Integrand size = 17, antiderivative size = 1 \[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.72 (sec) , antiderivative size = 1733, normalized size of antiderivative = 1733.00 \[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx =\text {Too large to display} \] Input:
Integrate[Sec[a + b*x]^3*Sec[c + b*x]^3,x]
Output:
((2*I)*ArcTan[Tan[a + b*x]]*(2 + Cos[2*a - 2*c])*Csc[a - c]^5)/b - ((2*I)* ArcTan[Tan[c + b*x]]*(2 + Cos[2*a - 2*c])*Csc[a - c]^5)/b - ((2 + Cos[2*a - 2*c])*Csc[a - c]^5*Log[Cos[a + b*x]^2])/b + ((2 + Cos[2*a - 2*c])*Csc[a - c]^5*Log[Cos[c + b*x]^2])/b + (Csc[a/2 - c/2]^3*Sec[a/2 - c/2]^3*Sec[a + b*x]^2)/(16*b) - (Csc[a/2 - c/2]^3*Sec[a/2 - c/2]^3*Sec[c + b*x]^2)/(16*b ) - (3*Csc[a/2 - c/2]^4*Sec[a/2 - c/2]^4*Sec[a + b*x]*(-Sin[a - c - b*x] + Sin[a - c + b*x]))/(32*b*(Cos[a/2] - Sin[a/2])*(Cos[a/2] + Sin[a/2])) - ( 3*Csc[a/2 - c/2]^4*Sec[a/2 - c/2]^4*Sec[c + b*x]*(-Sin[a - c - b*x] + Sin[ a - c + b*x]))/(32*b*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) + x*((-4 *I)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 - (I*Cos[c]^2)/(Cos[c]*Sin[a] - Cos[ a]*Sin[c])^5 - (I*Cos[a]^2*Cos[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 - ( 3*Cos[a]*Cos[c]^2*Sin[a])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + ((3*I)*Cos[c ]^2*Sin[a]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 - (2*Cos[c]*Sin[c])/(Cos[c ]*Sin[a] - Cos[a]*Sin[c])^5 + (2*Cos[a]^2*Cos[c]*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 - ((6*I)*Cos[a]*Cos[c]*Sin[a]*Sin[c])/(Cos[c]*Sin[a] - Co s[a]*Sin[c])^5 - (6*Cos[c]*Sin[a]^2*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c] )^5 + (I*Sin[c]^2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + (I*Cos[a]^2*Sin[c]^ 2)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + (3*Cos[a]*Sin[a]*Sin[c]^2)/(Cos[c]* Sin[a] - Cos[a]*Sin[c])^5 - ((3*I)*Sin[a]^2*Sin[c]^2)/(Cos[c]*Sin[a] - Cos [a]*Sin[c])^5 - (4*I)/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^5 - (I*Cos[a]^...
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.
| \(\displaystyle \int \sec ^3(a+b x) \sec ^3(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sec ^3(a+b x) \sec ^3(b x+c)dx\) |
Input:
Int[Sec[a + b*x]^3*Sec[c + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 17.78 (sec) , antiderivative size = 492, normalized size of antiderivative = 492.00
| method | result | size |
| default | \(\frac {\frac {\frac {\tan \left (b x +a \right )^{2} \sin \left (a \right ) \cos \left (c \right )}{2}-\frac {\tan \left (b x +a \right )^{2} \cos \left (a \right ) \sin \left (c \right )}{2}-3 \tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )-3 \tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{4}}+\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 (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{5} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {-4 \cos \left (c \right )^{3} \sin \left (a \right )^{2} \cos \left (a \right )-4 \cos \left (c \right )^{3} \cos \left (a \right )^{3}-4 \cos \left (c \right )^{2} \sin \left (c \right ) \sin \left (a \right )^{3}-4 \cos \left (c \right )^{2} \sin \left (c \right ) \cos \left (a \right )^{2} \sin \left (a \right )-4 \cos \left (c \right ) \sin \left (c \right )^{2} \cos \left (a \right ) \sin \left (a \right )^{2}-4 \cos \left (c \right ) \sin \left (c \right )^{2} \cos \left (a \right )^{3}-4 \sin \left (c \right )^{3} \sin \left (a \right )^{3}-4 \sin \left (c \right )^{3} \sin \left (a \right ) \cos \left (a \right )^{2}}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{5} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}-\frac {\left (-2 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-6 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-8 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )-6 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-2 \sin \left (c \right )^{2} \cos \left (a \right )^{2}\right ) \ln \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{5}}}{b}\) | \(492\) |
| 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}\) | \(817\) |
Input:
int(sec(b*x+a)^3*sec(b*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(1/(sin(a)*cos(c)-cos(a)*sin(c))^4*(1/2*tan(b*x+a)^2*sin(a)*cos(c)-1/2 *tan(b*x+a)^2*cos(a)*sin(c)-3*tan(b*x+a)*cos(a)*cos(c)-3*tan(b*x+a)*sin(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))^5/(tan(b*x+a)*sin(a)*cos (c)-tan(b*x+a)*cos(a)*sin(c)+cos(a)*cos(c)+sin(a)*sin(c))^2-(-4*cos(c)^3*s in(a)^2*cos(a)-4*cos(c)^3*cos(a)^3-4*cos(c)^2*sin(c)*sin(a)^3-4*cos(c)^2*s in(c)*cos(a)^2*sin(a)-4*cos(c)*sin(c)^2*cos(a)*sin(a)^2-4*cos(c)*sin(c)^2* cos(a)^3-4*sin(c)^3*sin(a)^3-4*sin(c)^3*sin(a)*cos(a)^2)/(sin(a)*cos(c)-co s(a)*sin(c))^5/(tan(b*x+a)*sin(a)*cos(c)-tan(b*x+a)*cos(a)*sin(c)+cos(a)*c os(c)+sin(a)*sin(c))-(-2*cos(c)^2*sin(a)^2-6*cos(a)^2*cos(c)^2-8*cos(a)*co s(c)*sin(a)*sin(c)-6*sin(a)^2*sin(c)^2-2*sin(c)^2*cos(a)^2)/(sin(a)*cos(c) -cos(a)*sin(c))^5*ln(tan(b*x+a)*sin(a)*cos(c)-tan(b*x+a)*cos(a)*sin(c)+cos (a)*cos(c)+sin(a)*sin(c)))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.13 (sec) , antiderivative size = 597, normalized size of antiderivative = 597.00 \[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx =\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^3,x, algorithm="fricas")
Output:
1/2*(24*(cos(-a + c)^4 - cos(-a + c)^2)*cos(b*x + c)^4 - cos(-a + c)^4 - 2 *(8*cos(-a + c)^4 - 7*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 - (cos(-a + c)^3 - cos(-a + c))*cos(b* x + c))*sin(b*x + c)*sin(-a + c) + 2*cos(-a + c)^2 + 2*(2*(2*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^3*sin(b*x + c)*sin(-a + c) + (4*cos(-a + c)^4 - 1)*cos(b*x + c)^4 - (2*cos(-a + c)^4 - cos(-a + c)^2 - 1)*cos(b*x + c)^ 2)*log(cos(b*x + c)^2) - 2*(2*(2*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c) ^3*sin(b*x + c)*sin(-a + c) + (4*cos(-a + c)^4 - 1)*cos(b*x + c)^4 - (2*co s(-a + c)^4 - cos(-a + c)^2 - 1)*cos(b*x + c)^2)*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)) - 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*sin(b*x + c) - ((2*b*cos(-a + c)^6 - 5*b*cos(-a + c)^4 + 4*b*cos(-a + c)^2 - b)*cos(b*x + c)^4 - (b*cos(-a + c)^6 - 3*b*cos(-a + c)^4 + 3*b*co s(-a + c)^2 - b)*cos(b*x + c)^2)*sin(-a + c))
\[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx=\int \sec ^{3}{\left (a + b x \right )} \sec ^{3}{\left (b x + c \right )}\, dx \] Input:
integrate(sec(b*x+a)**3*sec(b*x+c)**3,x)
Output:
Integral(sec(a + b*x)**3*sec(b*x + c)**3, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 61.56 (sec) , antiderivative size = 1929031, normalized size of antiderivative = 1929031.00 \[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^3,x, algorithm="maxima")
