Integrand size = 27, antiderivative size = 174 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=-\frac {(i a-b) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {(i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}} \]
-(I*a-b)*arctanh((a+b*cot(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d+(I* a+b)*arctanh((a+b*cot(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d-4/3*a*b /(a^2+b^2)/d/(a+b*cot(d*x+c))^(3/2)-2*b*(3*a^2-b^2)/(a^2+b^2)^2/d/(a+b*cot (d*x+c))^(1/2)
Time = 3.16 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.45 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=\frac {b \left (\frac {3 \left (a^3-3 a b^2+3 a^2 \sqrt {-b^2}+\left (-b^2\right )^{3/2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}+\frac {3 \left (-a^3+3 a b^2+3 a^2 \sqrt {-b^2}+\left (-b^2\right )^{3/2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}-\frac {4 a \left (a^2+b^2\right )}{(a+b \cot (c+d x))^{3/2}}+\frac {6 \left (-3 a^2+b^2\right )}{\sqrt {a+b \cot (c+d x)}}\right )}{3 \left (a^2+b^2\right )^2 d} \]
(b*((3*(a^3 - 3*a*b^2 + 3*a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) + (3*(-a^3 + 3*a*b^2 + 3*a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b* Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) - ( 4*a*(a^2 + b^2))/(a + b*Cot[c + d*x])^(3/2) + (6*(-3*a^2 + b^2))/Sqrt[a + b*Cot[c + d*x]]))/(3*(a^2 + b^2)^2*d)
Time = 0.91 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 4012, 25, 3042, 4012, 3042, 4022, 3042, 4020, 25, 73, 221}
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.
| \(\displaystyle \int \frac {b \cot (c+d x)-a}{(a+b \cot (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {-a-b \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int -\frac {a^2-2 b \cot (c+d x) a-b^2}{(a+b \cot (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {a^2-2 b \cot (c+d x) a-b^2}{(a+b \cot (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {a^2+2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}+\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {1}{2} (a-i b)^3 \int \frac {1-i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {i \cot (c+d x)+1}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {1}{2} (a-i b)^3 \int \frac {i \tan \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {i (a-i b)^3 \int -\frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}-\frac {i (a+i b)^3 \int -\frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {i (a+i b)^3 \int \frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}-\frac {i (a-i b)^3 \int \frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {-\frac {(a-i b)^3 \int \frac {1}{-\frac {i \cot ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \cot (c+d x)}}{b d}-\frac {(a+i b)^3 \int \frac {1}{\frac {i \cot ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \cot (c+d x)}}{b d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {-\frac {(a-i b)^3 \arctan \left (\frac {\cot (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {(a+i b)^3 \arctan \left (\frac {\cot (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}}{a^2+b^2}}{a^2+b^2}\) |
(-4*a*b)/(3*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^(3/2)) - ((-(((a + I*b)^3*A rcTan[Cot[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) - ((a - I*b)^3*ArcTa n[Cot[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d))/(a^2 + b^2) + (2*b*(3*a^ 2 - b^2))/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]]))/(a^2 + b^2)
3.2.6.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(3054\) vs. \(2(150)=300\).
Time = 0.12 (sec) , antiderivative size = 3055, normalized size of antiderivative = 17.56
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(3055\) |
| default | \(\text {Expression too large to display}\) | \(3055\) |
| parts | \(\text {Expression too large to display}\) | \(4473\) |
-6/d*b/(a^2+b^2)^2/(a+b*cot(d*x+c))^(1/2)*a^2+1/4/d*b^5/(a^2+b^2)^(7/2)*ln ((a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^ 2)^(1/2)-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/4/d*b^5/(a^2+b^2)^(7/2)*ln(b*c ot(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2) ^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b^5/(a^2+b^2)^3/(2*(a^2+b^2)^(1/ 2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/ 2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))-1/d*b^5/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)- 2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2) )/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))+1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)- 2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)) /(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^7-5/d*b^3/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^( 1/2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^( 1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3+1/4/d/b/(a^2+b^2)^(7/2)*ln((a+b*c ot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2 )-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^6-5/4/d*b^3/(a^2+b^2)^(7/2)*ln((a+b*c ot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2 )-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+7/d*b^5/(a^2+b^2)^(7/2)/(2*(a^2+b^2 )^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2* a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b ^2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/...
Leaf count of result is larger than twice the leaf count of optimal. 3922 vs. \(2 (141) = 282\).
