\(\int \csc ^3(a+b x) \csc ^3(c+b x) \, dx\) [107]

Optimal result

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

0
 

Mathematica [C] (verified)

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

Time = 4.09 (sec) , antiderivative size = 347, normalized size of antiderivative = 347.00 \[ \int \csc ^3(a+b x) \csc ^3(c+b x) \, dx=\frac {\csc ^4(a-c) \left (64 i \arctan (\tan (a+b x)) (2+\cos (2 (a-c))) \csc (a-c)-64 i \arctan (\tan (c+b x)) (2+\cos (2 (a-c))) \csc (a-c)-32 (2+\cos (2 (a-c))) \csc (a-c) \log \left (\sin ^2(a+b x)\right )+32 (2+\cos (2 (a-c))) \csc (a-c) \log \left (\sin ^2(c+b x)\right )+\csc (a) \csc (c) \csc ^2(a+b x) \csc ^2(c+b x) (-15 \sin (2 a)+3 \sin (2 (a-2 c))-3 \sin (4 a-2 c)-15 \sin (2 c)-16 \sin (2 b x)+10 \sin (2 a-2 c-2 b x)-4 \sin (2 (a-2 c-b x))+8 \sin (2 (a+b x))-3 \sin (4 (a+b x))-10 \sin (2 (a-c+b x))+8 \sin (2 (c+b x))-3 \sin (4 (c+b x))+8 \sin (2 (a+c+b x))+6 \sin (2 (a+2 b x))+2 \sin (4 a+2 b x)+4 \sin (4 a-2 c+2 b x)+6 \sin (2 (c+2 b x))-6 \sin (2 (a+c+2 b x))+2 \sin (4 c+2 b x))\right )}{32 b} \] Input:

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

Output:

(Csc[a - c]^4*((64*I)*ArcTan[Tan[a + b*x]]*(2 + Cos[2*(a - c)])*Csc[a - c] 
 - (64*I)*ArcTan[Tan[c + b*x]]*(2 + Cos[2*(a - c)])*Csc[a - c] - 32*(2 + C 
os[2*(a - c)])*Csc[a - c]*Log[Sin[a + b*x]^2] + 32*(2 + Cos[2*(a - c)])*Cs 
c[a - c]*Log[Sin[c + b*x]^2] + Csc[a]*Csc[c]*Csc[a + b*x]^2*Csc[c + b*x]^2 
*(-15*Sin[2*a] + 3*Sin[2*(a - 2*c)] - 3*Sin[4*a - 2*c] - 15*Sin[2*c] - 16* 
Sin[2*b*x] + 10*Sin[2*a - 2*c - 2*b*x] - 4*Sin[2*(a - 2*c - b*x)] + 8*Sin[ 
2*(a + b*x)] - 3*Sin[4*(a + b*x)] - 10*Sin[2*(a - c + b*x)] + 8*Sin[2*(c + 
 b*x)] - 3*Sin[4*(c + b*x)] + 8*Sin[2*(a + c + b*x)] + 6*Sin[2*(a + 2*b*x) 
] + 2*Sin[4*a + 2*b*x] + 4*Sin[4*a - 2*c + 2*b*x] + 6*Sin[2*(c + 2*b*x)] - 
 6*Sin[2*(a + c + 2*b*x)] + 2*Sin[4*c + 2*b*x])))/(32*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.

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

\(\Big \downarrow \) 7299

\(\displaystyle \int \csc ^3(a+b x) \csc ^3(b x+c)dx\)

Input:

Int[Csc[a + b*x]^3*Csc[c + b*x]^3,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 = 2.73 (sec) , antiderivative size = 749, normalized size of antiderivative = 749.00