Output:
-16*(600*((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*c) - (cos(8 *a) - 4*cos(6*a + 2*c) + cos(8*c))*sin(5*a + 3*c) - 6*cos(4*a + 4*c)*sin(5 *a + 3*c) + 6*cos(5*a + 3*c)*sin(4*a + 4*c))*cos(6*a + 4*c)^2 + 600*((sin( 8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*c) + (sin(8*a) - 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))*cos(3*a + 5*c) + 4*(sin(5*a + 3*c) + sin(3*a + 5*c))*cos(2*a + 6*c) - (cos(8*a) - 4*cos(6*a + 2*c) + cos(8*c) )*sin(5*a + 3*c) - 6*cos(4*a + 4*c)*sin(5*a + 3*c) + 6*cos(5*a + 3*c)*sin( 4*a + 4*c) - (cos(8*a) - 4*cos(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*s in(3*a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c))*sin(2*a + 6*c))*cos(4* a + 6*c)^2 + 150*((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*c) + (sin(8*a) - 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))*cos(3*a + 5* c) + 4*(sin(5*a + 3*c) + sin(3*a + 5*c))*cos(2*a + 6*c) - (cos(8*a) - 4*co s(6*a + 2*c) + cos(8*c))*sin(5*a + 3*c) - 6*cos(4*a + 4*c)*sin(5*a + 3*c) + 6*cos(5*a + 3*c)*sin(4*a + 4*c) - (cos(8*a) - 4*cos(6*a + 2*c) + 6*cos(4 *a + 4*c) + cos(8*c))*sin(3*a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c)) *sin(2*a + 6*c))*cos(2*a + 8*c)^2 + 600*((sin(8*a) - 4*sin(6*a + 2*c) + si n(8*c))*cos(5*a + 3*c) - (cos(8*a) - 4*cos(6*a + 2*c) + cos(8*c))*sin(5*a + 3*c) - 6*cos(4*a + 4*c)*sin(5*a + 3*c) + 6*cos(5*a + 3*c)*sin(4*a + 4*c) )*sin(6*a + 4*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 + si...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 10230, normalized size of antiderivative = 10230.00 \[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^3,x, algorithm="giac")
Output:
1/64*(4*(3*tan(1/2*a)^10*tan(1/2*c)^10 + 7*tan(1/2*a)^10*tan(1/2*c)^8 + 16 *tan(1/2*a)^9*tan(1/2*c)^9 + 7*tan(1/2*a)^8*tan(1/2*c)^10 + 6*tan(1/2*a)^1 0*tan(1/2*c)^6 + 32*tan(1/2*a)^9*tan(1/2*c)^7 + 59*tan(1/2*a)^8*tan(1/2*c) ^8 + 32*tan(1/2*a)^7*tan(1/2*c)^9 + 6*tan(1/2*a)^6*tan(1/2*c)^10 + 6*tan(1 /2*a)^10*tan(1/2*c)^4 + 142*tan(1/2*a)^8*tan(1/2*c)^6 + 64*tan(1/2*a)^7*ta n(1/2*c)^7 + 142*tan(1/2*a)^6*tan(1/2*c)^8 + 6*tan(1/2*a)^4*tan(1/2*c)^10 + 7*tan(1/2*a)^10*tan(1/2*c)^2 - 32*tan(1/2*a)^9*tan(1/2*c)^3 + 142*tan(1/ 2*a)^8*tan(1/2*c)^4 + 396*tan(1/2*a)^6*tan(1/2*c)^6 + 142*tan(1/2*a)^4*tan (1/2*c)^8 - 32*tan(1/2*a)^3*tan(1/2*c)^9 + 7*tan(1/2*a)^2*tan(1/2*c)^10 + 3*tan(1/2*a)^10 - 16*tan(1/2*a)^9*tan(1/2*c) + 59*tan(1/2*a)^8*tan(1/2*c)^ 2 - 64*tan(1/2*a)^7*tan(1/2*c)^3 + 396*tan(1/2*a)^6*tan(1/2*c)^4 + 396*tan (1/2*a)^4*tan(1/2*c)^6 - 64*tan(1/2*a)^3*tan(1/2*c)^7 + 59*tan(1/2*a)^2*ta n(1/2*c)^8 - 16*tan(1/2*a)*tan(1/2*c)^9 + 3*tan(1/2*c)^10 + 7*tan(1/2*a)^8 - 32*tan(1/2*a)^7*tan(1/2*c) + 142*tan(1/2*a)^6*tan(1/2*c)^2 + 396*tan(1/ 2*a)^4*tan(1/2*c)^4 + 142*tan(1/2*a)^2*tan(1/2*c)^6 - 32*tan(1/2*a)*tan(1/ 2*c)^7 + 7*tan(1/2*c)^8 + 6*tan(1/2*a)^6 + 142*tan(1/2*a)^4*tan(1/2*c)^2 + 64*tan(1/2*a)^3*tan(1/2*c)^3 + 142*tan(1/2*a)^2*tan(1/2*c)^4 + 6*tan(1/2* c)^6 + 6*tan(1/2*a)^4 + 32*tan(1/2*a)^3*tan(1/2*c) + 59*tan(1/2*a)^2*tan(1 /2*c)^2 + 32*tan(1/2*a)*tan(1/2*c)^3 + 6*tan(1/2*c)^4 + 7*tan(1/2*a)^2 + 1 6*tan(1/2*a)*tan(1/2*c) + 7*tan(1/2*c)^2 + 3)*log(abs(2*tan(b*x + a)*ta...