Time = 0.42 (sec) , antiderivative size = 3922, normalized size of antiderivative = 22.54 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
-1/6*(3*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d*cos(2*d*x + 2*c) - 2*(a^5*b + 2 *a^3*b^3 + a*b^5)*d*sin(2*d*x + 2*c) - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) *d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 + (a^10 + 5*a^8*b^2 + 1 0*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2*sqrt(-(49*a^12*b^2 - 490*a^ 10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/ ((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^1 0*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d ^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2 ))*log(-(7*a^8*b - 28*a^6*b^3 - 14*a^4*b^5 + 20*a^2*b^7 - b^9)*sqrt((b*cos (2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) + ((a^14 - a^12* b^2 - 19*a^10*b^4 - 45*a^8*b^6 - 45*a^6*b^8 - 19*a^4*b^10 - a^2*b^12 + b^1 4)*d^3*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 5 11*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120 *a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 4 5*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) + 4*(7*a^9*b^2 - 42*a^7*b^4 + 56*a^ 5*b^6 - 22*a^3*b^8 + a*b^10)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a *b^6 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2 *sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4 *b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14* b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a...
\[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=- \int \frac {a}{a^{2} \sqrt {a + b \cot {\left (c + d x \right )}} + 2 a b \sqrt {a + b \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )} + b^{2} \sqrt {a + b \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}}\, dx - \int \left (- \frac {b \cot {\left (c + d x \right )}}{a^{2} \sqrt {a + b \cot {\left (c + d x \right )}} + 2 a b \sqrt {a + b \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )} + b^{2} \sqrt {a + b \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}}\right )\, dx \]
-Integral(a/(a**2*sqrt(a + b*cot(c + d*x)) + 2*a*b*sqrt(a + b*cot(c + d*x) )*cot(c + d*x) + b**2*sqrt(a + b*cot(c + d*x))*cot(c + d*x)**2), x) - Inte gral(-b*cot(c + d*x)/(a**2*sqrt(a + b*cot(c + d*x)) + 2*a*b*sqrt(a + b*cot (c + d*x))*cot(c + d*x) + b**2*sqrt(a + b*cot(c + d*x))*cot(c + d*x)**2), x)
\[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=\int { \frac {b \cot \left (d x + c\right ) - a}{{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=\int { \frac {b \cot \left (d x + c\right ) - a}{{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Time = 29.37 (sec) , antiderivative size = 8438, normalized size of antiderivative = 48.49 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
(log(((-(4*a^7*d^2 - (320*a^6*b^8*d^4 - 16*a^4*b^10*d^4 - 1760*a^8*b^6*d^4 + 1600*a^10*b^4*d^4 - 400*a^12*b^2*d^4)^(1/2) + 20*a^3*b^4*d^2 - 40*a^5*b ^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4 *d^4 + 5*a^8*b^2*d^4))^(1/2)*((a + b*cot(c + d*x))^(1/2)*(320*a^6*b^14*d^3 - 16*a^2*b^18*d^3 + 1024*a^8*b^12*d^3 + 1440*a^10*b^10*d^3 + 1024*a^12*b^ 8*d^3 + 320*a^14*b^6*d^3 - 16*a^18*b^2*d^3) - ((-(4*a^7*d^2 - (320*a^6*b^8 *d^4 - 16*a^4*b^10*d^4 - 1760*a^8*b^6*d^4 + 1600*a^10*b^4*d^4 - 400*a^12*b ^2*d^4)^(1/2) + 20*a^3*b^4*d^2 - 40*a^5*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5* a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((- (4*a^7*d^2 - (320*a^6*b^8*d^4 - 16*a^4*b^10*d^4 - 1760*a^8*b^6*d^4 + 1600* a^10*b^4*d^4 - 400*a^12*b^2*d^4)^(1/2) + 20*a^3*b^4*d^2 - 40*a^5*b^2*d^2)/ (a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5 *a^8*b^2*d^4))^(1/2)*(a + b*cot(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b ^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 161 28*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6 *d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 - 32*a*b^21*d^4 - 160*a^3*b^ 19*d^4 - 128*a^5*b^17*d^4 + 896*a^7*b^15*d^4 + 3136*a^9*b^13*d^4 + 4928*a^ 11*b^11*d^4 + 4480*a^13*b^9*d^4 + 2432*a^15*b^7*d^4 + 736*a^17*b^5*d^4 + 9 6*a^19*b^3*d^4))/4))/4 + 16*a^4*b^15*d^2 + 96*a^6*b^13*d^2 + 240*a^8*b^11* d^2 + 320*a^10*b^9*d^2 + 240*a^12*b^7*d^2 + 96*a^14*b^5*d^2 + 16*a^16*b...