Integrand size = 17, antiderivative size = 1 \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.88 (sec) , antiderivative size = 1972, normalized size of antiderivative = 1972.00 \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx =\text {Too large to display} \] Input:
Integrate[Sec[a + b*x]^3*Sec[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[Cos[a + b*x]^2])/(2*b) + (3*(3 + 2*Cos[2* a - 2*c])*Csc[a - c]^7*Log[Cos[c + b*x]^2])/(2*b) + (Csc[a - c]^6*Sec[a]*S ec[c]*Sec[a + b*x]^2*Sec[c + b*x]^4*(220*Sin[2*a] - 4*Sin[4*a - 6*c] - 4*S in[6*a - 6*c] - 76*Sin[2*a - 4*c] - 36*Sin[4*a - 4*c] - 76*Sin[2*a - 2*c] + 52*Sin[4*a - 2*c] + 248*Sin[2*c] - 263*Sin[2*b*x] + Sin[2*a - 6*c - 6*b* x] + 10*Sin[2*a - 6*c - 4*b*x] + 2*Sin[4*a - 6*c - 4*b*x] + 50*Sin[2*a - 4 *c - 4*b*x] - 18*Sin[2*a - 6*c - 2*b*x] - 7*Sin[4*a - 6*c - 2*b*x] + Sin[6 *a - 6*c - 2*b*x] + 3*Sin[2*a - 4*c - 2*b*x] + 19*Sin[4*a - 4*c - 2*b*x] + 171*Sin[2*a - 2*c - 2*b*x] - 141*Sin[2*a + 2*b*x] + 42*Sin[4*a + 2*b*x] - 6*Sin[4*a - 4*c + 2*b*x] - 6*Sin[6*a - 4*c + 2*b*x] - 140*Sin[2*a - 2*c + 2*b*x] - 80*Sin[4*a - 2*c + 2*b*x] - 77*Sin[2*c + 2*b*x] + 144*Sin[2*a + 2*c + 2*b*x] + 96*Sin[4*c + 2*b*x] - 106*Sin[2*a + 4*b*x] - 66*Sin[4*a + 4 *b*x] - 4*Sin[4*a - 2*c + 4*b*x] - 4*Sin[6*a - 2*c + 4*b*x] - 138*Sin[2*c + 4*b*x] - 106*Sin[2*a + 2*c + 4*b*x] + 12*Sin[4*a + 2*c + 4*b*x] - 54*Sin [4*c + 4*b*x] + 36*Sin[2*a + 4*c + 4*b*x] + 12*Sin[6*c + 4*b*x] - Sin[4*a + 6*b*x] - Sin[6*a + 6*b*x] - 29*Sin[2*a + 2*c + 6*b*x] - 19*Sin[4*a + 2*c + 6*b*x] - 29*Sin[4*c + 6*b*x] - 29*Sin[2*a + 4*c + 6*b*x] - 11*Sin[6*c + 6*b*x]))/(256*b) + x*((-9*I)/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^7 - ((3*I...
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.
\(\displaystyle \int \sec ^3(a+b x) \sec ^5(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sec ^3(a+b x) \sec ^5(b x+c)dx\) |
Input:
Int[Sec[a + b*x]^3*Sec[c + b*x]^5,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 68.42 (sec) , antiderivative size = 1201, normalized size of antiderivative = 1201.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(1201\) |
risch | \(\text {Expression too large to display}\) | \(1214\) |
Input:
int(sec(b*x+a)^3*sec(b*x+c)^5,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/(-sin(a)*cos(c)+cos(a)*sin(c))^6*(1/2*tan(b*x+a)^2*cos(a)*sin(c)-1 /2*tan(b*x+a)^2*sin(a)*cos(c)+5*tan(b*x+a)*sin(a)*sin(c)+5*tan(b*x+a)*cos( a)*cos(c))-1/3*(-6*sin(c)^5*sin(a)^5-12*sin(c)^5*sin(a)^3*cos(a)^2-6*sin(c )^5*sin(a)*cos(a)^4-6*cos(c)*sin(c)^4*sin(a)^4*cos(a)-12*sin(c)^4*cos(c)*s in(a)^2*cos(a)^3-6*sin(c)^4*cos(c)*cos(a)^5-12*sin(c)^3*cos(c)^2*sin(a)^5- 24*cos(c)^2*sin(c)^3*sin(a)^3*cos(a)^2-12*sin(c)^3*cos(c)^2*sin(a)*cos(a)^ 4-12*sin(c)^2*cos(c)^3*sin(a)^4*cos(a)-24*cos(c)^3*sin(c)^2*sin(a)^2*cos(a )^3-12*sin(c)^2*cos(c)^3*cos(a)^5-6*sin(c)*cos(c)^4*sin(a)^5-12*sin(c)*cos (c)^4*sin(a)^3*cos(a)^2-6*cos(c)^4*sin(c)*sin(a)*cos(a)^4-6*cos(c)^5*sin(a )^4*cos(a)-12*cos(c)^5*sin(a)^2*cos(a)^3-6*cos(c)^5*cos(a)^5)/(-sin(a)*cos (c)+cos(a)*sin(c))^6/(sin(a)*cos(c)-cos(a)*sin(c))/(tan(b*x+a)*sin(a)*cos( c)-tan(b*x+a)*cos(a)*sin(c)+cos(a)*cos(c)+sin(a)*sin(c))^3+1/2*(-15*sin(a) ^4*sin(c)^4-18*cos(a)^2*sin(a)^2*sin(c)^4-3*cos(a)^4*sin(c)^4-24*cos(a)*si n(a)^3*cos(c)*sin(c)^3-24*cos(a)^3*sin(a)*cos(c)*sin(c)^3-18*sin(a)^4*cos( c)^2*sin(c)^2-36*cos(a)^2*sin(a)^2*cos(c)^2*sin(c)^2-18*cos(a)^4*cos(c)^2* sin(c)^2-24*cos(a)*sin(a)^3*cos(c)^3*sin(c)-24*cos(a)^3*cos(c)^3*sin(a)*si n(c)-3*sin(a)^4*cos(c)^4-18*cos(a)^2*sin(a)^2*cos(c)^4-15*cos(a)^4*cos(c)^ 4)/(-sin(a)*cos(c)+cos(a)*sin(c))^6/(sin(a)*cos(c)-cos(a)*sin(c))/(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+ 1/4*(-sin(c)^6*sin(a)^6-3*sin(c)^6*sin(a)^4*cos(a)^2-3*sin(c)^6*sin(a)^...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.17 (sec) , antiderivative size = 761, normalized size of antiderivative = 761.00 \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx =\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(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 + cos(-a + c)^6 - 2*(4*cos(-a + c)^6 + 34*cos(-a + c)^4 - 35*cos(-a + c)^2 - 3)*cos(b*x + c)^4 - 3*cos(-a + c)^4 - (2*cos(-a + c)^6 - cos(-a + c)^4 - 4*cos(-a + c)^2 + 3)*cos(b*x + c)^2 + 2*(2*(4*cos(-a + c)^5 + 24*co s(-a + c)^3 - 13*cos(-a + c))*cos(b*x + c)^5 - 10*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)^3 + (cos(-a + c)^5 - 2*cos(-a + c)^3 + cos(-a + c))*cos (b*x + c))*sin(b*x + c)*sin(-a + c) + 3*cos(-a + c)^2 + 6*(2*(4*cos(-a + c )^3 + cos(-a + c))*cos(b*x + c)^5*sin(b*x + c)*sin(-a + c) + (8*cos(-a + c )^4 - 2*cos(-a + c)^2 - 1)*cos(b*x + c)^6 - (4*cos(-a + c)^4 - 3*cos(-a + c)^2 - 1)*cos(b*x + c)^4)*log(cos(b*x + c)^2) - 6*(2*(4*cos(-a + c)^3 + co s(-a + c))*cos(b*x + c)^5*sin(b*x + c)*sin(-a + c) + (8*cos(-a + c)^4 - 2* cos(-a + c)^2 - 1)*cos(b*x + c)^6 - (4*cos(-a + c)^4 - 3*cos(-a + c)^2 - 1 )*cos(b*x + c)^4)*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)^9 - 4*b*cos(-a + c)^7 + 6 *b*cos(-a + c)^5 - 4*b*cos(-a + c)^3 + b*cos(-a + c))*cos(b*x + c)^5*sin(b *x + c) - ((2*b*cos(-a + c)^8 - 7*b*cos(-a + c)^6 + 9*b*cos(-a + c)^4 - 5* b*cos(-a + c)^2 + b)*cos(b*x + c)^6 - (b*cos(-a + c)^8 - 4*b*cos(-a + c)^6 + 6*b*cos(-a + c)^4 - 4*b*cos(-a + c)^2 + b)*cos(b*x + c)^4)*sin(-a + c))
Timed out. \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx=\text {Timed out} \] Input:
integrate(sec(b*x+a)**3*sec(b*x+c)**5,x)
Output:
Timed out
Timed out. \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx=\text {Timed out} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^5,x, algorithm="maxima")
