\(\int \cot ^3(c+b x) \sin (a+b x) \, dx\) [236]

Optimal result

Integrand size = 15, antiderivative size = 74 \[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {3 \text {arctanh}(\cos (c+b x)) \sin (a-c)}{2 b}-\frac {\cot (c+b x) \csc (c+b x) \sin (a-c)}{2 b}-\frac {\sin (a+b x)}{b} \] Output:

-cos(a-c)*csc(b*x+c)/b+3/2*arctanh(cos(b*x+c))*sin(a-c)/b-1/2*cot(b*x+c)*c 
sc(b*x+c)*sin(a-c)/b-sin(b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\frac {12 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \sin (a-c)+\csc ^2(c+b x) (2 \sin (a-2 c-b x)-5 \sin (a+b x)+\sin (a+2 c+3 b x))}{4 b} \] Input:

Integrate[Cot[c + b*x]^3*Sin[a + b*x],x]
 

Output:

(12*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Sin[a - c] + Csc[c + b*x]^2*(2*S 
in[a - 2*c - b*x] - 5*Sin[a + b*x] + Sin[a + 2*c + 3*b*x]))/(4*b)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {5089, 3042, 3091, 3042, 4257, 5088, 3042, 25, 3086, 24, 5089, 3042, 3117, 4257}

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.

\(\displaystyle \int \sin (a+b x) \cot ^3(b x+c) \, dx\)

\(\Big \downarrow \) 5089

\(\displaystyle \int \cos (a+b x) \cot ^2(c+b x)dx+\sin (a-c) \int \cot ^2(c+b x) \csc (c+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (a+b x) \cot ^2(c+b x)dx+\sin (a-c) \int \sec \left (c+b x-\frac {\pi }{2}\right ) \tan \left (c+b x-\frac {\pi }{2}\right )^2dx\)

\(\Big \downarrow \) 3091

\(\displaystyle \int \cos (a+b x) \cot ^2(c+b x)dx+\sin (a-c) \left (-\frac {1}{2} \int \csc (c+b x)dx-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (a+b x) \cot ^2(c+b x)dx+\sin (a-c) \left (-\frac {1}{2} \int \csc (c+b x)dx-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 4257

\(\displaystyle \int \cos (a+b x) \cot ^2(c+b x)dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 5088

\(\displaystyle -\int \cot (c+b x) \sin (a+b x)dx+\cos (a-c) \int \cot (c+b x) \csc (c+b x)dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \cot (c+b x) \sin (a+b x)dx+\cos (a-c) \int -\sec \left (c+b x-\frac {\pi }{2}\right ) \tan \left (c+b x-\frac {\pi }{2}\right )dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \cot (c+b x) \sin (a+b x)dx-\cos (a-c) \int \sec \left (\frac {1}{2} (2 c-\pi )+b x\right ) \tan \left (\frac {1}{2} (2 c-\pi )+b x\right )dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 3086

\(\displaystyle -\frac {\cos (a-c) \int 1d\csc (c+b x)}{b}-\int \cot (c+b x) \sin (a+b x)dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle -\int \cot (c+b x) \sin (a+b x)dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc (b x+c)}{b}\)

\(\Big \downarrow \) 5089

\(\displaystyle -\sin (a-c) \int \csc (c+b x)dx-\int \cos (a+b x)dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc (b x+c)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\sin (a-c) \int \csc (c+b x)dx-\int \sin \left (a+b x+\frac {\pi }{2}\right )dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc (b x+c)}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle -\sin (a-c) \int \csc (c+b x)dx+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc (b x+c)}{b}-\frac {\sin (a+b x)}{b}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\sin (a-c) \text {arctanh}(\cos (b x+c))}{b}+\sin (a-c) \left (\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc (b x+c)}{b}-\frac {\sin (a+b x)}{b}\)

Input:

Int[Cot[c + b*x]^3*Sin[a + b*x],x]
 

Output:

-((Cos[a - c]*Csc[c + b*x])/b) + (ArcTanh[Cos[c + b*x]]*Sin[a - c])/b + (A 
rcTanh[Cos[c + b*x]]/(2*b) - (Cot[c + b*x]*Csc[c + b*x])/(2*b))*Sin[a - c] 
 - Sin[a + b*x]/b
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 3042

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

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[a/f   Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 
), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 
] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])
 

rule 3091

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m 
 + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1))   Int[(a*Sec[e + f*x])^m*( 
b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & 
& NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 4257

