Integrand size = 17, antiderivative size = 1 \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.84 (sec) , antiderivative size = 1966, normalized size of antiderivative = 1966.00 \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx =\text {Too large to display} \] Input:
Integrate[Csc[a + b*x]^3*Csc[c + b*x]^5,x]
Output:
((3*I)*ArcTan[Tan[a + b*x]]*(3 + 2*Cos[2*a - 2*c])*Csc[a - c]^7)/b - ((3*I )*ArcTan[Tan[c + b*x]]*(3 + 2*Cos[2*a - 2*c])*Csc[a - c]^7)/b - (3*(3 + 2* Cos[2*a - 2*c])*Csc[a - c]^7*Log[Sin[a + b*x]^2])/(2*b) + (3*(3 + 2*Cos[2* a - 2*c])*Csc[a - c]^7*Log[Sin[c + b*x]^2])/(2*b) + x*((-9*I)/(Cos[c]*Sin[ a] - Cos[a]*Sin[c])^7 + ((3*I)*Cos[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 - ((9*I)*Cos[a]^2*Cos[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + (9*Cot[a] )/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + (3*Cos[c]^2*Cot[a])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + (3*Cos[a]^2*Cos[c]^2*Cot[a])/(Cos[c]*Sin[a] - Cos[a]*S in[c])^7 - (9*Cos[a]*Cos[c]^2*Sin[a])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + ((3*I)*Cos[c]^2*Sin[a]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + (6*Cos[c]*Si n[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + (18*Cos[a]^2*Cos[c]*Sin[c])/(Cos [c]*Sin[a] - Cos[a]*Sin[c])^7 - ((6*I)*Cos[c]*Cot[a]*Sin[c])/(Cos[c]*Sin[a ] - Cos[a]*Sin[c])^7 + ((6*I)*Cos[a]^2*Cos[c]*Cot[a]*Sin[c])/(Cos[c]*Sin[a ] - Cos[a]*Sin[c])^7 - ((18*I)*Cos[a]*Cos[c]*Sin[a]*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 - (6*Cos[c]*Sin[a]^2*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*S in[c])^7 - ((3*I)*Sin[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 + ((9*I)*Cos [a]^2*Sin[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 - (3*Cot[a]*Sin[c]^2)/(C os[c]*Sin[a] - Cos[a]*Sin[c])^7 - (3*Cos[a]^2*Cot[a]*Sin[c]^2)/(Cos[c]*Sin [a] - Cos[a]*Sin[c])^7 + (9*Cos[a]*Sin[a]*Sin[c]^2)/(Cos[c]*Sin[a] - Cos[a ]*Sin[c])^7 - ((3*I)*Sin[a]^2*Sin[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])...
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.
| \(\displaystyle \int \csc ^3(a+b x) \csc ^5(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \csc ^3(a+b x) \csc ^5(b x+c)dx\) |
Input:
Int[Csc[a + b*x]^3*Csc[c + b*x]^5,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 7.97 (sec) , antiderivative size = 1222, normalized size of antiderivative = 1222.00
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1222\) |
| default | \(\text {Expression too large to display}\) | \(1795\) |
Input:
int(csc(b*x+a)^3*csc(b*x+c)^5,x,method=_RETURNVERBOSE)
Output:
32*I/(exp(2*I*(b*x+a))-1)^2/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^4/(exp(2*I*a) -exp(2*I*c))^6/b*(-2*exp(I*(2*b*x+19*a+3*c))+18*exp(I*(8*b*x+13*a+15*c))-2 48*exp(I*(6*b*x+15*a+11*c))-12*exp(5*I*(2*b*x+3*a+3*c))-4*exp(I*(2*b*x+11* a+11*c))-50*exp(I*(2*b*x+17*a+5*c))-36*exp(I*(10*b*x+17*a+13*c))+42*exp(I* (8*b*x+19*a+9*c))+6*exp(I*(4*b*x+11*a+13*c))+exp(I*(4*b*x+21*a+3*c))+exp(I *(17*a+3*c))+19*exp(I*(4*b*x+19*a+5*c))-220*exp(I*(6*b*x+17*a+9*c))+263*ex p(I*(4*b*x+15*a+9*c))+exp(I*(11*a+9*c))-12*exp(I*(10*b*x+19*a+11*c))+29*ex p(5*I*(3*a+c))+96*exp(I*(8*b*x+15*a+13*c))-76*exp(I*(6*b*x+13*a+13*c))+140 *exp(I*(4*b*x+13*a+11*c))+171*exp(I*(4*b*x+17*a+7*c))-106*exp(I*(2*b*x+13* a+9*c))+144*exp(I*(8*b*x+17*a+11*c))-4*exp(I*(6*b*x+11*a+15*c))-138*exp(I* (2*b*x+15*a+7*c))-52*exp(I*(6*b*x+19*a+7*c))+29*exp(I*(13*a+7*c)))+384*I*l n(exp(2*I*(b*x+a))-1)/(exp(14*I*a)-7*exp(2*I*(6*a+c))+21*exp(2*I*(5*a+2*c) )-35*exp(2*I*(4*a+3*c))+35*exp(2*I*(3*a+4*c))-21*exp(2*I*(2*a+5*c))+7*exp( 2*I*(a+6*c))-exp(14*I*c))/b*exp(I*(5*c+9*a))+1152*I*ln(exp(2*I*(b*x+a))-1) /(exp(14*I*a)-7*exp(2*I*(6*a+c))+21*exp(2*I*(5*a+2*c))-35*exp(2*I*(4*a+3*c ))+35*exp(2*I*(3*a+4*c))-21*exp(2*I*(2*a+5*c))+7*exp(2*I*(a+6*c))-exp(14*I *c))/b*exp(7*I*(a+c))+384*I*ln(exp(2*I*(b*x+a))-1)/(exp(14*I*a)-7*exp(2*I* (6*a+c))+21*exp(2*I*(5*a+2*c))-35*exp(2*I*(4*a+3*c))+35*exp(2*I*(3*a+4*c)) -21*exp(2*I*(2*a+5*c))+7*exp(2*I*(a+6*c))-exp(14*I*c))/b*exp(I*(5*a+9*c))- 384*I*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))/(exp(14*I*a)-7*exp(2*I*(6*a+c...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.15 (sec) , antiderivative size = 1195, normalized size of antiderivative = 1195.00 \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*csc(b*x+c)^5,x, algorithm="fricas")
Output:
-1/4*(8*(2*cos(-a + c)^6 + 11*cos(-a + c)^4 - 13*cos(-a + c)^2)*cos(b*x + c)^6 - 7*cos(-a + c)^6 - 2*(20*cos(-a + c)^6 + 98*cos(-a + c)^4 - 121*cos( -a + c)^2 + 3)*cos(b*x + c)^4 - 18*cos(-a + c)^4 + 3*(10*cos(-a + c)^6 + 4 3*cos(-a + c)^4 - 56*cos(-a + c)^2 + 3)*cos(b*x + c)^2 + 2*(2*(4*cos(-a + c)^5 + 24*cos(-a + c)^3 - 13*cos(-a + c))*cos(b*x + c)^5 - 2*(8*cos(-a + c )^5 + 43*cos(-a + c)^3 - 21*cos(-a + c))*cos(b*x + c)^3 + 3*(3*cos(-a + c) ^5 + 12*cos(-a + c)^3 - 5*cos(-a + c))*cos(b*x + c))*sin(b*x + c)*sin(-a + c) + 27*cos(-a + c)^2 + 6*((8*cos(-a + c)^4 - 2*cos(-a + c)^2 - 1)*cos(b* x + c)^6 - (20*cos(-a + c)^4 - 3*cos(-a + c)^2 - 2)*cos(b*x + c)^4 - 4*cos (-a + c)^4 + (16*cos(-a + c)^4 - 1)*cos(b*x + c)^2 + 2*((4*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^5 - 2*(4*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^3 + (4*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) - 6*((8*cos(-a + c)^4 - 2*cos(-a + c)^2 - 1)*cos(b*x + c)^6 - (20*cos(-a + c)^4 - 3*cos(-a + c) ^2 - 2)*cos(b*x + c)^4 - 4*cos(-a + c)^4 + (16*cos(-a + c)^4 - 1)*cos(b*x + c)^2 + 2*((4*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^5 - 2*(4*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^3 + (4*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)*c os(-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)) - 2)/(2*((b*cos(...
