\(\int \cot ^3(c+b x) \sin ^2(a+b x) \, dx\) [242]

Optimal result

Integrand size = 17, antiderivative size = 1 \[ \int \cot ^3(c+b x) \sin ^2(a+b x) \, dx=0 \] Output:

0
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 6.28 (sec) , antiderivative size = 422, normalized size of antiderivative = 422.00 \[ \int \cot ^3(c+b x) \sin ^2(a+b x) \, dx=-\frac {i \arctan (\tan (c+b x)) (-1+3 \cos (2 a-2 c))}{2 b}+\frac {\cos (2 a) \cos (2 b x)}{4 b}+\frac {(\cos (2 a-2 c-b x)-\cos (2 a-2 c+b x)) \csc (c) \csc (c+b x)}{2 b}+\frac {(-1+3 \cos (2 a-2 c)) \log \left (\sin ^2(c+b x)\right )}{4 b}-3 x \cos (a-c) \sin (a-c)-\frac {\csc ^2(c+b x) \sin ^2(a-c)}{2 b}+x \left (-\frac {i}{2}-\frac {3}{4} i \cos ^2(a)+\frac {9}{4} i \cos ^2(a) \cos ^2(c)+\frac {\cot (c)}{2}-\frac {3}{4} \cos ^2(a) \cot (c)+\frac {1}{2} (-1+3 \cos (2 a-2 c)) \cot (c)-\frac {3}{4} \cos ^2(a) \cos ^2(c) \cot (c)-\frac {3}{2} \cos (a) \sin (a)-\frac {9}{2} \cos (a) \cos ^2(c) \sin (a)+\frac {3}{2} i \cos (a) \cot (c) \sin (a)-\frac {3}{2} i \cos (a) \cos ^2(c) \cot (c) \sin (a)+\frac {3}{4} i \sin ^2(a)-\frac {9}{4} i \cos ^2(c) \sin ^2(a)+\frac {3}{4} \cot (c) \sin ^2(a)+\frac {3}{4} \cos ^2(c) \cot (c) \sin ^2(a)+\frac {9}{4} \cos ^2(a) \cos (c) \sin (c)+\frac {9}{2} i \cos (a) \cos (c) \sin (a) \sin (c)-\frac {9}{4} \cos (c) \sin ^2(a) \sin (c)-\frac {3}{4} i \cos ^2(a) \sin ^2(c)+\frac {3}{2} \cos (a) \sin (a) \sin ^2(c)+\frac {3}{4} i \sin ^2(a) \sin ^2(c)\right )-\frac {\sin (2 a) \sin (2 b x)}{4 b} \] Input:

Integrate[Cot[c + b*x]^3*Sin[a + b*x]^2,x]
 

Output:

((-1/2*I)*ArcTan[Tan[c + b*x]]*(-1 + 3*Cos[2*a - 2*c]))/b + (Cos[2*a]*Cos[ 
2*b*x])/(4*b) + ((Cos[2*a - 2*c - b*x] - Cos[2*a - 2*c + b*x])*Csc[c]*Csc[ 
c + b*x])/(2*b) + ((-1 + 3*Cos[2*a - 2*c])*Log[Sin[c + b*x]^2])/(4*b) - 3* 
x*Cos[a - c]*Sin[a - c] - (Csc[c + b*x]^2*Sin[a - c]^2)/(2*b) + x*(-1/2*I 
- ((3*I)/4)*Cos[a]^2 + ((9*I)/4)*Cos[a]^2*Cos[c]^2 + Cot[c]/2 - (3*Cos[a]^ 
2*Cot[c])/4 + ((-1 + 3*Cos[2*a - 2*c])*Cot[c])/2 - (3*Cos[a]^2*Cos[c]^2*Co 
t[c])/4 - (3*Cos[a]*Sin[a])/2 - (9*Cos[a]*Cos[c]^2*Sin[a])/2 + ((3*I)/2)*C 
os[a]*Cot[c]*Sin[a] - ((3*I)/2)*Cos[a]*Cos[c]^2*Cot[c]*Sin[a] + ((3*I)/4)* 
Sin[a]^2 - ((9*I)/4)*Cos[c]^2*Sin[a]^2 + (3*Cot[c]*Sin[a]^2)/4 + (3*Cos[c] 
^2*Cot[c]*Sin[a]^2)/4 + (9*Cos[a]^2*Cos[c]*Sin[c])/4 + ((9*I)/2)*Cos[a]*Co 
s[c]*Sin[a]*Sin[c] - (9*Cos[c]*Sin[a]^2*Sin[c])/4 - ((3*I)/4)*Cos[a]^2*Sin 
[c]^2 + (3*Cos[a]*Sin[a]*Sin[c]^2)/2 + ((3*I)/4)*Sin[a]^2*Sin[c]^2) - (Sin 
[2*a]*Sin[2*b*x])/(4*b)
 

