5*a^4*b - 6*a*b^4 - 3*b^5 + 2*a^2*b^3 + 10*a^3*b^2) )/(3*a^4*(a - b)^2) + (2*tan(x/2)^4*(2*a*b^2 + 5*a^2*b - 4*a^3 - 3*b^3))/a ^4 + (2*tan(x/2)^2*(6*a*b^3 - 14*a^3*b + 3*a^4 + 9*b^4 - 20*a^2*b^2))/(3*a ^4*(a - b)))/(4*a*b^3 + 4*a^3*b + tan(x/2)^4*(6*a^4 + 6*b^4 - 12*a^2*b^2) + tan(x/2)^2*(8*a*b^3 - 8*a^3*b - 4*a^4 + 4*b^4) - tan(x/2)^6*(8*a*b^3 - 8 *a^3*b + 4*a^4 - 4*b^4) + a^4 + b^4 + tan(x/2)^8*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2) + 6*a^2*b^2)
Time = 0.18 (sec) , antiderivative size = 1653, normalized size of antiderivative = 16.53 \[ \int \frac {1}{(a \cot (x)+b \csc (x))^5} \, dx =\text {Too large to display} \] Input:
int(1/(a*cot(x)+b*csc(x))^5,x)
Output:
(48*cos(x)*log(tan(x/2)**2 + 1)*sin(x)**2*a**5*b + 96*cos(x)*log(tan(x/2)* *2 + 1)*sin(x)**2*a**4*b**2 + 48*cos(x)*log(tan(x/2)**2 + 1)*sin(x)**2*a** 3*b**3 - 48*cos(x)*log(tan(x/2)**2 + 1)*a**5*b - 96*cos(x)*log(tan(x/2)**2 + 1)*a**4*b**2 - 96*cos(x)*log(tan(x/2)**2 + 1)*a**3*b**3 - 96*cos(x)*log (tan(x/2)**2 + 1)*a**2*b**4 - 48*cos(x)*log(tan(x/2)**2 + 1)*a*b**5 - 48*c os(x)*log(tan(x/2)**2*a - tan(x/2)**2*b - a - b)*sin(x)**2*a**5*b - 96*cos (x)*log(tan(x/2)**2*a - tan(x/2)**2*b - a - b)*sin(x)**2*a**4*b**2 - 48*co s(x)*log(tan(x/2)**2*a - tan(x/2)**2*b - a - b)*sin(x)**2*a**3*b**3 + 48*c os(x)*log(tan(x/2)**2*a - tan(x/2)**2*b - a - b)*a**5*b + 96*cos(x)*log(ta n(x/2)**2*a - tan(x/2)**2*b - a - b)*a**4*b**2 + 96*cos(x)*log(tan(x/2)**2 *a - tan(x/2)**2*b - a - b)*a**3*b**3 + 96*cos(x)*log(tan(x/2)**2*a - tan( x/2)**2*b - a - b)*a**2*b**4 + 48*cos(x)*log(tan(x/2)**2*a - tan(x/2)**2*b - a - b)*a*b**5 - 12*cos(x)*sin(x)**2*a**5*b + 56*cos(x)*sin(x)**2*a**4*b **2 + 52*cos(x)*sin(x)**2*a**3*b**3 + 20*cos(x)*a**5*b - 40*cos(x)*a**4*b* *2 + 8*cos(x)*a**3*b**3 + 24*cos(x)*a**2*b**4 - 12*cos(x)*a*b**5 - 12*log( tan(x/2)**2 + 1)*sin(x)**4*a**6 - 24*log(tan(x/2)**2 + 1)*sin(x)**4*a**5*b - 12*log(tan(x/2)**2 + 1)*sin(x)**4*a**4*b**2 + 24*log(tan(x/2)**2 + 1)*s in(x)**2*a**6 + 48*log(tan(x/2)**2 + 1)*sin(x)**2*a**5*b + 96*log(tan(x/2) **2 + 1)*sin(x)**2*a**4*b**2 + 144*log(tan(x/2)**2 + 1)*sin(x)**2*a**3*b** 3 + 72*log(tan(x/2)**2 + 1)*sin(x)**2*a**2*b**4 - 12*log(tan(x/2)**2 + ...