Integrand size = 17, antiderivative size = 1 \[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.67 (sec) , antiderivative size = 1451, normalized size of antiderivative = 1451.00 \[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx =\text {Too large to display} \] Input:
Integrate[Csc[c + b*x]^3*Sec[a + b*x]^3,x]
Output:
((-2*I)*ArcTan[Tan[a + b*x]]*(-2 + Cos[2*a - 2*c])*Sec[a - c]^5)/b + ((2*I )*ArcTan[Tan[c + b*x]]*(-2 + Cos[2*a - 2*c])*Sec[a - c]^5)/b + ((-2 + Cos[ 2*a - 2*c])*Log[Cos[a + b*x]^2]*Sec[a - c]^5)/b - ((-2 + Cos[2*a - 2*c])*L og[Sin[c + b*x]^2]*Sec[a - c]^5)/b + (3*(Cos[a - c - b*x] - Cos[a - c + b* x])*Csc[c/2]*Csc[c + b*x]*Sec[c/2])/(4*b*(Cos[a/2 - c/2] - Sin[a/2 - c/2]) ^4*(Cos[a/2 - c/2] + Sin[a/2 - c/2])^4) - (3*(-Cos[a - c - b*x] + Cos[a - c + b*x])*Sec[a + b*x])/(2*b*(Cos[a/2] - Sin[a/2])*(Cos[a/2] + Sin[a/2])*( Cos[a/2 - c/2] - Sin[a/2 - c/2])^4*(Cos[a/2 - c/2] + Sin[a/2 - c/2])^4) - Csc[c + b*x]^2/(2*b*(Cos[a/2 - c/2] - Sin[a/2 - c/2])^3*(Cos[a/2 - c/2] + Sin[a/2 - c/2])^3) + Sec[a + b*x]^2/(2*b*(Cos[a/2 - c/2] - Sin[a/2 - c/2]) ^3*(Cos[a/2 - c/2] + Sin[a/2 - c/2])^3) + x*((I*Cos[a]^2)/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 + (I*Cos[c]^2)/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 - ((2*I )*Cos[a]^2*Cos[c]^2)/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 - (4*Cot[c])/(Cos[a ]*Cos[c] + Sin[a]*Sin[c])^5 + (Cos[a]^2*Cot[c])/(Cos[a]*Cos[c] + Sin[a]*Si n[c])^5 + (Cos[a]^2*Cos[c]^2*Cot[c])/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 + ( 2*Cos[a]*Sin[a])/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 + (9*Cos[a]*Cos[c]^2*Si n[a])/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 - ((2*I)*Cos[a]*Cot[c]*Sin[a])/(Co s[a]*Cos[c] + Sin[a]*Sin[c])^5 + ((2*I)*Cos[a]*Cos[c]^2*Cot[c]*Sin[a])/(Co s[a]*Cos[c] + Sin[a]*Sin[c])^5 - (I*Sin[a]^2)/(Cos[a]*Cos[c] + Sin[a]*Sin[ c])^5 - (Cot[c]*Sin[a]^2)/(Cos[a]*Cos[c] + Sin[a]*Sin[c])^5 - (Cos[c]^2...
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.