Rubi [F]

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.

\(\displaystyle \int \sin ^2(a+b x) \cot ^3(b x+c) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin ^2(a+b x) \cot ^3(b x+c)dx\)

Input:

Int[Cot[c + b*x]^3*Sin[a + b*x]^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.77 (sec) , antiderivative size = 230, normalized size of antiderivative = 230.00

method	result	size
risch	\(\frac {3 i x \,{\mathrm e}^{2 i \left (a -c \right )}}{2}+\frac {i x}{2}+\frac {{\mathrm e}^{2 i \left (b x +a \right )}}{8 b}+\frac {{\mathrm e}^{-2 i \left (b x +a \right )}}{8 b}-3 i \cos \left (2 a -2 c \right ) x -\frac {3 i \cos \left (2 a -2 c \right ) a}{b}+\frac {i a}{b}+\frac {-3 \,{\mathrm e}^{2 i \left (b x +3 a \right )}+2 \,{\mathrm e}^{2 i \left (b x +2 a +c \right )}+{\mathrm e}^{2 i \left (b x +a +2 c \right )}+2 \,{\mathrm e}^{2 i \left (3 a -c \right )}-2 \,{\mathrm e}^{2 i \left (a +c \right )}}{2 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right )}{2 b}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}\)	\(230\)

Input:

int(cot(b*x+c)^3*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

3/2*I*x*exp(2*I*(a-c))+1/2*I*x+1/8/b*exp(2*I*(b*x+a))+1/8/b*exp(-2*I*(b*x+ 
a))-3*I*cos(2*a-2*c)*x-3*I/b*cos(2*a-2*c)*a+I/b*a+1/2/b/(-exp(2*I*(b*x+a+c 
))+exp(2*I*a))^2*(-3*exp(2*I*(b*x+3*a))+2*exp(2*I*(b*x+2*a+c))+exp(2*I*(b* 
x+a+2*c))+2*exp(2*I*(3*a-c))-2*exp(2*I*(a+c)))-1/2/b*ln(exp(2*I*(b*x+a))-e 
xp(2*I*(a-c)))+3/2/b*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))*cos(2*a-2*c)
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.10 (sec) , antiderivative size = 419, normalized size of antiderivative = 419.00 \[ \int \cot ^3(c+b x) \sin ^2(a+b x) \, dx=\frac {4 \, \cos \left (b x + a\right )^{4} \cos \left (-2 \, a + 2 \, c\right ) + 2 \, {\left (4 \, \cos \left (-2 \, a + 2 \, c\right )^{2} - 2 \, \cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \cos \left (b x + a\right )^{2} - 4 \, \cos \left (-2 \, a + 2 \, c\right )^{2} - {\left (2 \, {\left (3 \, \cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - 2 \, {\left (3 \, \cos \left (-2 \, a + 2 \, c\right )^{2} - \cos \left (-2 \, a + 2 \, c\right )\right )} \cos \left (b x + a\right )^{2} + 3 \, \cos \left (-2 \, a + 2 \, c\right )^{2} + 2 \, \cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}{\cos \left (-2 \, a + 2 \, c\right ) + 1}\right ) + 2 \, {\left (6 \, {\left (b x \cos \left (-2 \, a + 2 \, c\right )^{2} - b x\right )} \cos \left (b x + a\right ) - {\left (2 \, \cos \left (b x + a\right )^{3} + {\left (4 \, \cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \cos \left (b x + a\right )\right )} \sin \left (-2 \, a + 2 \, c\right )\right )} \sin \left (b x + a\right ) + 6 \, {\left (2 \, b x \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - b x \cos \left (-2 \, a + 2 \, c\right ) - b x\right )} \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 7}{4 \, {\left (2 \, b \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, b \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - b \cos \left (-2 \, a + 2 \, c\right ) - b\right )}} \] Input:

integrate(cot(b*x+c)^3*sin(b*x+a)^2,x, algorithm="fricas")
 