Timed out. \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)**3*csc(b*x+c)**5,x)
Output:
Timed out
Timed out. \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)^3*csc(b*x+c)^5,x, algorithm="maxima")
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 7.00 (sec) , antiderivative size = 57794, normalized size of antiderivative = 57794.00 \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*csc(b*x+c)^5,x, algorithm="giac")
Output:
1/512*(12*(5*tan(1/2*a)^16*tan(1/2*c)^16 + 14*tan(1/2*a)^16*tan(1/2*c)^14 + 52*tan(1/2*a)^15*tan(1/2*c)^15 + 14*tan(1/2*a)^14*tan(1/2*c)^16 + 6*tan( 1/2*a)^16*tan(1/2*c)^12 + 172*tan(1/2*a)^15*tan(1/2*c)^13 + 244*tan(1/2*a) ^14*tan(1/2*c)^14 + 172*tan(1/2*a)^13*tan(1/2*c)^15 + 6*tan(1/2*a)^12*tan( 1/2*c)^16 - 10*tan(1/2*a)^16*tan(1/2*c)^10 + 132*tan(1/2*a)^15*tan(1/2*c)^ 11 + 836*tan(1/2*a)^14*tan(1/2*c)^12 + 884*tan(1/2*a)^13*tan(1/2*c)^13 + 8 36*tan(1/2*a)^12*tan(1/2*c)^14 + 132*tan(1/2*a)^11*tan(1/2*c)^15 - 10*tan( 1/2*a)^10*tan(1/2*c)^16 - 100*tan(1/2*a)^15*tan(1/2*c)^9 + 996*tan(1/2*a)^ 14*tan(1/2*c)^10 + 2012*tan(1/2*a)^13*tan(1/2*c)^11 + 3284*tan(1/2*a)^12*t an(1/2*c)^12 + 2012*tan(1/2*a)^11*tan(1/2*c)^13 + 996*tan(1/2*a)^10*tan(1/ 2*c)^14 - 100*tan(1/2*a)^9*tan(1/2*c)^15 + 10*tan(1/2*a)^16*tan(1/2*c)^6 - 100*tan(1/2*a)^15*tan(1/2*c)^7 + 2820*tan(1/2*a)^13*tan(1/2*c)^9 + 4084*t an(1/2*a)^12*tan(1/2*c)^10 + 8212*tan(1/2*a)^11*tan(1/2*c)^11 + 4084*tan(1 /2*a)^10*tan(1/2*c)^12 + 2820*tan(1/2*a)^9*tan(1/2*c)^13 - 100*tan(1/2*a)^ 7*tan(1/2*c)^15 + 10*tan(1/2*a)^6*tan(1/2*c)^16 - 6*tan(1/2*a)^16*tan(1/2* c)^4 + 132*tan(1/2*a)^15*tan(1/2*c)^5 - 996*tan(1/2*a)^14*tan(1/2*c)^6 + 2 820*tan(1/2*a)^13*tan(1/2*c)^7 + 15500*tan(1/2*a)^11*tan(1/2*c)^9 + 5140*t an(1/2*a)^10*tan(1/2*c)^10 + 15500*tan(1/2*a)^9*tan(1/2*c)^11 + 2820*tan(1 /2*a)^7*tan(1/2*c)^13 - 996*tan(1/2*a)^6*tan(1/2*c)^14 + 132*tan(1/2*a)^5* tan(1/2*c)^15 - 6*tan(1/2*a)^4*tan(1/2*c)^16 - 14*tan(1/2*a)^16*tan(1/2...
Timed out. \[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(sin(a + b*x)^3*sin(c + b*x)^5),x)
Output:
\text{Hanged}
\[ \int \csc ^3(a+b x) \csc ^5(c+b x) \, dx=\int \csc \left (b x +c \right )^{5} \csc \left (b x +a \right )^{3}d x \] Input:
int(csc(b*x+a)^3*csc(b*x+c)^5,x)
Output:
int(csc(b*x + c)**5*csc(a + b*x)**3,x)