Integrand size = 17, antiderivative size = 1 \[ \int \csc ^3(a+b x) \csc ^3(c+b x) \, dx=0 \] Output:
0
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)
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
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)*...
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))
\[ \int \csc ^3(a+b x) \csc ^3(c+b x) \, dx=\int \csc ^{3}{\left (a + b x \right )} \csc ^{3}{\left (b x + c \right )}\, dx \] 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)
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
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...
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}
\[ \int \csc ^3(a+b x) \csc ^3(c+b x) \, dx=\int \csc \left (b x +c \right )^{3} \csc \left (b x +a \right )^{3}d x \] 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)