\(\int \frac {1}{(a+b \cot ^2(c+d x))^2} \, dx\) [6]

Optimal result

Integrand size = 14, antiderivative size = 97 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {x}{(a-b)^2}+\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^2 d}+\frac {b \cot (c+d x)}{2 a (a-b) d \left (a+b \cot ^2(c+d x)\right )} \]

[Out]

x/(a-b)^2+1/2*b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)+1/2*(3*a-b)*arctan(cot(d*x+c)*b^(1/2)/a^(1/2))*b^(1/2)
/a^(3/2)/(a-b)^2/d

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3742, 425, 536, 209, 211} \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {\sqrt {b} (3 a-b) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^2}+\frac {b \cot (c+d x)}{2 a d (a-b) \left (a+b \cot ^2(c+d x)\right )}+\frac {x}{(a-b)^2} \]

[In]

Int[(a + b*Cot[c + d*x]^2)^(-2),x]

[Out]

x/(a - b)^2 + ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*d) + (b*Cot[c +
d*x])/(2*a*(a - b)*d*(a + b*Cot[c + d*x]^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {b \cot (c+d x)}{2 a (a-b) d \left (a+b \cot ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {2 a-b-b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\cot (c+d x)\right )}{2 a (a-b) d} \\ & = \frac {b \cot (c+d x)}{2 a (a-b) d \left (a+b \cot ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{(a-b)^2 d}+\frac {((3 a-b) b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cot (c+d x)\right )}{2 a (a-b)^2 d} \\ & = \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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {-2 \arctan (\cot (c+d x))+\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+\frac {(a-b) b \cot (c+d x)}{a \left (a+b \cot ^2(c+d x)\right )}}{2 (a-b)^2 d} \]

[In]

Integrate[(a + b*Cot[c + d*x]^2)^(-2),x]

[Out]

(-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)

Maple [A] (verified)

Time = 0.12 (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	\(\frac {x}{a^{2}-2 a b +b^{2}}-\frac {i b \left ({\mathrm e}^{2 i \left (d x +c \right )} a +{\mathrm e}^{2 i \left (d x +c \right )} b -a +b \right )}{d a \left (-a +b \right )^{2} \left (-a \,{\mathrm e}^{4 i \left (d x +c \right )}+b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{2 i \left (d x +c \right )} a +2 \,{\mathrm e}^{2 i \left (d x +c \right )} b -a +b \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{4 a \left (a -b \right )^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b}{4 a^{2} \left (a -b \right )^{2} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{4 a \left (a -b \right )^{2} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b}{4 a^{2} \left (a -b \right )^{2} d}\)	\(335\)

[In]

int(1/(a+b*cot(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (85) = 170\).

Time = 0.31 (sec) , antiderivative size = 534, normalized size of antiderivative = 5.51 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\left [\frac {8 \, {\left (a^{2} - a b\right )} d x \cos \left (2 \, d x + 2 \, c\right ) - 8 \, {\left (a^{2} + a b\right )} d x + {\left (3 \, a^{2} + 2 \, a b - b^{2} - {\left (3 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + 4 \, {\left (a^{2} - a b - {\left (a^{2} + a b\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (2 \, d x + 2 \, c\right ) + a^{2} - 6 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}\right ) - 4 \, {\left (a b - b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{8 \, {\left ({\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} d \cos \left (2 \, d x + 2 \, c\right ) - {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right )} d\right )}}, \frac {4 \, {\left (a^{2} - a b\right )} d x \cos \left (2 \, d x + 2 \, c\right ) - 4 \, {\left (a^{2} + a b\right )} d x - {\left (3 \, a^{2} + 2 \, a b - b^{2} - {\left (3 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a + b\right )} \sqrt {\frac {b}{a}}}{2 \, b \sin \left (2 \, d x + 2 \, c\right )}\right ) - 2 \, {\left (a b - b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, {\left ({\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} d \cos \left (2 \, d x + 2 \, c\right ) - {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right )} d\right )}}\right ] \]

[In]

integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[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)*co
s(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))*sq
rt(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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2125 vs. \(2 (78) = 156\).

Time = 9.07 (sec) , antiderivative size = 2125, normalized size of antiderivative = 21.91 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cot(d*x+c)**2)**2,x)

[Out]

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*cot(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*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) - 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)**2 + 4*a**2*b**2*d*sqrt(-a/b) + 4*a*b**3*d*sqrt(-a/b)*cot(c + d*
x)**2) + 2*a*b*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)*co
t(c + d*x)**2) + 3*a*b*log(-sqrt(-a/b) + cot(c + d*x))*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)**2 + 4*a**2*b**2*d*sqrt(
-a/b) + 4*a*b**3*d*sqrt(-a/b)*cot(c + d*x)**2) - a*b*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*b*log(sqrt(-a/b) + cot(c + d*x))*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*sqr
t(-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) + a*b*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) - 2*b
**2*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) - b**2*log(-sqrt(-a/b) + cot(c + d*x))*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)**2 + 4*a**2*b**2*d*sqrt(-a/b) + 4*a*
b**3*d*sqrt(-a/b)*cot(c + d*x)**2) + b**2*log(sqrt(-a/b) + cot(c + d*x))*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)**2 + 4
*a**2*b**2*d*sqrt(-a/b) + 4*a*b**3*d*sqrt(-a/b)*cot(c + d*x)**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {\frac {b \tan \left (d x + c\right )}{a^{2} b - a b^{2} + {\left (a^{3} - a^{2} b\right )} \tan \left (d x + c\right )^{2}} - \frac {{\left (3 \, a b - b^{2}\right )} \arctan \left (\frac {a \tan \left (d x + c\right )}{\sqrt {a b}}\right )}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b}} + \frac {2 \, {\left (d x + c\right )}}{a^{2} - 2 \, a b + b^{2}}}{2 \, d} \]

[In]

integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (d x + c\right )}{\sqrt {a b}}\right )\right )} {\left (3 \, a b - b^{2}\right )}}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b}} - \frac {2 \, {\left (d x + c\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {b \tan \left (d x + c\right )}{{\left (a \tan \left (d x + c\right )^{2} + b\right )} {\left (a^{2} - a b\right )}}}{2 \, d} \]

[In]

integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-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

Mupad [B] (verification not implemented)

Time = 12.40 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^2} \, dx=\frac {\frac {a\,x}{{\left (a-b\right )}^2}+\frac {b\,x\,{\mathrm {cot}\left (c+d\,x\right )}^2}{{\left (a-b\right )}^2}+\frac {b\,\mathrm {cot}\left (c+d\,x\right )}{2\,a\,d\,\left (a-b\right )}}{b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a}+\frac {\mathrm {atan}\left (\frac {b\,\mathrm {cot}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )\,\left (3\,a\,b-b^2\right )}{\sqrt {a\,b}\,\left (2\,a^3\,d-a\,b\,\left (4\,a\,d-2\,b\,d\right )\right )} \]

[In]

int(1/(a + b*cot(c + d*x)^2)^2,x)

[Out]

((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)))