Output:

1/4*(4*cos(b*x + a)^4*cos(-2*a + 2*c) + 2*(4*cos(-2*a + 2*c)^2 - 2*cos(-2* 
a + 2*c) - 5)*cos(b*x + a)^2 - 4*cos(-2*a + 2*c)^2 - (2*(3*cos(-2*a + 2*c) 
 - 1)*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - 2*(3*cos(-2*a + 2*c)^2 - 
 cos(-2*a + 2*c))*cos(b*x + a)^2 + 3*cos(-2*a + 2*c)^2 + 2*cos(-2*a + 2*c) 
 - 1)*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) - cos(-2*a + 2*c) - 1)/(cos(-2*a + 2*c) + 1)) + 2*(6*(b*x 
*cos(-2*a + 2*c)^2 - b*x)*cos(b*x + a) - (2*cos(b*x + a)^3 + (4*cos(-2*a + 
 2*c) - 1)*cos(b*x + a))*sin(-2*a + 2*c))*sin(b*x + a) + 6*(2*b*x*cos(b*x 
+ a)^2*cos(-2*a + 2*c) - b*x*cos(-2*a + 2*c) - b*x)*sin(-2*a + 2*c) - cos( 
-2*a + 2*c) + 7)/(2*b*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*b*cos(b*x + a)*si 
n(b*x + a)*sin(-2*a + 2*c) - b*cos(-2*a + 2*c) - b)
 

Sympy [F]

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

integrate(cot(b*x+c)**3*sin(b*x+a)**2,x)
 

Output:

Integral(sin(a + b*x)**2*cot(b*x + c)**3, x)
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.13 (sec) , antiderivative size = 2160, normalized size of antiderivative = 2160.00 \[ \int \cot ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(cot(b*x+c)^3*sin(b*x+a)^2,x, algorithm="maxima")
 

Output:

1/8*((cos(6*b*x + 2*a + 6*c) - 2*cos(4*b*x + 2*a + 4*c) + cos(2*b*x + 2*a 
+ 2*c))*cos(8*b*x + 4*a + 6*c) + 2*(2*cos(4*b*x + 2*a + 4*c) - cos(2*b*x + 
 2*a + 2*c))*cos(6*b*x + 4*a + 4*c) - (24*b*x*sin(4*b*x + 6*c) - 12*b*x*si 
n(2*b*x + 4*c) + 2*cos(6*b*x + 4*a + 4*c) + 11*cos(4*b*x + 4*a + 2*c) - 8* 
cos(4*b*x + 2*a + 4*c) - 5*cos(4*b*x + 6*c) - 8*cos(2*b*x + 4*a) + 10*cos( 
2*b*x + 4*c) - cos(2*c))*cos(6*b*x + 2*a + 6*c) - 12*(b*x*sin(6*b*x + 2*a 
+ 6*c) - 2*b*x*sin(4*b*x + 2*a + 4*c) + b*x*sin(2*b*x + 2*a + 2*c))*cos(6* 
b*x + 8*c) - 2*(12*b*x*sin(2*b*x + 4*c) - 11*cos(4*b*x + 4*a + 2*c) + 8*co 
s(2*b*x + 4*a) - 4*cos(2*b*x + 2*a + 2*c) - 10*cos(2*b*x + 4*c) + cos(2*c) 
)*cos(4*b*x + 2*a + 4*c) - 16*cos(4*b*x + 2*a + 4*c)^2 - (48*b*x*sin(4*b*x 
 + 2*a + 4*c) - 24*b*x*sin(2*b*x + 2*a + 2*c) + 10*cos(4*b*x + 2*a + 4*c) 
- 5*cos(2*b*x + 2*a + 2*c))*cos(4*b*x + 6*c) + (8*cos(2*b*x + 4*a) + cos(2 
*c))*cos(2*b*x + 2*a + 2*c) - 11*cos(4*b*x + 4*a + 2*c)*cos(2*b*x + 2*a + 
2*c) - 2*(6*b*x*sin(2*b*x + 2*a + 2*c) + 5*cos(2*b*x + 2*a + 2*c))*cos(2*b 
*x + 4*c) + 2*((3*cos(-2*a + 2*c) - 1)*cos(6*b*x + 2*a + 6*c)^2 + 4*(3*cos 
(-2*a + 2*c) - 1)*cos(4*b*x + 2*a + 4*c)^2 - 4*(3*cos(-2*a + 2*c) - 1)*cos 
(4*b*x + 2*a + 4*c)*cos(2*b*x + 2*a + 2*c) + (3*cos(-2*a + 2*c) - 1)*cos(2 
*b*x + 2*a + 2*c)^2 + (3*cos(-2*a + 2*c) - 1)*sin(6*b*x + 2*a + 6*c)^2 + 4 
*(3*cos(-2*a + 2*c) - 1)*sin(4*b*x + 2*a + 4*c)^2 - 4*(3*cos(-2*a + 2*c) - 
 1)*sin(4*b*x + 2*a + 4*c)*sin(2*b*x + 2*a + 2*c) + (3*cos(-2*a + 2*c) ...
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.20 (sec) , antiderivative size = 2856, normalized size of antiderivative = 2856.00 \[ \int \cot ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(cot(b*x+c)^3*sin(b*x+a)^2,x, algorithm="giac")
 

