\(\int \cot (a+b x) \cot (c+b x) \, dx\) [141]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \cot (a+b x) \cot (c+b x) \, dx=-x-\frac {\cot (a-c) \log (\sin (a+b x))}{b}+\frac {\cot (a-c) \log (\sin (c+b x))}{b} \]

[Out]

-x-cot(a-c)*ln(sin(b*x+a))/b+cot(a-c)*ln(sin(b*x+c))/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4709, 4707, 3556} \[ \int \cot (a+b x) \cot (c+b x) \, dx=-\frac {\cot (a-c) \log (\sin (a+b x))}{b}+\frac {\cot (a-c) \log (\sin (b x+c))}{b}-x \]

[In]

Int[Cot[a + b*x]*Cot[c + b*x],x]

[Out]

-x - (Cot[a - c]*Log[Sin[a + b*x]])/b + (Cot[a - c]*Log[Sin[c + b*x]])/b

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4707

Int[Csc[(a_.) + (b_.)*(x_)]*Csc[(c_) + (d_.)*(x_)], x_Symbol] :> Dist[Csc[(b*c - a*d)/b], Int[Cot[a + b*x], x]
, x] - Dist[Csc[(b*c - a*d)/d], Int[Cot[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ
[b*c - a*d, 0]

Rule 4709

Int[Cot[(a_.) + (b_.)*(x_)]*Cot[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[(-b)*(x/d), x] + Dist[Cos[(b*c - a*d)/d]
, Int[Csc[a + b*x]*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]

Rubi steps \begin{align*} \text {integral}& = -x+\cos (a-c) \int \csc (a+b x) \csc (c+b x) \, dx \\ & = -x-\cot (a-c) \int \cot (a+b x) \, dx+\cot (a-c) \int \cot (c+b x) \, dx \\ & = -x-\frac {\cot (a-c) \log (\sin (a+b x))}{b}+\frac {\cot (a-c) \log (\sin (c+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \cot (a+b x) \cot (c+b x) \, dx=-x+\frac {\cot (a-c) (-\log (\sin (a+b x))+\log (\sin (c+b x)))}{b} \]

[In]

Integrate[Cot[a + b*x]*Cot[c + b*x],x]

[Out]

-x + (Cot[a - c]*(-Log[Sin[a + b*x]] + Log[Sin[c + b*x]]))/b

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 4.54

method	result	size
risch	\(-x -\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}-\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) {\mathrm e}^{2 i c}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i c}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}\)	\(177\)

[In]

int(cot(b*x+a)*cot(b*x+c),x,method=_RETURNVERBOSE)

[Out]

-x-I/b/(exp(2*I*a)-exp(2*I*c))*ln(exp(2*I*(b*x+a))-1)*exp(2*I*a)-I/b/(exp(2*I*a)-exp(2*I*c))*ln(exp(2*I*(b*x+a
))-1)*exp(2*I*c)+I/b/(exp(2*I*a)-exp(2*I*c))*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))*exp(2*I*a)+I/b/(exp(2*I*a)-ex
p(2*I*c))*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))*exp(2*I*c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (39) = 78\).

Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.03 \[ \int \cot (a+b x) \cot (c+b x) \, dx=-\frac {2 \, b x \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {\cos \left (2 \, b x + 2 \, c\right ) \cos \left (-2 \, a + 2 \, c\right ) + \sin \left (2 \, b x + 2 \, c\right ) \sin \left (-2 \, a + 2 \, c\right ) - 1}{\cos \left (-2 \, a + 2 \, c\right ) + 1}\right ) + {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, c\right ) + \frac {1}{2}\right )}{2 \, b \sin \left (-2 \, a + 2 \, c\right )} \]

[In]

integrate(cot(b*x+a)*cot(b*x+c),x, algorithm="fricas")

[Out]

-1/2*(2*b*x*sin(-2*a + 2*c) - (cos(-2*a + 2*c) + 1)*log(-(cos(2*b*x + 2*c)*cos(-2*a + 2*c) + sin(2*b*x + 2*c)*
sin(-2*a + 2*c) - 1)/(cos(-2*a + 2*c) + 1)) + (cos(-2*a + 2*c) + 1)*log(-1/2*cos(2*b*x + 2*c) + 1/2))/(b*sin(-
2*a + 2*c))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1982 vs. \(2 (31) = 62\).

