Integrand size = 14, antiderivative size = 97 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {x}{(a-b)^2}+\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^2 d}+\frac {b \cot (c+d x)}{2 a (a-b) d \left (a+b \cot ^2(c+d x)\right )} \] Output:
x/(a-b)^2+1/2*(3*a-b)*b^(1/2)*arctan(b^(1/2)*cot(d*x+c)/a^(1/2))/a^(3/2)/( a-b)^2/d+1/2*b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)
Time = 0.55 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {-2 \arctan (\cot (c+d x))+\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+\frac {(a-b) b \cot (c+d x)}{a \left (a+b \cot ^2(c+d x)\right )}}{2 (a-b)^2 d} \] Input:
Integrate[(a + b*Cot[c + d*x]^2)^(-2),x]
Output:
(-2*ArcTan[Cot[c + d*x]] + ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x] )/Sqrt[a]])/a^(3/2) + ((a - b)*b*Cot[c + d*x])/(a*(a + b*Cot[c + d*x]^2))) /(2*(a - b)^2*d)
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4144, 316, 397, 216, 218}
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.
| \(\displaystyle \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )^2}dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^2}d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle -\frac {\frac {\int \frac {-b \cot ^2(c+d x)+2 a-b}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )}d\cot (c+d x)}{2 a (a-b)}-\frac {b \cot (c+d x)}{2 a (a-b) \left (a+b \cot ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\frac {2 a \int \frac {1}{\cot ^2(c+d x)+1}d\cot (c+d x)}{a-b}-\frac {b (3 a-b) \int \frac {1}{b \cot ^2(c+d x)+a}d\cot (c+d x)}{a-b}}{2 a (a-b)}-\frac {b \cot (c+d x)}{2 a (a-b) \left (a+b \cot ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\frac {\frac {2 a \arctan (\cot (c+d x))}{a-b}-\frac {b (3 a-b) \int \frac {1}{b \cot ^2(c+d x)+a}d\cot (c+d x)}{a-b}}{2 a (a-b)}-\frac {b \cot (c+d x)}{2 a (a-b) \left (a+b \cot ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {2 a \arctan (\cot (c+d x))}{a-b}-\frac {\sqrt {b} (3 a-b) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)}}{2 a (a-b)}-\frac {b \cot (c+d x)}{2 a (a-b) \left (a+b \cot ^2(c+d x)\right )}}{d}\) |
Input:
Int[(a + b*Cot[c + d*x]^2)^(-2),x]
Output:
-((((2*a*ArcTan[Cot[c + d*x]])/(a - b) - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b ]*Cot[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)))/(2*a*(a - b)) - (b*Cot[c + d* x])/(2*a*(a - b)*(a + b*Cot[c + d*x]^2)))/d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )}{\left (a -b \right )^{2}}+\frac {b \left (\frac {\left (a -b \right ) \cot \left (d x +c \right )}{2 a \left (a +b \cot \left (d x +c \right )^{2}\right )}+\frac {\left (3 a -b \right ) \arctan \left (\frac {b \cot \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a -b \right )^{2}}}{d}\) | \(99\) |
| default | \(\frac {-\frac {\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )}{\left (a -b \right )^{2}}+\frac {b \left (\frac {\left (a -b \right ) \cot \left (d x +c \right )}{2 a \left (a +b \cot \left (d x +c \right )^{2}\right )}+\frac {\left (3 a -b \right ) \arctan \left (\frac {b \cot \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a -b \right )^{2}}}{d}\) | \(99\) |
| risch | \(\frac {x}{a^{2}-2 a b +b^{2}}-\frac {i b \left ({\mathrm e}^{2 i \left (d x +c \right )} a +{\mathrm e}^{2 i \left (d x +c \right )} b -a +b \right )}{d a \left (-a +b \right )^{2} \left (-{\mathrm e}^{4 i \left (d x +c \right )} a +b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{2 i \left (d x +c \right )} a +2 \,{\mathrm e}^{2 i \left (d x +c \right )} b -a +b \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{4 a \left (a -b \right )^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b}{4 a^{2} \left (a -b \right )^{2} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{4 a \left (a -b \right )^{2} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b}{4 a^{2} \left (a -b \right )^{2} d}\) | \(335\) |
Input:
int(1/(a+b*cot(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/(a-b)^2*(1/2*Pi-arccot(cot(d*x+c)))+1/(a-b)^2*b*(1/2*(a-b)/a*cot(d *x+c)/(a+b*cot(d*x+c)^2)+1/2*(3*a-b)/a/(a*b)^(1/2)*arctan(b*cot(d*x+c)/(a* b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (85) = 170\).