Timed out. \[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^3*cos(c + b*x)^3),x)
Output:
\text{Hanged}
\[ \int \sec ^3(a+b x) \sec ^3(c+b x) \, dx=\text {too large to display} \] Input:
int(sec(b*x+a)^3*sec(b*x+c)^3,x)
Output:
( - 8*cos(b*x + c)**2*cos(a + b*x)**2*sin(a + b*x) - 5*cos(b*x + c)**2*cos (a + b*x)*sin(a + b*x) - 14*cos(b*x + c)**2*sin(a + b*x)**3 + 14*cos(b*x + c)**2*sin(a + b*x) + 42*cos(b*x + c)*cos(a + b*x)**2*sin(b*x + c)**2*sin( a + b*x) - 11*cos(b*x + c)*cos(a + b*x)**2*sin(b*x + c) - 42*cos(b*x + c)* cos(a + b*x)**2*sin(a + b*x) + 8*cos(b*x + c)*cos(a + b*x)*int(cos(b*x + c )/(cos(a + b*x)*sin(b*x + c)**2*sin(a + b*x)**2 - cos(a + b*x)*sin(b*x + c )**2 - cos(a + b*x)*sin(a + b*x)**2 + cos(a + b*x)),x)*sin(b*x + c)**2*sin (a + b*x)**2*b - 8*cos(b*x + c)*cos(a + b*x)*int(cos(b*x + c)/(cos(a + b*x )*sin(b*x + c)**2*sin(a + b*x)**2 - cos(a + b*x)*sin(b*x + c)**2 - cos(a + b*x)*sin(a + b*x)**2 + cos(a + b*x)),x)*sin(b*x + c)**2*b - 8*cos(b*x + c )*cos(a + b*x)*int(cos(b*x + c)/(cos(a + b*x)*sin(b*x + c)**2*sin(a + b*x) **2 - cos(a + b*x)*sin(b*x + c)**2 - cos(a + b*x)*sin(a + b*x)**2 + cos(a + b*x)),x)*sin(a + b*x)**2*b + 8*cos(b*x + c)*cos(a + b*x)*int(cos(b*x + c )/(cos(a + b*x)*sin(b*x + c)**2*sin(a + b*x)**2 - cos(a + b*x)*sin(b*x + c )**2 - cos(a + b*x)*sin(a + b*x)**2 + cos(a + b*x)),x)*b + 8*cos(b*x + c)* cos(a + b*x)*int(cos(b*x + c)/(sin(b*x + c)**2*sin(a + b*x)**2 - sin(b*x + c)**2 - sin(a + b*x)**2 + 1),x)*sin(b*x + c)**2*sin(a + b*x)**2*b - 8*cos (b*x + c)*cos(a + b*x)*int(cos(b*x + c)/(sin(b*x + c)**2*sin(a + b*x)**2 - sin(b*x + c)**2 - sin(a + b*x)**2 + 1),x)*sin(b*x + c)**2*b - 8*cos(b*x + c)*cos(a + b*x)*int(cos(b*x + c)/(sin(b*x + c)**2*sin(a + b*x)**2 - si...