method	result	size
default	\(\text {Expression too large to display}\)	\(749\)
risch	\(\frac {16 i \left (2 \,{\mathrm e}^{i \left (6 b x +13 a +5 c \right )}+8 \,{\mathrm e}^{i \left (6 b x +11 a +7 c \right )}+2 \,{\mathrm e}^{3 i \left (2 b x +3 a +3 c \right )}-3 \,{\mathrm e}^{i \left (4 b x +13 a +3 c \right )}-15 \,{\mathrm e}^{i \left (4 b x +11 a +5 c \right )}-15 \,{\mathrm e}^{i \left (4 b x +9 a +7 c \right )}-3 \,{\mathrm e}^{i \left (4 b x +7 a +9 c \right )}+10 \,{\mathrm e}^{i \left (2 b x +11 a +3 c \right )}+16 \,{\mathrm e}^{i \left (2 b x +9 a +5 c \right )}+10 \,{\mathrm e}^{i \left (2 b x +7 a +7 c \right )}-6 \,{\mathrm e}^{3 i \left (3 a +c \right )}-6 \,{\mathrm e}^{i \left (7 a +5 c \right )}\right )}{\left ({\mathrm e}^{2 i \left (b x +a +c \right )}-{\mathrm e}^{2 i a}\right )^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2} \left (-{\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{4} b}+\frac {32 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{i \left (7 a +3 c \right )}}{\left ({\mathrm e}^{10 i a}-5 \,{\mathrm e}^{2 i \left (4 a +c \right )}+10 \,{\mathrm e}^{2 i \left (3 a +2 c \right )}-10 \,{\mathrm e}^{2 i \left (2 a +3 c \right )}+5 \,{\mathrm e}^{2 i \left (a +4 c \right )}-{\mathrm e}^{10 i c}\right ) b}+\frac {128 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{5 i \left (a +c \right )}}{\left ({\mathrm e}^{10 i a}-5 \,{\mathrm e}^{2 i \left (4 a +c \right )}+10 \,{\mathrm e}^{2 i \left (3 a +2 c \right )}-10 \,{\mathrm e}^{2 i \left (2 a +3 c \right )}+5 \,{\mathrm e}^{2 i \left (a +4 c \right )}-{\mathrm e}^{10 i c}\right ) b}+\frac {32 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{i \left (3 a +7 c \right )}}{\left ({\mathrm e}^{10 i a}-5 \,{\mathrm e}^{2 i \left (4 a +c \right )}+10 \,{\mathrm e}^{2 i \left (3 a +2 c \right )}-10 \,{\mathrm e}^{2 i \left (2 a +3 c \right )}+5 \,{\mathrm e}^{2 i \left (a +4 c \right )}-{\mathrm e}^{10 i c}\right ) b}-\frac {32 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{i \left (7 a +3 c \right )}}{\left ({\mathrm e}^{10 i a}-5 \,{\mathrm e}^{2 i \left (4 a +c \right )}+10 \,{\mathrm e}^{2 i \left (3 a +2 c \right )}-10 \,{\mathrm e}^{2 i \left (2 a +3 c \right )}+5 \,{\mathrm e}^{2 i \left (a +4 c \right )}-{\mathrm e}^{10 i c}\right ) b}-\frac {128 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{5 i \left (a +c \right )}}{\left ({\mathrm e}^{10 i a}-5 \,{\mathrm e}^{2 i \left (4 a +c \right )}+10 \,{\mathrm e}^{2 i \left (3 a +2 c \right )}-10 \,{\mathrm e}^{2 i \left (2 a +3 c \right )}+5 \,{\mathrm e}^{2 i \left (a +4 c \right )}-{\mathrm e}^{10 i c}\right ) b}-\frac {32 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{i \left (3 a +7 c \right )}}{\left ({\mathrm e}^{10 i a}-5 \,{\mathrm e}^{2 i \left (4 a +c \right )}+10 \,{\mathrm e}^{2 i \left (3 a +2 c \right )}-10 \,{\mathrm e}^{2 i \left (2 a +3 c \right )}+5 \,{\mathrm e}^{2 i \left (a +4 c \right )}-{\mathrm e}^{10 i c}\right ) b}\)	\(825\)

Input:

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

Output:

1/b*(-(-3*cos(a)*cos(c)-3*sin(a)*sin(c))/(sin(a)*cos(c)-cos(a)*sin(c))^4/t 
an(b*x+a)+1/(sin(a)*cos(c)-cos(a)*sin(c))^5*(-2*cos(c)^2*sin(a)^2-6*cos(a) 
^2*cos(c)^2-8*cos(a)*cos(c)*sin(a)*sin(c)-6*sin(a)^2*sin(c)^2-2*sin(c)^2*c 
os(a)^2)*ln(tan(b*x+a))+1/2/(sin(a)*cos(c)-cos(a)*sin(c))^3/tan(b*x+a)^2-( 
2*cos(c)^3*sin(a)^2*cos(a)+6*cos(c)^3*cos(a)^3+2*cos(c)^2*sin(c)*sin(a)^3+ 
14*cos(c)^2*sin(c)*cos(a)^2*sin(a)+14*cos(c)*sin(c)^2*cos(a)*sin(a)^2+2*co 
s(c)*sin(c)^2*cos(a)^3+6*sin(c)^3*sin(a)^3+2*sin(c)^3*sin(a)*cos(a)^2)/(si 
n(a)*cos(c)-cos(a)*sin(c))^5/(-cos(a)*cos(c)-sin(a)*sin(c))*ln(-tan(b*x+a) 
*sin(a)*sin(c)-tan(b*x+a)*cos(a)*cos(c)+sin(a)*cos(c)-cos(a)*sin(c))+1/2*( 
sin(a)^4*cos(c)^4+2*cos(a)^2*sin(a)^2*cos(c)^4+cos(a)^4*cos(c)^4+2*sin(a)^ 
4*cos(c)^2*sin(c)^2+4*cos(a)^2*sin(a)^2*cos(c)^2*sin(c)^2+2*cos(a)^4*cos(c 
)^2*sin(c)^2+sin(a)^4*sin(c)^4+2*cos(a)^2*sin(a)^2*sin(c)^4+cos(a)^4*sin(c 
)^4)/(sin(a)*cos(c)-cos(a)*sin(c))^3/(cos(a)*cos(c)+sin(a)*sin(c))/(-cos(a 
)*cos(c)-sin(a)*sin(c))/(-tan(b*x+a)*sin(a)*sin(c)-tan(b*x+a)*cos(a)*cos(c 
)+sin(a)*cos(c)-cos(a)*sin(c))^2-(sin(a)^4*cos(c)^4-2*cos(a)^2*sin(a)^2*co 
s(c)^4-3*cos(a)^4*cos(c)^4-8*cos(a)*sin(a)^3*cos(c)^3*sin(c)-8*cos(a)^3*co 
s(c)^3*sin(a)*sin(c)-2*sin(a)^4*cos(c)^2*sin(c)^2-4*cos(a)^2*sin(a)^2*cos( 
c)^2*sin(c)^2-2*cos(a)^4*cos(c)^2*sin(c)^2-8*cos(a)*sin(a)^3*cos(c)*sin(c) 
^3-8*cos(a)^3*sin(a)*cos(c)*sin(c)^3-3*sin(a)^4*sin(c)^4-2*cos(a)^2*sin(a) 
^2*sin(c)^4+cos(a)^4*sin(c)^4)/(sin(a)*cos(c)-cos(a)*sin(c))^4/(cos(a)*...
 

Fricas [C] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 772, normalized size of antiderivative = 772.00 \[ \int \csc ^3(a+b x) \csc ^3(c+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(24*(cos(-a + c)^4 - cos(-a + c)^2)*cos(b*x + c)^4 + 7*cos(-a + c)^4 - 
 2*(16*cos(-a + c)^4 - 17*cos(-a + c)^2 + 1)*cos(b*x + c)^2 + 4*(3*(2*cos( 
-a + c)^3 - cos(-a + c))*cos(b*x + c)^3 - (5*cos(-a + c)^3 - 2*cos(-a + c) 
)*cos(b*x + c))*sin(b*x + c)*sin(-a + c) - 8*cos(-a + c)^2 + 2*((4*cos(-a 
+ c)^4 - 1)*cos(b*x + c)^4 + 2*cos(-a + c)^4 - (6*cos(-a + c)^4 + cos(-a + 
 c)^2 - 1)*cos(b*x + c)^2 + 2*((2*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c 
)^3 - (2*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c))*sin(b*x + c)*sin(-a + 
c) + cos(-a + c)^2)*log(-1/4*cos(b*x + c)^2 + 1/4) - 2*((4*cos(-a + c)^4 - 
 1)*cos(b*x + c)^4 + 2*cos(-a + c)^4 - (6*cos(-a + c)^4 + cos(-a + c)^2 - 
1)*cos(b*x + c)^2 + 2*((2*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^3 - (2 
*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c))*sin(b*x + c)*sin(-a + c) + cos 
(-a + c)^2)*log(-(2*cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) + (2 
*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2)/(cos(-a + c)^2 + 2*cos 
(-a + c) + 1)) + 1)/(2*((b*cos(-a + c)^7 - 3*b*cos(-a + c)^5 + 3*b*cos(-a 
+ c)^3 - b*cos(-a + c))*cos(b*x + c)^3 - (b*cos(-a + c)^7 - 3*b*cos(-a + c 
)^5 + 3*b*cos(-a + c)^3 - b*cos(-a + c))*cos(b*x + c))*sin(b*x + c) - (b*c 
os(-a + c)^6 + (2*b*cos(-a + c)^6 - 5*b*cos(-a + c)^4 + 4*b*cos(-a + c)^2 
- b)*cos(b*x + c)^4 - 2*b*cos(-a + c)^4 - (3*b*cos(-a + c)^6 - 7*b*cos(-a 
+ c)^4 + 5*b*cos(-a + c)^2 - b)*cos(b*x + c)^2 + b*cos(-a + c)^2)*sin(-a + 
 c))
 