Time = 0.12 (sec) , antiderivative size = 534, normalized size of antiderivative = 5.51 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\left [\frac {8 \, {\left (a^{2} - a b\right )} d x \cos \left (2 \, d x + 2 \, c\right ) - 8 \, {\left (a^{2} + a b\right )} d x + {\left (3 \, a^{2} + 2 \, a b - b^{2} - {\left (3 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + 4 \, {\left (a^{2} - a b - {\left (a^{2} + a b\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (2 \, d x + 2 \, c\right ) + a^{2} - 6 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}\right ) - 4 \, {\left (a b - b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{8 \, {\left ({\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} d \cos \left (2 \, d x + 2 \, c\right ) - {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right )} d\right )}}, \frac {4 \, {\left (a^{2} - a b\right )} d x \cos \left (2 \, d x + 2 \, c\right ) - 4 \, {\left (a^{2} + a b\right )} d x - {\left (3 \, a^{2} + 2 \, a b - b^{2} - {\left (3 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a + b\right )} \sqrt {\frac {b}{a}}}{2 \, b \sin \left (2 \, d x + 2 \, c\right )}\right ) - 2 \, {\left (a b - b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, {\left ({\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} d \cos \left (2 \, d x + 2 \, c\right ) - {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right )} d\right )}}\right ] \] Input:
integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm="fricas")
Output:
[1/8*(8*(a^2 - a*b)*d*x*cos(2*d*x + 2*c) - 8*(a^2 + a*b)*d*x + (3*a^2 + 2* a*b - b^2 - (3*a^2 - 4*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(2*d*x + 2*c)^2 + 4*(a^2 - a*b - (a^2 + a*b)*cos(2*d*x + 2*c))*sqrt(-b/a)*sin(2*d*x + 2*c) + a^2 - 6*a*b + b^2 - 2*(a^2 - b^2)*cos( 2*d*x + 2*c))/((a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))) - 4*(a*b - b^2)*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^ 2 + a*b^3)*d), 1/4*(4*(a^2 - a*b)*d*x*cos(2*d*x + 2*c) - 4*(a^2 + a*b)*d*x - (3*a^2 + 2*a*b - b^2 - (3*a^2 - 4*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(b/a )*arctan(1/2*((a + b)*cos(2*d*x + 2*c) - a + b)*sqrt(b/a)/(b*sin(2*d*x + 2 *c))) - 2*(a*b - b^2)*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^ 3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^2 + a*b^3)*d)]
Leaf count of result is larger than twice the leaf count of optimal. 2125 vs. \(2 (78) = 156\).
Time = 6.91 (sec) , antiderivative size = 2125, normalized size of antiderivative = 21.91 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*cot(d*x+c)**2)**2,x)
Output:
Piecewise((zoo*x/cot(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (x/a**2, Eq(b , 0)), ((x - 1/(d*cot(c + d*x)) + 1/(3*d*cot(c + d*x)**3))/b**2, Eq(a, 0)) , (3*d*x*cot(c + d*x)**4/(8*b**2*d*cot(c + d*x)**4 + 16*b**2*d*cot(c + d*x )**2 + 8*b**2*d) + 6*d*x*cot(c + d*x)**2/(8*b**2*d*cot(c + d*x)**4 + 16*b* *2*d*cot(c + d*x)**2 + 8*b**2*d) + 3*d*x/(8*b**2*d*cot(c + d*x)**4 + 16*b* *2*d*cot(c + d*x)**2 + 8*b**2*d) - 3*cot(c + d*x)**3/(8*b**2*d*cot(c + d*x )**4 + 16*b**2*d*cot(c + d*x)**2 + 8*b**2*d) - 5*cot(c + d*x)/(8*b**2*d*co t(c + d*x)**4 + 16*b**2*d*cot(c + d*x)**2 + 8*b**2*d), Eq(a, b)), (x/(a + b*cot(c)**2)**2, Eq(d, 0)), (4*a**2*d*x*sqrt(-a/b)/(4*a**4*d*sqrt(-a/b) + 4*a**3*b*d*sqrt(-a/b)*cot(c + d*x)**2 - 8*a**3*b*d*sqrt(-a/b) - 8*a**2*b** 2*d*sqrt(-a/b)*cot(c + d*x)**2 + 4*a**2*b**2*d*sqrt(-a/b) + 4*a*b**3*d*sqr t(-a/b)*cot(c + d*x)**2) + 3*a**2*log(-sqrt(-a/b) + cot(c + d*x))/(4*a**4* d*sqrt(-a/b) + 4*a**3*b*d*sqrt(-a/b)*cot(c + d*x)**2 - 8*a**3*b*d*sqrt(-a/ b) - 8*a**2*b**2*d*sqrt(-a/b)*cot(c + d*x)**2 + 4*a**2*b**2*d*sqrt(-a/b) + 4*a*b**3*d*sqrt(-a/b)*cot(c + d*x)**2) - 3*a**2*log(sqrt(-a/b) + cot(c + d*x))/(4*a**4*d*sqrt(-a/b) + 4*a**3*b*d*sqrt(-a/b)*cot(c + d*x)**2 - 8*a** 3*b*d*sqrt(-a/b) - 8*a**2*b**2*d*sqrt(-a/b)*cot(c + d*x)**2 + 4*a**2*b**2* d*sqrt(-a/b) + 4*a*b**3*d*sqrt(-a/b)*cot(c + d*x)**2) + 4*a*b*d*x*sqrt(-a/ b)*cot(c + d*x)**2/(4*a**4*d*sqrt(-a/b) + 4*a**3*b*d*sqrt(-a/b)*cot(c + d* x)**2 - 8*a**3*b*d*sqrt(-a/b) - 8*a**2*b**2*d*sqrt(-a/b)*cot(c + d*x)**...
