3.2.2 \(\int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx\) [102]

3.2.2.1 Optimal result

Integrand size = 25, antiderivative size = 138 \[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {(i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {2 (A b-a B)}{\left (a^2+b^2\right ) d \sqrt {a+b \cot (c+d x)}} \]

output

(I*A+B)*arctanh((a+b*cot(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d-(I*A 
-B)*arctanh((a+b*cot(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d+2*(A*b-B 
*a)/(a^2+b^2)/d/(a+b*cot(d*x+c))^(1/2)
 

3.2.2.2 Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=-\frac {\frac {\left (a A b+A b \sqrt {-b^2}+b^2 B-a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}-\frac {\left (a A b-A b \sqrt {-b^2}+b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}+\frac {2 (-A b+a B)}{\sqrt {a+b \cot (c+d x)}}}{\left (a^2+b^2\right ) d} \]

input

Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]
 

output

-((((a*A*b + A*b*Sqrt[-b^2] + b^2*B - a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*C 
ot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - (( 
a*A*b - A*b*Sqrt[-b^2] + b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Cot[c 
+ d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*(-(A 
*b) + a*B))/Sqrt[a + b*Cot[c + d*x]])/((a^2 + b^2)*d))
 

3.2.2.3 Rubi [A] (warning: unable to verify)

Time = 0.69 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {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.

\(\displaystyle \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A-B \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {\int \frac {a A+b B-(A b-a B) \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}+\frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a A+b B-(a B-A b) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}\)

\(\Big \downarrow \) 4022

\(\displaystyle \frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {1}{2} (a-i b) (A+i B) \int \frac {1-i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}}dx+\frac {1}{2} (a+i b) (A-i B) \int \frac {i \cot (c+d x)+1}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {1}{2} (a+i b) (A-i B) \int \frac {1-i \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {1}{2} (a-i b) (A+i B) \int \frac {i \tan \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {i (a-i b) (A+i B) \int -\frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}-\frac {i (a+i b) (A-i B) \int -\frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}}{a^2+b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {\frac {i (a+i b) (A-i B) \int \frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}-\frac {i (a-i b) (A+i B) \int \frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}}{a^2+b^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {-\frac {(a-i b) (A+i B) \int \frac {1}{-\frac {i \cot ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \cot (c+d x)}}{b d}-\frac {(a+i b) (A-i B) \int \frac {1}{\frac {i \cot ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \cot (c+d x)}}{b d}}{a^2+b^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 (A b-a B)}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}+\frac {-\frac {(a+i b) (A-i B) \arctan \left (\frac {\cot (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {(a-i b) (A+i B) \arctan \left (\frac {\cot (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}}{a^2+b^2}\)

input

Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]
 

output

(-(((a + I*b)*(A - I*B)*ArcTan[Cot[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b] 
*d)) - ((a - I*b)*(A + I*B)*ArcTan[Cot[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + 
I*b]*d))/(a^2 + b^2) + (2*(A*b - a*B))/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d 
*x]])
 

3.2.2.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

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]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4012

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 
]
 

rule 4020

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]
 

rule 4022

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]
 

3.2.2.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3682\) vs. \(2(118)=236\).

Time = 0.15 (sec) , antiderivative size = 3683, normalized size of antiderivative = 26.69

method	result	size
parts	\(\text {Expression too large to display}\)	\(3683\)
derivativedivides	\(\text {Expression too large to display}\)	\(7951\)
default	\(\text {Expression too large to display}\)	\(7951\)

input

int((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

output

A*(-1/4/d/b/(a^2+b^2)^2*ln(b*cot(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)*a^3-1/ 
4/d*b/(a^2+b^2)^2*ln(b*cot(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)*a+1/4/d/b/(a 
^2+b^2)^(5/2)*ln(b*cot(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)*a^4-1/4/d*b^3/(a 
^2+b^2)^(5/2)*ln(b*cot(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/(a^2+b^2)^ 
(3/2)/(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/d*b/(a^2+b^ 
2)^(3/2)/(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)^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^2-1/d/b/(a^2+b^ 
2)^(5/2)/(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^5+1/d*b^3/(a 
^2+b^2)^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))-3/d*b^3/(a^2+ 
b^2)^(5/2)/(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-4/d*b/(...
 

3.2.2.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4572 vs. \(2 (112) = 224\).