Time = 11.46 (sec) , antiderivative size = 7404, normalized size of antiderivative = 189.85 \[ \int \cot (a+b x) \cot (c+b x) \, dx=\text {Too large to display} \]

[In]

integrate(cot(b*x+a)*cot(b*x+c),x)

[Out]

Piecewise((x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)), Eq(b, 0) & Eq(a, atan(tan(c)) + pi*floor((c - pi
/2)/pi))), (-b*x*tan(c)**5/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*ta
n(c) + b*tan(b*x)) - b*x*tan(c)**4*tan(b*x)/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**
2*tan(b*x) + b*tan(c) + b*tan(b*x)) + b*x*tan(c)**3/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*
tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) + b*x*tan(c)**2*tan(b*x)/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b
*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - 2*log(tan(c) + tan(b*x))*tan(c)**4/(b*tan(c)**5
+ b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - 2*log(tan(c) + tan
(b*x))*tan(c)**3*tan(b*x)/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan
(c) + b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**4/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*t
an(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**3*tan(b*x)/(b*tan(c)**5 + b*tan(c)**
4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - tan(c)**6/(b*tan(c)**5 + b*tan(
c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - tan(c)**4/(b*tan(c)**5 + b*
tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)), Eq(a, atan(tan(c)) + pi*
floor((c - pi/2)/pi))), (x/(cot(a)*cot(c) + zoo*cot(a) + zoo*cot(c) + zoo), Eq(b, 0)), (-2*b*x*tan(a)**3*tan(c
)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*ta
n(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*b*x*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*
tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3
- 2*b*tan(c)) + 2*b*x*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2
*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*b*x*tan(a)*tan(c)**2
/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)
**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(a) + tan(b*x))*tan(a)**3*tan(c)**3/(2*b*tan(a)**3*t
an(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a)
- 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(a) + tan(b*x))*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a
)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*
b*tan(c)) + 2*log(tan(c) + tan(b*x))*tan(a)**3*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)
**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log
(tan(c) + tan(b*x))*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*
tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(
a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*t
an(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)*tan(c)**3/(2*b*tan(a)
**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*ta
n(a) - 2*b*tan(c)**3 - 2*b*tan(c)), True)) + Piecewise((zoo*x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)),
Eq(b, 0) & Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))), (b*x*tan(c)**5/(b*tan(c)**5 + b*tan(c)**4*tan(b*x)
+ 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) + b*x*tan(c)**4*tan(b*x)/(b*tan(c)**5 + b*t
an(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - b*x*tan(c)**3/(b*tan(c)*
*5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - b*x*tan(c)**2*ta
n(b*x)/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) +
2*log(tan(c) + tan(b*x))*tan(c)**4/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*
x) + b*tan(c) + b*tan(b*x)) + 2*log(tan(c) + tan(b*x))*tan(c)**3*tan(b*x)/(b*tan(c)**5 + b*tan(c)**4*tan(b*x)
+ 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**4/(b*tan(c)**
5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c) + b*tan(b*x)) - log(tan(b*x)**2 +
1)*tan(c)**3*tan(b*x)/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*tan(c)
+ b*tan(b*x)) - tan(c)**4/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) + b*ta
n(c) + b*tan(b*x)) - tan(c)**2/(b*tan(c)**5 + b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) +
b*tan(c) + b*tan(b*x)), Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))), (zoo*x/(cot(a)*cot(c) + zoo*cot(a) + z
oo*cot(c) + zoo), Eq(b, 0)), (2*b*x*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)*
*2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*b*x*
tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c)
+ 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*b*x*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(
c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2
*b*tan(c)**3 - 2*b*tan(c)) + 2*b*x*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*t
an(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(
a) + tan(b*x))*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a
)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(a) + tan(b*x))*tan(a
)*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a
)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log(tan(c) + tan(b*x))*tan(a)**3*tan(c)/(2*b*tan(a)
**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*ta
n(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log(tan(c) + tan(b*x))*tan(a)*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan
(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 -
2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*t
an(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + log(tan(b*
x)**2 + 1)*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2
*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)), True))*cot(a)*cot(c) - Piecewise((z
oo*x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)), Eq(b, 0) & Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))
), (4*b*x*tan(c)**4/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan
(c) + 2*b*tan(b*x)) + 4*b*x*tan(c)**3*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*t
an(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) - 2*log(tan(c) + tan(b*x))*tan(c)**5/(2*b*tan(c)**5 + 2*b*tan(c
)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) - 2*log(tan(c) + tan(b*x))
*tan(c)**4*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan
(c) + 2*b*tan(b*x)) + 2*log(tan(c) + tan(b*x))*tan(c)**3/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)*
*3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + 2*log(tan(c) + tan(b*x))*tan(c)**2*tan(b*x)/(2*b*ta
n(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + log(t
an(b*x)**2 + 1)*tan(c)**5/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2
*b*tan(c) + 2*b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**4*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) +
4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**3/(2*b*tan(
c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) - log(tan
(b*x)**2 + 1)*tan(c)**2*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b
*x) + 2*b*tan(c) + 2*b*tan(b*x)) + 2*tan(c)**5/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*t
an(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + 2*tan(c)**3/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan
(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)), Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))),
(zoo*x/(cot(a)*cot(c) + zoo*cot(a) + zoo*cot(c) + zoo), Eq(b, 0)), (2*b*x*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(
c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2
*b*tan(c)**3 - 2*b*tan(c)) - 2*b*x*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*t
an(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log(tan(
a) + tan(b*x))*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*ta
n(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log(tan(a) + tan(b*x))*ta
n(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b
*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(c) + tan(b*x))*tan(a)**3*tan(c)**2/(2
*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2
+ 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(c) + tan(b*x))*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)*
*2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*
tan(c)**3 - 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 -
2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c
)) - log(tan(b*x)**2 + 1)*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)*
*3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - log(tan(b*x)**2
+ 1)*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c
) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)*tan(c)**2/(2
*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2
+ 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)), True))*cot(a) - Piecewise((zoo*x/(zoo*cot(c) + zoo + cot(c)/tan(c
) + zoo/tan(c)), Eq(b, 0) & Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))), (4*b*x*tan(c)**4/(2*b*tan(c)**5 +
2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + 4*b*x*tan(c)**3
*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*
tan(b*x)) - 2*log(tan(c) + tan(b*x))*tan(c)**5/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*t
an(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) - 2*log(tan(c) + tan(b*x))*tan(c)**4*tan(b*x)/(2*b*tan(c)**5 +
2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + 2*log(tan(c) +
tan(b*x))*tan(c)**3/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan
(c) + 2*b*tan(b*x)) + 2*log(tan(c) + tan(b*x))*tan(c)**2*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*
b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**5/(2*b*tan(c)
**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + log(tan(b
*x)**2 + 1)*tan(c)**4*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x
) + 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**3/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*
tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**2*tan(b*x)/(2*b
*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(b*x)) + 2*
tan(c)**5/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*
tan(b*x)) + 2*tan(c)**3/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) + 2*b
*tan(c) + 2*b*tan(b*x)), Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))), (zoo*x/(cot(a)*cot(c) + zoo*cot(a) +
zoo*cot(c) + zoo), Eq(b, 0)), (2*b*x*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2
*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*b*x*ta
n(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b
*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log(tan(a) + tan(b*x))*tan(a)**2*tan(c)**3/(2
*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2
+ 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + 2*log(tan(a) + tan(b*x))*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)*
*2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*
tan(c)**3 - 2*b*tan(c)) - 2*log(tan(c) + tan(b*x))*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**
3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*t
an(c)) - 2*log(tan(c) + tan(b*x))*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*ta
n(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) + log(tan(b*x
)**2 + 1)*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)*
*2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)**2*t
an(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a
)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)**2*tan(c)/(2*b*tan(a)**3*
tan(c)**2 + 2*b*tan(a)**3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a)
- 2*b*tan(c)**3 - 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**
3 - 2*b*tan(a)**2*tan(c)**3 - 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*t
an(c)), True))*cot(c)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (39) = 78\).

