\(\int \csc ^2(a+b x) \csc ^4(c+b x) \, dx\) [106]

Optimal result

Integrand size = 17, antiderivative size = 1 \[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=0 \] Output:

0
 

Mathematica [C] (verified)

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

Time = 4.28 (sec) , antiderivative size = 328, normalized size of antiderivative = 328.00 \[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=\frac {\csc ^4(a-c) \left (-192 i \arctan (\tan (a+b x)) \cot (a-c)+192 i \arctan (\tan (c+b x)) \cot (a-c)+96 \cot (a-c) \log \left (\sin ^2(a+b x)\right )-96 \cot (a-c) \log \left (\sin ^2(c+b x)\right )+\csc (a) \csc (c) \csc (a+b x) \csc ^3(c+b x) (15 \sin (2 a)-3 \sin (2 (a-2 c))+9 \sin (2 (a-c))+3 \sin (4 (a-c))+18 \sin (2 c)+25 \sin (2 b x)-\sin (4 a-4 c-2 b x)-7 \sin (2 a-2 c-2 b x)-2 \sin (2 (a-2 c-b x))-16 \sin (2 (a+b x))+\sin (4 (a+b x))+3 \sin (2 (a-c+b x))-7 \sin (2 (c+b x))+4 \sin (4 (c+b x))-6 \sin (2 (a+c+b x))-\sin (2 (a+2 b x))-3 \sin (4 a-2 c+2 b x)-10 \sin (2 (c+2 b x))+7 \sin (2 (a+c+2 b x))-6 \sin (4 c+2 b x)+\sin (2 (a-2 (c+b x))))\right )}{48 b} \] Input:

Integrate[Csc[a + b*x]^2*Csc[c + b*x]^4,x]
 

Output:

(Csc[a - c]^4*((-192*I)*ArcTan[Tan[a + b*x]]*Cot[a - c] + (192*I)*ArcTan[T 
an[c + b*x]]*Cot[a - c] + 96*Cot[a - c]*Log[Sin[a + b*x]^2] - 96*Cot[a - c 
]*Log[Sin[c + b*x]^2] + Csc[a]*Csc[c]*Csc[a + b*x]*Csc[c + b*x]^3*(15*Sin[ 
2*a] - 3*Sin[2*(a - 2*c)] + 9*Sin[2*(a - c)] + 3*Sin[4*(a - c)] + 18*Sin[2 
*c] + 25*Sin[2*b*x] - Sin[4*a - 4*c - 2*b*x] - 7*Sin[2*a - 2*c - 2*b*x] - 
2*Sin[2*(a - 2*c - b*x)] - 16*Sin[2*(a + b*x)] + Sin[4*(a + b*x)] + 3*Sin[ 
2*(a - c + b*x)] - 7*Sin[2*(c + b*x)] + 4*Sin[4*(c + b*x)] - 6*Sin[2*(a + 
c + b*x)] - Sin[2*(a + 2*b*x)] - 3*Sin[4*a - 2*c + 2*b*x] - 10*Sin[2*(c + 
2*b*x)] + 7*Sin[2*(a + c + 2*b*x)] - 6*Sin[4*c + 2*b*x] + Sin[2*(a - 2*(c 
+ b*x))])))/(48*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.

\(\displaystyle \int \csc ^2(a+b x) \csc ^4(b x+c) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \csc ^2(a+b x) \csc ^4(b x+c)dx\)

Input:

Int[Csc[a + b*x]^2*Csc[c + b*x]^4,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 = 3.85 (sec) , antiderivative size = 627, normalized size of antiderivative = 627.00

method	result	size
risch	\(\frac {32 i \left (6 \,{\mathrm e}^{2 i \left (3 b x +6 a +4 c \right )}+6 \,{\mathrm e}^{2 i \left (3 b x +5 a +5 c \right )}-15 \,{\mathrm e}^{2 i \left (2 b x +6 a +3 c \right )}-18 \,{\mathrm e}^{2 i \left (2 b x +5 a +4 c \right )}-3 \,{\mathrm e}^{2 i \left (2 b x +4 a +5 c \right )}+{\mathrm e}^{2 i \left (b x +7 a +c \right )}+7 \,{\mathrm e}^{2 i \left (b x +6 a +2 c \right )}+25 \,{\mathrm e}^{2 i \left (b x +5 a +3 c \right )}+3 \,{\mathrm e}^{2 i \left (b x +4 a +4 c \right )}-{\mathrm e}^{2 i \left (6 a +c \right )}-10 \,{\mathrm e}^{2 i \left (5 a +2 c \right )}-{\mathrm e}^{2 i \left (4 a +3 c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3} \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )^{4} b}+\frac {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (3 a +2 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 {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (2 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 {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (3 a +2 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 {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (2 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}\)	\(627\)
default	\(\text {Expression too large to display}\)	\(675\)

Input:

int(csc(b*x+a)^2*csc(b*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

32/3*I/(exp(2*I*(b*x+a))-1)/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^3/(exp(2*I*a) 
-exp(2*I*c))^4/b*(6*exp(2*I*(3*b*x+6*a+4*c))+6*exp(2*I*(3*b*x+5*a+5*c))-15 
*exp(2*I*(2*b*x+6*a+3*c))-18*exp(2*I*(2*b*x+5*a+4*c))-3*exp(2*I*(2*b*x+4*a 
+5*c))+exp(2*I*(b*x+7*a+c))+7*exp(2*I*(b*x+6*a+2*c))+25*exp(2*I*(b*x+5*a+3 
*c))+3*exp(2*I*(b*x+4*a+4*c))-exp(2*I*(6*a+c))-10*exp(2*I*(5*a+2*c))-exp(2 
*I*(4*a+3*c)))+64*I*ln(exp(2*I*(b*x+a))-1)/(exp(10*I*a)-5*exp(2*I*(4*a+c)) 
+10*exp(2*I*(3*a+2*c))-10*exp(2*I*(2*a+3*c))+5*exp(2*I*(a+4*c))-exp(10*I*c 
))/b*exp(2*I*(3*a+2*c))+64*I*ln(exp(2*I*(b*x+a))-1)/(exp(10*I*a)-5*exp(2*I 
*(4*a+c))+10*exp(2*I*(3*a+2*c))-10*exp(2*I*(2*a+3*c))+5*exp(2*I*(a+4*c))-e 
xp(10*I*c))/b*exp(2*I*(2*a+3*c))-64*I*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))/ 
(exp(10*I*a)-5*exp(2*I*(4*a+c))+10*exp(2*I*(3*a+2*c))-10*exp(2*I*(2*a+3*c) 
)+5*exp(2*I*(a+4*c))-exp(10*I*c))/b*exp(2*I*(3*a+2*c))-64*I*ln(exp(2*I*(b* 
x+a))-exp(2*I*(a-c)))/(exp(10*I*a)-5*exp(2*I*(4*a+c))+10*exp(2*I*(3*a+2*c) 
)-10*exp(2*I*(2*a+3*c))+5*exp(2*I*(a+4*c))-exp(10*I*c))/b*exp(2*I*(2*a+3*c 
))
 

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 624, normalized size of antiderivative = 624.00 \[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx =\text {Too large to display} \] Input:

integrate(csc(b*x+a)^2*csc(b*x+c)^4,x, algorithm="fricas")
 

Output:

-1/3*(4*(cos(-a + c)^4 + cos(-a + c)^2 - 2)*cos(b*x + c)^4 + 3*cos(-a + c) 
^4 - 6*(cos(-a + c)^4 + cos(-a + c)^2 - 2)*cos(b*x + c)^2 + 2*(2*(cos(-a + 
 c)^3 + 2*cos(-a + c))*cos(b*x + c)^3 - 3*(cos(-a + c)^3 + cos(-a + c))*co 
s(b*x + c))*sin(b*x + c)*sin(-a + c) + 6*(cos(b*x + c)^4*cos(-a + c)^2 - 2 
*cos(b*x + c)^2*cos(-a + c)^2 + (cos(b*x + c)^3*cos(-a + c) - cos(b*x + c) 
*cos(-a + c))*sin(b*x + c)*sin(-a + c) + cos(-a + c)^2)*log(-1/4*cos(b*x + 
 c)^2 + 1/4) - 6*(cos(b*x + c)^4*cos(-a + c)^2 - 2*cos(b*x + c)^2*cos(-a + 
 c)^2 + (cos(b*x + c)^3*cos(-a + c) - cos(b*x + c)*cos(-a + 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)/(c 
os(-a + c)^2 + 2*cos(-a + c) + 1)) - 3)/(((b*cos(-a + c)^6 - 3*b*cos(-a + 
c)^4 + 3*b*cos(-a + c)^2 - b)*cos(b*x + c)^3 - (b*cos(-a + c)^6 - 3*b*cos( 
-a + c)^4 + 3*b*cos(-a + c)^2 - b)*cos(b*x + c))*sin(b*x + c) - (b*cos(-a 
+ c)^5 + (b*cos(-a + c)^5 - 2*b*cos(-a + c)^3 + b*cos(-a + c))*cos(b*x + c 
)^4 - 2*b*cos(-a + c)^3 - 2*(b*cos(-a + c)^5 - 2*b*cos(-a + c)^3 + b*cos(- 
a + c))*cos(b*x + c)^2 + b*cos(-a + c))*sin(-a + c))
 

Sympy [F]

\[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=\int \csc ^{2}{\left (a + b x \right )} \csc ^{4}{\left (b x + c \right )}\, dx \] Input:

integrate(csc(b*x+a)**2*csc(b*x+c)**4,x)
 

Output:

Integral(csc(a + b*x)**2*csc(b*x + c)**4, x)
 

Maxima [C] (verification not implemented)

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

Time = 80.19 (sec) , antiderivative size = 2049662, normalized size of antiderivative = 2049662.00 \[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+a)^2*csc(b*x+c)^4,x, algorithm="maxima")
 

Output:

32/3*(25*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) - (cos(8*a) + cos(8*c))*sin 
(6*a + 2*c))*cos(8*a + 2*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) 
+ 2*(5*sin(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) - (cos(8* 
a) + cos(8*c))*sin(6*a + 2*c) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos( 
8*c))*sin(4*a + 4*c))*cos(6*a + 4*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6* 
a + 2*c) + 2*(5*sin(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) 
+ (sin(8*a) + 46*sin(4*a + 4*c) + sin(8*c))*cos(2*a + 6*c) - (cos(8*a) + c 
os(8*c))*sin(6*a + 2*c) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos(8*c))* 
sin(4*a + 4*c) - (cos(8*a) + 46*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c)) 
*cos(4*a + 6*c)^2 + 25*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 2*(5*sin(8* 
a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) + (sin(8*a) + 46*sin(4 
*a + 4*c) + sin(8*c))*cos(2*a + 6*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2*c 
) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos(8*c))*sin(4*a + 4*c) - (cos( 
8*a) + 46*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c))*cos(2*a + 8*c)^2 + 25 
*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2 
*c))*sin(8*a + 2*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 2*(5*s 
in(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) - (cos(8*a) + cos 
(8*c))*sin(6*a + 2*c) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos(8*c))*si 
n(4*a + 4*c))*sin(6*a + 4*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) 
 + 2*(5*sin(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) + (si...
 