Output:

-1/2*(12*(tan(1/2*a)^4*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2* 
a)^4*tan(1/2*c) + 6*tan(1/2*a)^3*tan(1/2*c)^2 - 6*tan(1/2*a)^2*tan(1/2*c)^ 
3 + tan(1/2*a)*tan(1/2*c)^4 - tan(1/2*a)^3 + 6*tan(1/2*a)^2*tan(1/2*c) - 6 
*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*c)^3 + tan(1/2*a) - tan(1/2*c))*b*x/(ta 
n(1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan 
(1/2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2* 
tan(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) + (tan(1/2*a)^4*tan(1/2*c)^4 - 10*tan(1 
/2*a)^4*tan(1/2*c)^2 + 24*tan(1/2*a)^3*tan(1/2*c)^3 - 10*tan(1/2*a)^2*tan( 
1/2*c)^4 + tan(1/2*a)^4 - 24*tan(1/2*a)^3*tan(1/2*c) + 52*tan(1/2*a)^2*tan 
(1/2*c)^2 - 24*tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*c)^4 - 10*tan(1/2*a)^2 + 
24*tan(1/2*a)*tan(1/2*c) - 10*tan(1/2*c)^2 + 1)*log(tan(b*x)^2 + 1)/(tan(1 
/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1/ 
2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*tan 
(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) - 2*(tan(1/2*a)^4*tan(1/2*c)^6 - 11*tan(1/ 
2*a)^4*tan(1/2*c)^4 + 24*tan(1/2*a)^3*tan(1/2*c)^5 - 10*tan(1/2*a)^2*tan(1 
/2*c)^6 + 11*tan(1/2*a)^4*tan(1/2*c)^2 - 48*tan(1/2*a)^3*tan(1/2*c)^3 + 62 
*tan(1/2*a)^2*tan(1/2*c)^4 - 24*tan(1/2*a)*tan(1/2*c)^5 + tan(1/2*c)^6 - t 
an(1/2*a)^4 + 24*tan(1/2*a)^3*tan(1/2*c) - 62*tan(1/2*a)^2*tan(1/2*c)^2 + 
48*tan(1/2*a)*tan(1/2*c)^3 - 11*tan(1/2*c)^4 + 10*tan(1/2*a)^2 - 24*tan(1/ 
2*a)*tan(1/2*c) + 11*tan(1/2*c)^2 - 1)*log(abs(tan(b*x)*tan(1/2*c)^2 - ...
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Hanged} \] Input:

int(cot(c + b*x)^3*sin(a + b*x)^2,x)
 

Output:

\text{Hanged}
 

Reduce [F]

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

int(cot(b*x+c)^3*sin(b*x+a)^2,x)
 

Output:

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