Time = 0.25 (sec) , antiderivative size = 549, normalized size of antiderivative = 14.08 \[ \int \cot (a+b x) \cot (c+b x) \, dx=-\frac {{\left (2 \, b \cos \left (2 \, a\right ) \cos \left (2 \, c\right ) - b \cos \left (2 \, c\right )^{2} + 2 \, b \sin \left (2 \, a\right ) \sin \left (2 \, c\right ) - b \sin \left (2 \, c\right )^{2} - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} b\right )} x + {\left (\cos \left (2 \, a\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\right )} \arctan \left (\sin \left (b x\right ) + \sin \left (a\right ), \cos \left (b x\right ) - \cos \left (a\right )\right ) + {\left (\cos \left (2 \, a\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\right )} \arctan \left (\sin \left (b x\right ) - \sin \left (a\right ), \cos \left (b x\right ) + \cos \left (a\right )\right ) - {\left (\cos \left (2 \, a\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\right )} \arctan \left (\sin \left (b x\right ) + \sin \left (c\right ), \cos \left (b x\right ) - \cos \left (c\right )\right ) - {\left (\cos \left (2 \, a\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\right )} \arctan \left (\sin \left (b x\right ) - \sin \left (c\right ), \cos \left (b x\right ) + \cos \left (c\right )\right ) - {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) - {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) + {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right )}{2 \, b \cos \left (2 \, a\right ) \cos \left (2 \, c\right ) - b \cos \left (2 \, c\right )^{2} + 2 \, b \sin \left (2 \, a\right ) \sin \left (2 \, c\right ) - b \sin \left (2 \, c\right )^{2} - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} b} \]

[In]

integrate(cot(b*x+a)*cot(b*x+c),x, algorithm="maxima")

[Out]

-((2*b*cos(2*a)*cos(2*c) - b*cos(2*c)^2 + 2*b*sin(2*a)*sin(2*c) - b*sin(2*c)^2 - (cos(2*a)^2 + sin(2*a)^2)*b)*
x + (cos(2*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) + sin(a), cos(b*x) - cos(a)) + (cos(2
*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) - sin(a), cos(b*x) + cos(a)) - (cos(2*a)^2 - co
s(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) + sin(c), cos(b*x) - cos(c)) - (cos(2*a)^2 - cos(2*c)^2 +
sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) - sin(c), cos(b*x) + cos(c)) - (cos(2*c)*sin(2*a) - cos(2*a)*sin(2*
c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - (cos(2*c)*sin
(2*a) - cos(2*a)*sin(2*c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + si
n(a)^2) + (cos(2*c)*sin(2*a) - cos(2*a)*sin(2*c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 -
2*sin(b*x)*sin(c) + sin(c)^2) + (cos(2*c)*sin(2*a) - cos(2*a)*sin(2*c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(c) +
cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2))/(2*b*cos(2*a)*cos(2*c) - b*cos(2*c)^2 + 2*b*sin(2*a)*si
n(2*c) - b*sin(2*c)^2 - (cos(2*a)^2 + sin(2*a)^2)*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (39) = 78\).