Sympy [F]

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

integrate(csc(b*x+a)**3*csc(b*x+c)**3,x)
 

Output:

Integral(csc(a + b*x)**3*csc(b*x + c)**3, x)
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [C] (verification not implemented)

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

Time = 0.60 (sec) , antiderivative size = 13206, normalized size of antiderivative = 13206.00 \[ \int \csc ^3(a+b x) \csc ^3(c+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/16*((3*tan(1/2*a)^12*tan(1/2*c)^12 + 4*tan(1/2*a)^12*tan(1/2*c)^10 + 28* 
tan(1/2*a)^11*tan(1/2*c)^11 + 4*tan(1/2*a)^10*tan(1/2*c)^12 - tan(1/2*a)^1 
2*tan(1/2*c)^8 + 44*tan(1/2*a)^11*tan(1/2*c)^9 + 112*tan(1/2*a)^10*tan(1/2 
*c)^10 + 44*tan(1/2*a)^9*tan(1/2*c)^11 - tan(1/2*a)^8*tan(1/2*c)^12 - 8*ta 
n(1/2*a)^11*tan(1/2*c)^7 + 212*tan(1/2*a)^10*tan(1/2*c)^8 + 252*tan(1/2*a) 
^9*tan(1/2*c)^9 + 212*tan(1/2*a)^8*tan(1/2*c)^10 - 8*tan(1/2*a)^7*tan(1/2* 
c)^11 + tan(1/2*a)^12*tan(1/2*c)^4 - 8*tan(1/2*a)^11*tan(1/2*c)^5 + 536*ta 
n(1/2*a)^9*tan(1/2*c)^7 + 427*tan(1/2*a)^8*tan(1/2*c)^8 + 536*tan(1/2*a)^7 
*tan(1/2*c)^9 - 8*tan(1/2*a)^5*tan(1/2*c)^11 + tan(1/2*a)^4*tan(1/2*c)^12 
- 4*tan(1/2*a)^12*tan(1/2*c)^2 + 44*tan(1/2*a)^11*tan(1/2*c)^3 - 212*tan(1 
/2*a)^10*tan(1/2*c)^4 + 536*tan(1/2*a)^9*tan(1/2*c)^5 + 1648*tan(1/2*a)^7* 
tan(1/2*c)^7 + 536*tan(1/2*a)^5*tan(1/2*c)^9 - 212*tan(1/2*a)^4*tan(1/2*c) 
^10 + 44*tan(1/2*a)^3*tan(1/2*c)^11 - 4*tan(1/2*a)^2*tan(1/2*c)^12 - 3*tan 
(1/2*a)^12 + 28*tan(1/2*a)^11*tan(1/2*c) - 112*tan(1/2*a)^10*tan(1/2*c)^2 
+ 252*tan(1/2*a)^9*tan(1/2*c)^3 - 427*tan(1/2*a)^8*tan(1/2*c)^4 + 1648*tan 
(1/2*a)^7*tan(1/2*c)^5 + 1648*tan(1/2*a)^5*tan(1/2*c)^7 - 427*tan(1/2*a)^4 
*tan(1/2*c)^8 + 252*tan(1/2*a)^3*tan(1/2*c)^9 - 112*tan(1/2*a)^2*tan(1/2*c 
)^10 + 28*tan(1/2*a)*tan(1/2*c)^11 - 3*tan(1/2*c)^12 - 4*tan(1/2*a)^10 + 4 
4*tan(1/2*a)^9*tan(1/2*c) - 212*tan(1/2*a)^8*tan(1/2*c)^2 + 536*tan(1/2*a) 
^7*tan(1/2*c)^3 + 1648*tan(1/2*a)^5*tan(1/2*c)^5 + 536*tan(1/2*a)^3*tan...
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(1/(sin(a + b*x)^3*sin(c + b*x)^3),x)
 

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

int(csc(b*x + c)**3*csc(a + b*x)**3,x)