Giac [C] (verification not implemented)

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

Time = 1.49 (sec) , antiderivative size = 27104, normalized size of antiderivative = 27104.00 \[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+a)^2*csc(b*x+c)^4,x, algorithm="giac")
 

Output:

-1/48*(6*(tan(1/2*a)^12*tan(1/2*c)^12 + 2*tan(1/2*a)^12*tan(1/2*c)^10 + 8* 
tan(1/2*a)^11*tan(1/2*c)^11 + 2*tan(1/2*a)^10*tan(1/2*c)^12 - tan(1/2*a)^1 
2*tan(1/2*c)^8 + 24*tan(1/2*a)^11*tan(1/2*c)^9 + 20*tan(1/2*a)^10*tan(1/2* 
c)^10 + 24*tan(1/2*a)^9*tan(1/2*c)^11 - tan(1/2*a)^8*tan(1/2*c)^12 - 4*tan 
(1/2*a)^12*tan(1/2*c)^6 + 16*tan(1/2*a)^11*tan(1/2*c)^7 + 62*tan(1/2*a)^10 
*tan(1/2*c)^8 + 72*tan(1/2*a)^9*tan(1/2*c)^9 + 62*tan(1/2*a)^8*tan(1/2*c)^ 
10 + 16*tan(1/2*a)^7*tan(1/2*c)^11 - 4*tan(1/2*a)^6*tan(1/2*c)^12 - tan(1/ 
2*a)^12*tan(1/2*c)^4 - 16*tan(1/2*a)^11*tan(1/2*c)^5 + 88*tan(1/2*a)^10*ta 
n(1/2*c)^6 + 48*tan(1/2*a)^9*tan(1/2*c)^7 + 257*tan(1/2*a)^8*tan(1/2*c)^8 
+ 48*tan(1/2*a)^7*tan(1/2*c)^9 + 88*tan(1/2*a)^6*tan(1/2*c)^10 - 16*tan(1/ 
2*a)^5*tan(1/2*c)^11 - tan(1/2*a)^4*tan(1/2*c)^12 + 2*tan(1/2*a)^12*tan(1/ 
2*c)^2 - 24*tan(1/2*a)^11*tan(1/2*c)^3 + 62*tan(1/2*a)^10*tan(1/2*c)^4 - 4 
8*tan(1/2*a)^9*tan(1/2*c)^5 + 388*tan(1/2*a)^8*tan(1/2*c)^6 + 32*tan(1/2*a 
)^7*tan(1/2*c)^7 + 388*tan(1/2*a)^6*tan(1/2*c)^8 - 48*tan(1/2*a)^5*tan(1/2 
*c)^9 + 62*tan(1/2*a)^4*tan(1/2*c)^10 - 24*tan(1/2*a)^3*tan(1/2*c)^11 + 2* 
tan(1/2*a)^2*tan(1/2*c)^12 + tan(1/2*a)^12 - 8*tan(1/2*a)^11*tan(1/2*c) + 
20*tan(1/2*a)^10*tan(1/2*c)^2 - 72*tan(1/2*a)^9*tan(1/2*c)^3 + 257*tan(1/2 
*a)^8*tan(1/2*c)^4 - 32*tan(1/2*a)^7*tan(1/2*c)^5 + 592*tan(1/2*a)^6*tan(1 
/2*c)^6 - 32*tan(1/2*a)^5*tan(1/2*c)^7 + 257*tan(1/2*a)^4*tan(1/2*c)^8 - 7 
2*tan(1/2*a)^3*tan(1/2*c)^9 + 20*tan(1/2*a)^2*tan(1/2*c)^10 - 8*tan(1/2...
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=\text {Hanged} \] Input:

int(1/(sin(a + b*x)^2*sin(c + b*x)^4),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \csc ^2(a+b x) \csc ^4(c+b x) \, dx=\int \csc \left (b x +c \right )^{4} \csc \left (b x +a \right )^{2}d x \] Input:

int(csc(b*x+a)^2*csc(b*x+c)^4,x)
 

Output:

int(csc(b*x + c)**4*csc(a + b*x)**2,x)