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

rule 5088

Int[Cos[v_]*Cot[w_]^(n_.), x_Symbol] :> -Int[Sin[v]*Cot[w]^(n - 1), x] + Si 
mp[Cos[v - w]   Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - 
 w, x] && NeQ[w, v]
 

rule 5089

Int[Cot[w_]^(n_.)*Sin[v_], x_Symbol] :> Int[Cos[v]*Cot[w]^(n - 1), x] + Sim 
p[Sin[v - w]   Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - 
w, x] && NeQ[w, v]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.49

method	result	size
risch	\(\frac {i {\mathrm e}^{i \left (b x +a \right )}}{2 b}-\frac {i {\mathrm e}^{-i \left (b x +a \right )}}{2 b}+\frac {i \left (-3 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}-{\mathrm e}^{i \left (3 b x +3 a +4 c \right )}+{\mathrm e}^{i \left (b x +5 a \right )}+3 \,{\mathrm e}^{i \left (b x +3 a +2 c \right )}\right )}{2 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}\)	\(184\)

Input:

int(cot(b*x+c)^3*sin(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/2*I/b*exp(I*(b*x+a))-1/2*I/b*exp(-I*(b*x+a))+1/2*I/b/(-exp(2*I*(b*x+a+c) 
)+exp(2*I*a))^2*(-3*exp(I*(3*b*x+5*a+2*c))-exp(I*(3*b*x+3*a+4*c))+exp(I*(b 
*x+5*a))+3*exp(I*(b*x+3*a+2*c)))+3/2*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*sin 
(a-c)-3/2*ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*sin(a-c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (70) = 140\).

Time = 0.10 (sec) , antiderivative size = 372, normalized size of antiderivative = 5.03 \[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\frac {\frac {3 \, \sqrt {2} {\left (2 \, {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + {\left (2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \sin \left (-2 \, a + 2 \, c\right )\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - 4 \, {\left (4 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 3 \, \cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \sin \left (b x + a\right ) - 4 \, {\left (4 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )} \sin \left (-2 \, a + 2 \, c\right )}{8 \, {\left (2 \, b \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, b \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - b \cos \left (-2 \, a + 2 \, c\right ) - b\right )}} \] Input:

integrate(cot(b*x+c)^3*sin(b*x+a),x, algorithm="fricas")
 

Output:

1/8*(3*sqrt(2)*(2*(cos(-2*a + 2*c)^2 - 1)*cos(b*x + a)*sin(b*x + a) + (2*c 
os(b*x + a)^2*cos(-2*a + 2*c) - cos(-2*a + 2*c) - 1)*sin(-2*a + 2*c))*log( 
-(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2*a 
+ 2*c) - 2*sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a) - sin(b*x + a)*sin( 
-2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - cos(-2*a + 2*c) + 3)/(2*cos(b*x + 
 a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos( 
-2*a + 2*c) - 1))/sqrt(cos(-2*a + 2*c) + 1) - 4*(4*cos(b*x + a)^2*cos(-2*a 
 + 2*c) - 3*cos(-2*a + 2*c) - 5)*sin(b*x + a) - 4*(4*cos(b*x + a)^3 - 5*co 
s(b*x + a))*sin(-2*a + 2*c))/(2*b*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*b*cos 
(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - b*cos(-2*a + 2*c) - b)
 

Sympy [F]

\[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \cot ^{3}{\left (b x + c \right )}\, dx \] Input:

integrate(cot(b*x+c)**3*sin(b*x+a),x)
 

Output:

Integral(sin(a + b*x)*cot(b*x + c)**3, x)
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1254 vs. \(2 (70) = 140\).

Time = 0.08 (sec) , antiderivative size = 1254, normalized size of antiderivative = 16.95 \[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(cot(b*x+c)^3*sin(b*x+a),x, algorithm="maxima")
 

Output:

1/4*(2*(sin(5*b*x + a + 4*c) - 2*sin(3*b*x + a + 2*c) + sin(b*x + a))*cos( 
6*b*x + 2*a + 4*c) + 2*(5*sin(4*b*x + 2*a + 2*c) + 2*sin(4*b*x + 4*c) - 2* 
sin(2*b*x + 2*a) - 5*sin(2*b*x + 2*c))*cos(5*b*x + a + 4*c) + 10*(2*sin(3* 
b*x + a + 2*c) - sin(b*x + a))*cos(4*b*x + 2*a + 2*c) + 4*(2*sin(3*b*x + a 
 + 2*c) - sin(b*x + a))*cos(4*b*x + 4*c) + 4*(2*sin(2*b*x + 2*a) + 5*sin(2 
*b*x + 2*c))*cos(3*b*x + a + 2*c) - 3*(cos(5*b*x + a + 4*c)^2*sin(-a + c) 
+ 4*cos(3*b*x + a + 2*c)^2*sin(-a + c) - 4*cos(3*b*x + a + 2*c)*cos(b*x + 
a)*sin(-a + c) + cos(b*x + a)^2*sin(-a + c) + sin(5*b*x + a + 4*c)^2*sin(- 
a + c) + 4*sin(3*b*x + a + 2*c)^2*sin(-a + c) - 4*sin(3*b*x + a + 2*c)*sin 
(b*x + a)*sin(-a + c) + sin(b*x + a)^2*sin(-a + c) - 2*(2*cos(3*b*x + a + 
2*c)*sin(-a + c) - cos(b*x + a)*sin(-a + c))*cos(5*b*x + a + 4*c) - 2*(2*s 
in(3*b*x + a + 2*c)*sin(-a + c) - sin(b*x + a)*sin(-a + c))*sin(5*b*x + a 
+ 4*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) + 3*(cos(5*b*x + a + 4*c)^2*sin(-a + c) + 4*cos(3 
*b*x + a + 2*c)^2*sin(-a + c) - 4*cos(3*b*x + a + 2*c)*cos(b*x + a)*sin(-a 
 + c) + cos(b*x + a)^2*sin(-a + c) + sin(5*b*x + a + 4*c)^2*sin(-a + c) + 
4*sin(3*b*x + a + 2*c)^2*sin(-a + c) - 4*sin(3*b*x + a + 2*c)*sin(b*x + a) 
*sin(-a + c) + sin(b*x + a)^2*sin(-a + c) - 2*(2*cos(3*b*x + a + 2*c)*sin( 
-a + c) - cos(b*x + a)*sin(-a + c))*cos(5*b*x + a + 4*c) - 2*(2*sin(3*b*x 
+ a + 2*c)*sin(-a + c) - sin(b*x + a)*sin(-a + c))*sin(5*b*x + a + 4*c)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (70) = 140\).