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {\frac {b \tan \left (d x + c\right )}{a^{2} b - a b^{2} + {\left (a^{3} - a^{2} b\right )} \tan \left (d x + c\right )^{2}} - \frac {{\left (3 \, a b - b^{2}\right )} \arctan \left (\frac {a \tan \left (d x + c\right )}{\sqrt {a b}}\right )}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b}} + \frac {2 \, {\left (d x + c\right )}}{a^{2} - 2 \, a b + b^{2}}}{2 \, d} \] Input:
integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/2*(b*tan(d*x + c)/(a^2*b - a*b^2 + (a^3 - a^2*b)*tan(d*x + c)^2) - (3*a* b - b^2)*arctan(a*tan(d*x + c)/sqrt(a*b))/((a^3 - 2*a^2*b + a*b^2)*sqrt(a* b)) + 2*(d*x + c)/(a^2 - 2*a*b + b^2))/d
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (d x + c\right )}{\sqrt {a b}}\right )\right )} {\left (3 \, a b - b^{2}\right )}}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b}} - \frac {2 \, {\left (d x + c\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {b \tan \left (d x + c\right )}{{\left (a \tan \left (d x + c\right )^{2} + b\right )} {\left (a^{2} - a b\right )}}}{2 \, d} \] Input:
integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm="giac")
Output:
-1/2*((pi*floor((d*x + c)/pi + 1/2)*sgn(a) + arctan(a*tan(d*x + c)/sqrt(a* b)))*(3*a*b - b^2)/((a^3 - 2*a^2*b + a*b^2)*sqrt(a*b)) - 2*(d*x + c)/(a^2 - 2*a*b + b^2) - b*tan(d*x + c)/((a*tan(d*x + c)^2 + b)*(a^2 - a*b)))/d
Time = 9.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {\frac {a\,x}{{\left (a-b\right )}^2}+\frac {b\,x\,{\mathrm {cot}\left (c+d\,x\right )}^2}{{\left (a-b\right )}^2}+\frac {b\,\mathrm {cot}\left (c+d\,x\right )}{2\,a\,d\,\left (a-b\right )}}{b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a}+\frac {\mathrm {atan}\left (\frac {b\,\mathrm {cot}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )\,\left (3\,a\,b-b^2\right )}{\sqrt {a\,b}\,\left (2\,a^3\,d-a\,b\,\left (4\,a\,d-2\,b\,d\right )\right )} \] Input:
int(1/(a + b*cot(c + d*x)^2)^2,x)
Output:
((a*x)/(a - b)^2 + (b*x*cot(c + d*x)^2)/(a - b)^2 + (b*cot(c + d*x))/(2*a* d*(a - b)))/(a + b*cot(c + d*x)^2) + (atan((b*cot(c + d*x))/(a*b)^(1/2))*( 3*a*b - b^2))/((a*b)^(1/2)*(2*a^3*d - a*b*(4*a*d - 2*b*d)))
Time = 0.20 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\cot \left (d x +c \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \cot \left (d x +c \right )^{2} a b -\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\cot \left (d x +c \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \cot \left (d x +c \right )^{2} b^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\cot \left (d x +c \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\cot \left (d x +c \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) a b +2 \cot \left (d x +c \right )^{2} a^{2} b d x +\cot \left (d x +c \right ) a^{2} b -\cot \left (d x +c \right ) a \,b^{2}+2 a^{3} d x}{2 a^{2} d \left (\cot \left (d x +c \right )^{2} a^{2} b -2 \cot \left (d x +c \right )^{2} a \,b^{2}+\cot \left (d x +c \right )^{2} b^{3}+a^{3}-2 a^{2} b +a \,b^{2}\right )} \] Input:
int(1/(a+b*cot(d*x+c)^2)^2,x)
Output:
(3*sqrt(b)*sqrt(a)*atan((cot(c + d*x)*b)/(sqrt(b)*sqrt(a)))*cot(c + d*x)** 2*a*b - sqrt(b)*sqrt(a)*atan((cot(c + d*x)*b)/(sqrt(b)*sqrt(a)))*cot(c + d *x)**2*b**2 + 3*sqrt(b)*sqrt(a)*atan((cot(c + d*x)*b)/(sqrt(b)*sqrt(a)))*a **2 - sqrt(b)*sqrt(a)*atan((cot(c + d*x)*b)/(sqrt(b)*sqrt(a)))*a*b + 2*cot (c + d*x)**2*a**2*b*d*x + cot(c + d*x)*a**2*b - cot(c + d*x)*a*b**2 + 2*a* *3*d*x)/(2*a**2*d*(cot(c + d*x)**2*a**2*b - 2*cot(c + d*x)**2*a*b**2 + cot (c + d*x)**2*b**3 + a**3 - 2*a**2*b + a*b**2))