Time = 0.30 (sec) , antiderivative size = 348, normalized size of antiderivative = 8.92 \[ \int \cot (a+b x) \cot (c+b x) \, dx=-\frac {2 \, b x + \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{4} + 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{2} - 1\right )} \log \left ({\left | \tan \left (b x\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - \tan \left (b x\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{3} - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, a\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, \tan \left (\frac {1}{2} \, c\right )^{2} - 1\right )} \log \left ({\left | \tan \left (b x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x\right ) - 2 \, \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{4} - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 2 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, c\right )}}{2 \, b} \]

[In]

integrate(cot(b*x+a)*cot(b*x+c),x, algorithm="giac")

[Out]

-1/2*(2*b*x + (tan(1/2*a)^4*tan(1/2*c)^2 - tan(1/2*a)^4 + 4*tan(1/2*a)^3*tan(1/2*c) - 2*tan(1/2*a)^2*tan(1/2*c
)^2 + 2*tan(1/2*a)^2 - 4*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 - 1)*log(abs(tan(b*x)*tan(1/2*a)^2 - tan(b*x) -
2*tan(1/2*a)))/(tan(1/2*a)^4*tan(1/2*c) - tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^3 - 2*tan(1/2*a)^2*tan(1/2*c)
+ tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a) + tan(1/2*c)) - (tan(1/2*a)^2*tan(1/2*c)^4 - 2*tan(1/2*a)^2*tan(1/2*c)
^2 + 4*tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 + tan(1/2*a)^2 - 4*tan(1/2*a)*tan(1/2*c) + 2*tan(1/2*c)^2 - 1)*l
og(abs(tan(b*x)*tan(1/2*c)^2 - tan(b*x) - 2*tan(1/2*c)))/(tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)*tan(1/2*c)^4
- tan(1/2*a)^2*tan(1/2*c) + 2*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*c)^3 - tan(1/2*a) + tan(1/2*c)))/b

Mupad [B] (verification not implemented)

Time = 31.69 (sec) , antiderivative size = 207, normalized size of antiderivative = 5.31 \[ \int \cot (a+b x) \cot (c+b x) \, dx=-\frac {\frac {x}{2}+x\,\left ({\sin \left (a-c\right )}^2-\frac {1}{2}\right )}{{\sin \left (a-c\right )}^2}-\frac {\frac {\sin \left (2\,a-2\,c\right )\,\ln \left ({\sin \left (2\,a-2\,c\right )}^2\,2{}\mathrm {i}+{\sin \left (a+b\,x\right )}^2\,2{}\mathrm {i}-{\sin \left (3\,a-2\,c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a-4\,c\right )-\sin \left (6\,a-4\,c+2\,b\,x\right )+\sin \left (2\,a+2\,b\,x\right )\right )}{2}-\frac {\sin \left (2\,a-2\,c\right )\,\ln \left ({\sin \left (2\,a-2\,c\right )}^2\,2{}\mathrm {i}+{\sin \left (c+b\,x\right )}^2\,2{}\mathrm {i}-{\sin \left (2\,a-c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a-4\,c\right )-\sin \left (4\,a-2\,c+2\,b\,x\right )+\sin \left (2\,c+2\,b\,x\right )\right )}{2}}{b\,{\sin \left (a-c\right )}^2} \]

[In]

int(cot(a + b*x)*cot(c + b*x),x)

[Out]

- (x/2 + x*(sin(a - c)^2 - 1/2))/sin(a - c)^2 - ((sin(2*a - 2*c)*log(sin(4*a - 4*c) - sin(6*a - 4*c + 2*b*x) +
sin(2*a + 2*b*x) + sin(2*a - 2*c)^2*2i + sin(a + b*x)^2*2i - sin(3*a - 2*c + b*x)^2*2i))/2 - (sin(2*a - 2*c)*
log(sin(4*a - 4*c) - sin(4*a - 2*c + 2*b*x) + sin(2*c + 2*b*x) + sin(2*a - 2*c)^2*2i + sin(c + b*x)^2*2i - sin
(2*a - c + b*x)^2*2i))/2)/(b*sin(a - c)^2)