Time = 0.19 (sec) , antiderivative size = 870, normalized size of antiderivative = 11.76 \[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(cot(b*x+c)^3*sin(b*x+a),x, algorithm="giac")
 

Output:

1/4*(12*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)* 
tan(1/2*c) - tan(1/2*c)^2)*log(abs(tan(1/2*b*x)*tan(1/2*c) - 1))/(tan(1/2* 
a)^2*tan(1/2*c)^3 + tan(1/2*a)^2*tan(1/2*c) + tan(1/2*c)^3 + tan(1/2*c)) - 
 12*(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))*log(abs(tan(1/2*b*x) + tan(1/2*c)))/(tan(1/2*a)^2*tan(1/2*c)^2 + t 
an(1/2*a)^2 + tan(1/2*c)^2 + 1) + 8*(tan(1/2*b*x)*tan(1/2*a)^2 - tan(1/2*b 
*x) - 2*tan(1/2*a))/((tan(1/2*b*x)^2 + 1)*(tan(1/2*a)^2 + 1)) - (2*tan(1/2 
*b*x)^3*tan(1/2*a)*tan(1/2*c)^7 + tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*c)^7 
 + tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*c)^8 - 4*tan(1/2*b*x)^3*tan(1/2*a)^2* 
tan(1/2*c)^4 + 6*tan(1/2*b*x)^3*tan(1/2*a)*tan(1/2*c)^5 - 5*tan(1/2*b*x)^2 
*tan(1/2*a)^2*tan(1/2*c)^5 + 2*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*c)^6 - 4* 
tan(1/2*b*x)*tan(1/2*a)^2*tan(1/2*c)^6 - tan(1/2*b*x)^2*tan(1/2*c)^7 - 2*t 
an(1/2*b*x)*tan(1/2*a)*tan(1/2*c)^7 - 6*tan(1/2*b*x)^3*tan(1/2*a)*tan(1/2* 
c)^3 + 5*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*c)^3 + 4*tan(1/2*b*x)^3*tan(1 
/2*c)^4 - 22*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*c)^4 + 4*tan(1/2*b*x)*tan(1 
/2*a)^2*tan(1/2*c)^4 + 5*tan(1/2*b*x)^2*tan(1/2*c)^5 - 14*tan(1/2*b*x)*tan 
(1/2*a)*tan(1/2*c)^5 + 2*tan(1/2*a)^2*tan(1/2*c)^5 + 4*tan(1/2*b*x)*tan(1/ 
2*c)^6 + 2*tan(1/2*a)*tan(1/2*c)^6 - 2*tan(1/2*b*x)^3*tan(1/2*a)*tan(1/2*c 
) - tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*c) + 2*tan(1/2*b*x)^2*tan(1/2*a)*t 
an(1/2*c)^2 - 4*tan(1/2*b*x)*tan(1/2*a)^2*tan(1/2*c)^2 - 5*tan(1/2*b*x)...
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\text {Hanged} \] Input:

int(cot(c + b*x)^3*sin(a + b*x),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \cot ^3(c+b x) \sin (a+b x) \, dx=\frac {-\cos \left (b x +a \right ) \cot \left (b x +c \right )-\cot \left (b x +c \right )^{2} \sin \left (b x +a \right )-3 \left (\int \cot \left (b x +c \right ) \sin \left (b x +a \right )d x \right ) b -\sin \left (b x +a \right )}{2 b} \] Input:

int(cot(b*x+c)^3*sin(b*x+a),x)
 

Output:

( - cos(a + b*x)*cot(b*x + c) - cot(b*x + c)**2*sin(a + b*x) - 3*int(cot(b 
*x + c)*sin(a + b*x),x)*b - sin(a + b*x))/(2*b)