| \(\displaystyle \int \sec ^3(a+b x) \csc ^3(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sec ^3(a+b x) \csc ^3(b x+c)dx\) |
Input:
Int[Csc[c + b*x]^3*Sec[a + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 8.55 (sec) , antiderivative size = 483, normalized size of antiderivative = 483.00
| method | result | size |
| default | \(\frac {\frac {\frac {\tan \left (b x +a \right )^{2} \cos \left (a \right ) \cos \left (c \right )}{2}+\frac {\tan \left (b x +a \right )^{2} \sin \left (a \right ) \sin \left (c \right )}{2}+3 \tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-3 \tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4}}+\frac {\left (2 \cos \left (a \right )^{2} \cos \left (c \right )^{2}+6 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-8 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+6 \sin \left (c \right )^{2} \cos \left (a \right )^{2}+2 \sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{5}}-\frac {4 \cos \left (c \right )^{3} \sin \left (a \right ) \cos \left (a \right )^{2}+4 \cos \left (c \right )^{3} \sin \left (a \right )^{3}-4 \cos \left (c \right )^{2} \sin \left (c \right ) \cos \left (a \right )^{3}-4 \cos \left (c \right )^{2} \sin \left (c \right ) \sin \left (a \right )^{2} \cos \left (a \right )+4 \cos \left (c \right ) \sin \left (c \right )^{2} \sin \left (a \right ) \cos \left (a \right )^{2}+4 \cos \left (c \right ) \sin \left (c \right )^{2} \sin \left (a \right )^{3}-4 \sin \left (c \right )^{3} \cos \left (a \right )^{3}-4 \sin \left (c \right )^{3} \sin \left (a \right )^{2} \cos \left (a \right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{5} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}-\frac {\sin \left (a \right )^{4} \cos \left (c \right )^{4}+2 \cos \left (a \right )^{2} \sin \left (a \right )^{2} \cos \left (c \right )^{4}+\cos \left (a \right )^{4} \cos \left (c \right )^{4}+2 \sin \left (a \right )^{4} \cos \left (c \right )^{2} \sin \left (c \right )^{2}+4 \cos \left (a \right )^{2} \sin \left (a \right )^{2} \cos \left (c \right )^{2} \sin \left (c \right )^{2}+2 \cos \left (a \right )^{4} \cos \left (c \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{4} \sin \left (c \right )^{4}+2 \cos \left (a \right )^{2} \sin \left (a \right )^{2} \sin \left (c \right )^{4}+\cos \left (a \right )^{4} \sin \left (c \right )^{4}}{2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{5} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}}{b}\) | \(483\) |
| risch | \(-\frac {16 \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 \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (b x +a +c \right )}-{\mathrm e}^{2 i a}\right )^{2} \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{4} b}+\frac {32 \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 \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 \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}-\frac {32 \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 \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 \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}\) | \(804\) |
Input:
int(csc(b*x+c)^3*sec(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(1/(cos(a)*cos(c)+sin(a)*sin(c))^4*(1/2*tan(b*x+a)^2*cos(a)*cos(c)+1/2 *tan(b*x+a)^2*sin(a)*sin(c)+3*tan(b*x+a)*sin(a)*cos(c)-3*tan(b*x+a)*cos(a) *sin(c))+(2*cos(a)^2*cos(c)^2+6*cos(c)^2*sin(a)^2-8*cos(a)*cos(c)*sin(a)*s in(c)+6*sin(c)^2*cos(a)^2+2*sin(a)^2*sin(c)^2)/(cos(a)*cos(c)+sin(a)*sin(c ))^5*ln(tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)-sin(a)*cos(c)+co s(a)*sin(c))-(4*cos(c)^3*sin(a)*cos(a)^2+4*cos(c)^3*sin(a)^3-4*cos(c)^2*si n(c)*cos(a)^3-4*cos(c)^2*sin(c)*sin(a)^2*cos(a)+4*cos(c)*sin(c)^2*sin(a)*c os(a)^2+4*cos(c)*sin(c)^2*sin(a)^3-4*sin(c)^3*cos(a)^3-4*sin(c)^3*sin(a)^2 *cos(a))/(cos(a)*cos(c)+sin(a)*sin(c))^5/(tan(b*x+a)*cos(a)*cos(c)+tan(b*x +a)*sin(a)*sin(c)-sin(a)*cos(c)+cos(a)*sin(c))-1/2/(cos(a)*cos(c)+sin(a)*s in(c))^5*(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)/(tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)-sin(a)*c os(c)+cos(a)*sin(c))^2)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.12 (sec) , antiderivative size = 702, normalized size of antiderivative = 702.00 \[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sec(b*x+a)^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 - 15*cos(-a + c)^2)*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 - 3*cos(-a + c))*c os(b*x + c))*sin(b*x + c)*sin(-a + c) - 6*cos(-a + c)^2 + 2*((4*cos(-a + c )^4 - 8*cos(-a + c)^2 + 3)*cos(b*x + c)^4 + 2*cos(-a + c)^4 - (6*cos(-a + c)^4 - 13*cos(-a + c)^2 + 6)*cos(b*x + c)^2 + 2*((2*cos(-a + c)^3 - 3*cos( -a + c))*cos(b*x + c)^3 - (2*cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c))* sin(b*x + c)*sin(-a + c) - 5*cos(-a + c)^2 + 3)*log(-1/4*cos(b*x + c)^2 + 1/4) - 2*((4*cos(-a + c)^4 - 8*cos(-a + c)^2 + 3)*cos(b*x + c)^4 + 2*cos(- a + c)^4 - (6*cos(-a + c)^4 - 13*cos(-a + c)^2 + 6)*cos(b*x + c)^2 + 2*((2 *cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^3 - (2*cos(-a + c)^3 - 3*cos( -a + c))*cos(b*x + c))*sin(b*x + c)*sin(-a + c) - 5*cos(-a + c)^2 + 3)*log (4*(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 + 1)/(cos(-a + c)^2 + 2*cos(-a + c) + 1)))/(b*cos(-a + c)^7 - b*cos(-a + c)^5 + (2*b*cos(-a + c)^7 - b*cos(-a + c)^5)*cos(b*x + c)^4 - (3*b*cos(-a + c)^7 - 2*b*cos(-a + c)^5)*cos(b*x + c)^2 + 2*(b*cos(b*x + c)^3*cos(-a + c)^6 - b*cos(b*x + c)*cos(-a + c)^6)*s in(b*x + c)*sin(-a + c))
\[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx=\int \csc ^{3}{\left (b x + c \right )} \sec ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+c)**3*sec(b*x+a)**3,x)
Output:
Integral(csc(b*x + c)**3*sec(a + b*x)**3, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 103.29 (sec) , antiderivative size = 2494919, normalized size of antiderivative = 2494919.00 \[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sec(b*x+a)^3,x, algorithm="maxima")
Output:
16*(600*((cos(8*a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c) + 6*cos(5 *a + 3*c)*cos(4*a + 4*c) + (sin(8*a) + 4*sin(6*a + 2*c) + sin(8*c))*sin(5* a + 3*c) + 6*sin(5*a + 3*c)*sin(4*a + 4*c))*cos(6*a + 4*c)^2 + 600*((cos(8 *a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c) + 6*cos(5*a + 3*c)*cos(4 *a + 4*c) - (cos(8*a) + 4*cos(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*co s(3*a + 5*c) + 4*(cos(5*a + 3*c) - cos(3*a + 5*c))*cos(2*a + 6*c) + (sin(8 *a) + 4*sin(6*a + 2*c) + sin(8*c))*sin(5*a + 3*c) + 6*sin(5*a + 3*c)*sin(4 *a + 4*c) - (sin(8*a) + 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))*si n(3*a + 5*c) + 4*(sin(5*a + 3*c) - sin(3*a + 5*c))*sin(2*a + 6*c))*cos(4*a + 6*c)^2 + 36*(cos(10*a)^2 + 10*(cos(10*a) + cos(10*c))*cos(8*a + 2*c) + 25*cos(8*a + 2*c)^2 + 2*cos(10*a)*cos(10*c) + cos(10*c)^2 + sin(10*a)^2 + 10*(sin(10*a) + sin(10*c))*sin(8*a + 2*c) + 25*sin(8*a + 2*c)^2 + 2*sin(10 *a)*sin(10*c) + sin(10*c)^2)*cos(5*a + 3*c)*cos(4*a + 4*c) + 150*((cos(8*a ) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c) + 6*cos(5*a + 3*c)*cos(4*a + 4*c) - (cos(8*a) + 4*cos(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*cos( 3*a + 5*c) + 4*(cos(5*a + 3*c) - cos(3*a + 5*c))*cos(2*a + 6*c) + (sin(8*a ) + 4*sin(6*a + 2*c) + sin(8*c))*sin(5*a + 3*c) + 6*sin(5*a + 3*c)*sin(4*a + 4*c) - (sin(8*a) + 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))*sin( 3*a + 5*c) + 4*(sin(5*a + 3*c) - sin(3*a + 5*c))*sin(2*a + 6*c))*cos(2*a + 8*c)^2 + 600*((cos(8*a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c) ...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.51 (sec) , antiderivative size = 13903, normalized size of antiderivative = 13903.00 \[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sec(b*x+a)^3,x, algorithm="giac")
Output:
-1/8*(16*(tan(1/2*a)^12*tan(1/2*c)^11 - tan(1/2*a)^11*tan(1/2*c)^12 + 13*t an(1/2*a)^12*tan(1/2*c)^9 - 28*tan(1/2*a)^11*tan(1/2*c)^10 + 28*tan(1/2*a) ^10*tan(1/2*c)^11 - 13*tan(1/2*a)^9*tan(1/2*c)^12 + 34*tan(1/2*a)^12*tan(1 /2*c)^7 - 53*tan(1/2*a)^11*tan(1/2*c)^8 + 44*tan(1/2*a)^10*tan(1/2*c)^9 - 44*tan(1/2*a)^9*tan(1/2*c)^10 + 53*tan(1/2*a)^8*tan(1/2*c)^11 - 34*tan(1/2 *a)^7*tan(1/2*c)^12 + 34*tan(1/2*a)^12*tan(1/2*c)^5 - 8*tan(1/2*a)^10*tan( 1/2*c)^7 - 49*tan(1/2*a)^9*tan(1/2*c)^8 + 49*tan(1/2*a)^8*tan(1/2*c)^9 + 8 *tan(1/2*a)^7*tan(1/2*c)^10 - 34*tan(1/2*a)^5*tan(1/2*c)^12 + 13*tan(1/2*a )^12*tan(1/2*c)^3 + 53*tan(1/2*a)^11*tan(1/2*c)^4 - 8*tan(1/2*a)^10*tan(1/ 2*c)^5 - 118*tan(1/2*a)^8*tan(1/2*c)^7 + 118*tan(1/2*a)^7*tan(1/2*c)^8 + 8 *tan(1/2*a)^5*tan(1/2*c)^10 - 53*tan(1/2*a)^4*tan(1/2*c)^11 - 13*tan(1/2*a )^3*tan(1/2*c)^12 + tan(1/2*a)^12*tan(1/2*c) + 28*tan(1/2*a)^11*tan(1/2*c) ^2 + 44*tan(1/2*a)^10*tan(1/2*c)^3 + 49*tan(1/2*a)^9*tan(1/2*c)^4 - 118*ta n(1/2*a)^8*tan(1/2*c)^5 + 118*tan(1/2*a)^5*tan(1/2*c)^8 - 49*tan(1/2*a)^4* tan(1/2*c)^9 - 44*tan(1/2*a)^3*tan(1/2*c)^10 - 28*tan(1/2*a)^2*tan(1/2*c)^ 11 - tan(1/2*a)*tan(1/2*c)^12 + tan(1/2*a)^11 + 28*tan(1/2*a)^10*tan(1/2*c ) + 44*tan(1/2*a)^9*tan(1/2*c)^2 + 49*tan(1/2*a)^8*tan(1/2*c)^3 - 118*tan( 1/2*a)^7*tan(1/2*c)^4 + 118*tan(1/2*a)^4*tan(1/2*c)^7 - 49*tan(1/2*a)^3*ta n(1/2*c)^8 - 44*tan(1/2*a)^2*tan(1/2*c)^9 - 28*tan(1/2*a)*tan(1/2*c)^10 - tan(1/2*c)^11 + 13*tan(1/2*a)^9 + 53*tan(1/2*a)^8*tan(1/2*c) - 8*tan(1/...
Timed out. \[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^3*sin(c + b*x)^3),x)
Output:
\text{Hanged}
\[ \int \csc ^3(c+b x) \sec ^3(a+b x) \, dx=\int \csc \left (b x +c \right )^{3} \sec \left (b x +a \right )^{3}d x \] Input:
int(csc(b*x+c)^3*sec(b*x+a)^3,x)
Output:
int(csc(b*x + c)**3*sec(a + b*x)**3,x)