Time = 0.80 (sec) , antiderivative size = 4572, normalized size of antiderivative = 33.13 \[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

input

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x, algorithm="fricas")
 

output

-1/2*(((a^2*b + b^3)*d*cos(2*d*x + 2*c) + (a^3 + a*b^2)*d*sin(2*d*x + 2*c) 
 + (a^2*b + b^3)*d)*sqrt(-(6*A*B*a^2*b - 2*A*B*b^3 + (A^2 - B^2)*a^3 - 3*( 
A^2 - B^2)*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(4*A^2*B^ 
2*a^6 - 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 
+ 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 - 12*(A^3 
*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/((a^12 + 6*a^10*b^2 + 15* 
a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^ 
4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(2*(A^3*B + A*B^3)*a^3 - 3*(A^4 - B^4) 
*a^2*b - 6*(A^3*B + A*B^3)*a*b^2 + (A^4 - B^4)*b^3)*sqrt((b*cos(2*d*x + 2* 
c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) + ((A*a^8 + 2*B*a^7*b + 2*A 
*a^6*b^2 + 6*B*a^5*b^3 + 6*B*a^3*b^5 - 2*A*a^2*b^6 + 2*B*a*b^7 - A*b^8)*d^ 
3*sqrt(-(4*A^2*B^2*a^6 - 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 
+ 3*B^4)*a^4*b^2 + 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)* 
a^2*b^4 - 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/((a^12 + 
 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^ 
4)) + (2*A*B^2*a^5 - (7*A^2*B - 3*B^3)*a^4*b + 2*(3*A^3 - 7*A*B^2)*a^3*b^2 
 + 4*(4*A^2*B - B^3)*a^2*b^3 - 2*(A^3 - 4*A*B^2)*a*b^4 - (A^2*B - B^3)*b^5 
)*d)*sqrt(-(6*A*B*a^2*b - 2*A*B*b^3 + (A^2 - B^2)*a^3 - 3*(A^2 - B^2)*a*b^ 
2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(4*A^2*B^2*a^6 - 12*(A^3 
*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 + 40*(A^3*B ...
 

3.2.2.6 Sympy [F]

\[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\int \frac {A + B \cot {\left (c + d x \right )}}{\left (a + b \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))**(3/2),x)
 

output

Integral((A + B*cot(c + d*x))/(a + b*cot(c + d*x))**(3/2), x)
 

3.2.2.7 Maxima [F]

\[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\int { \frac {B \cot \left (d x + c\right ) + A}{{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x, algorithm="maxima")
 

output

integrate((B*cot(d*x + c) + A)/(b*cot(d*x + c) + a)^(3/2), x)
 

3.2.2.8 Giac [F]

\[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\int { \frac {B \cot \left (d x + c\right ) + A}{{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x, algorithm="giac")
 

output

integrate((B*cot(d*x + c) + A)/(b*cot(d*x + c) + a)^(3/2), x)
 

3.2.2.9 Mupad [B] (verification not implemented)

Time = 19.29 (sec) , antiderivative size = 5737, normalized size of antiderivative = 41.57 \[ \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

input

int((A + B*cot(c + d*x))/(a + b*cot(c + d*x))^(3/2),x)
 

output

(log((((a + b*cot(c + d*x))^(1/2)*(16*A^2*b^10*d^3 + 32*A^2*a^2*b^8*d^3 - 
32*A^2*a^6*b^4*d^3 - 16*A^2*a^8*b^2*d^3) + ((((96*A^4*a^2*b^4*d^4 - 16*A^4 
*b^6*d^4 - 144*A^4*a^4*b^2*d^4)^(1/2) - 4*A^2*a^3*d^2 + 12*A^2*a*b^2*d^2)/ 
(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(64*A*a*b^11*d^ 
4 - ((((96*A^4*a^2*b^4*d^4 - 16*A^4*b^6*d^4 - 144*A^4*a^4*b^2*d^4)^(1/2) - 
 4*A^2*a^3*d^2 + 12*A^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3* 
a^4*b^2*d^4))^(1/2)*(a + b*cot(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^ 
10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2 
*d^5))/4 + 256*A*a^3*b^9*d^4 + 384*A*a^5*b^7*d^4 + 256*A*a^7*b^5*d^4 + 64* 
A*a^9*b^3*d^4))/4)*(((96*A^4*a^2*b^4*d^4 - 16*A^4*b^6*d^4 - 144*A^4*a^4*b^ 
2*d^4)^(1/2) - 4*A^2*a^3*d^2 + 12*A^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^ 
2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - 8*A^3*b^9*d^2 - 24*A^3*a^2*b^7*d^2 
- 24*A^3*a^4*b^5*d^2 - 8*A^3*a^6*b^3*d^2)*(((96*A^4*a^2*b^4*d^4 - 16*A^4*b 
^6*d^4 - 144*A^4*a^4*b^2*d^4)^(1/2) - 4*A^2*a^3*d^2 + 12*A^2*a*b^2*d^2)/(a 
^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 + (log((((a + 
b*cot(c + d*x))^(1/2)*(16*A^2*b^10*d^3 + 32*A^2*a^2*b^8*d^3 - 32*A^2*a^6*b 
^4*d^3 - 16*A^2*a^8*b^2*d^3) + ((-((96*A^4*a^2*b^4*d^4 - 16*A^4*b^6*d^4 - 
144*A^4*a^4*b^2*d^4)^(1/2) + 4*A^2*a^3*d^2 - 12*A^2*a*b^2*d^2)/(a^6*d^4 + 
b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(64*A*a*b^11*d^4 - ((-((96 
*A^4*a^2*b^4*d^4 - 16*A^4*b^6*d^4 - 144*A^4*a^4*b^2*d^4)^(1/2) + 4*A^2*...