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.72 (sec) , antiderivative size = 34934, normalized size of antiderivative = 34934.00 \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^5,x, algorithm="giac")
Output:
1/128*(3*(5*tan(1/2*a)^14*tan(1/2*c)^14 + 19*tan(1/2*a)^14*tan(1/2*c)^12 + 32*tan(1/2*a)^13*tan(1/2*c)^13 + 19*tan(1/2*a)^12*tan(1/2*c)^14 + 25*tan( 1/2*a)^14*tan(1/2*c)^10 + 128*tan(1/2*a)^13*tan(1/2*c)^11 + 149*tan(1/2*a) ^12*tan(1/2*c)^12 + 128*tan(1/2*a)^11*tan(1/2*c)^13 + 25*tan(1/2*a)^10*tan (1/2*c)^14 + 15*tan(1/2*a)^14*tan(1/2*c)^8 + 160*tan(1/2*a)^13*tan(1/2*c)^ 9 + 479*tan(1/2*a)^12*tan(1/2*c)^10 + 512*tan(1/2*a)^11*tan(1/2*c)^11 + 47 9*tan(1/2*a)^10*tan(1/2*c)^12 + 160*tan(1/2*a)^9*tan(1/2*c)^13 + 15*tan(1/ 2*a)^8*tan(1/2*c)^14 + 15*tan(1/2*a)^14*tan(1/2*c)^6 + 825*tan(1/2*a)^12*t an(1/2*c)^8 + 640*tan(1/2*a)^11*tan(1/2*c)^9 + 2045*tan(1/2*a)^10*tan(1/2* c)^10 + 640*tan(1/2*a)^9*tan(1/2*c)^11 + 825*tan(1/2*a)^8*tan(1/2*c)^12 + 15*tan(1/2*a)^6*tan(1/2*c)^14 + 25*tan(1/2*a)^14*tan(1/2*c)^4 - 160*tan(1/ 2*a)^13*tan(1/2*c)^5 + 825*tan(1/2*a)^12*tan(1/2*c)^6 + 3915*tan(1/2*a)^10 *tan(1/2*c)^8 + 800*tan(1/2*a)^9*tan(1/2*c)^9 + 3915*tan(1/2*a)^8*tan(1/2* c)^10 + 825*tan(1/2*a)^6*tan(1/2*c)^12 - 160*tan(1/2*a)^5*tan(1/2*c)^13 + 25*tan(1/2*a)^4*tan(1/2*c)^14 + 19*tan(1/2*a)^14*tan(1/2*c)^2 - 128*tan(1/ 2*a)^13*tan(1/2*c)^3 + 479*tan(1/2*a)^12*tan(1/2*c)^4 - 640*tan(1/2*a)^11* tan(1/2*c)^5 + 3915*tan(1/2*a)^10*tan(1/2*c)^6 + 7725*tan(1/2*a)^8*tan(1/2 *c)^8 + 3915*tan(1/2*a)^6*tan(1/2*c)^10 - 640*tan(1/2*a)^5*tan(1/2*c)^11 + 479*tan(1/2*a)^4*tan(1/2*c)^12 - 128*tan(1/2*a)^3*tan(1/2*c)^13 + 19*tan( 1/2*a)^2*tan(1/2*c)^14 + 5*tan(1/2*a)^14 - 32*tan(1/2*a)^13*tan(1/2*c) ...
Timed out. \[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^3*cos(c + b*x)^5),x)
Output:
\text{Hanged}
\[ \int \sec ^3(a+b x) \sec ^5(c+b x) \, dx=\text {too large to display} \] Input:
int(sec(b*x+a)^3*sec(b*x+c)^5,x)
Output:
( - 3024*cos(b*x + c)**2*cos(a + b*x)**2*sin(b*x + c)**2*sin(a + b*x) + 28 56*cos(b*x + c)**2*cos(a + b*x)**2*sin(a + b*x) + 70400*cos(b*x + c)**2*co s(a + b*x)*sin(b*x + c)**3*sin(a + b*x)**2 - 70400*cos(b*x + c)**2*cos(a + b*x)*sin(b*x + c)**3 - 1857*cos(b*x + c)**2*cos(a + b*x)*sin(b*x + c)**2* sin(a + b*x) + 70400*cos(b*x + c)**2*cos(a + b*x)*sin(b*x + c)*sin(a + b*x )**2 - 70400*cos(b*x + c)**2*cos(a + b*x)*sin(b*x + c) + 1884*cos(b*x + c) **2*cos(a + b*x)*sin(a + b*x) - 175310*cos(b*x + c)**2*sin(b*x + c)**2*sin (a + b*x)**3 + 175310*cos(b*x + c)**2*sin(b*x + c)**2*sin(a + b*x) + 14080 0*cos(b*x + c)**2*sin(b*x + c)*sin(a + b*x)**2 - 140800*cos(b*x + c)**2*si n(b*x + c) + 187570*cos(b*x + c)**2*sin(a + b*x)**3 - 187570*cos(b*x + c)* *2*sin(a + b*x) + 13566*cos(b*x + c)*cos(a + b*x)**2*sin(b*x + c)**4*sin(a + b*x) - 10191*cos(b*x + c)*cos(a + b*x)**2*sin(b*x + c)**3 - 27132*cos(b *x + c)*cos(a + b*x)**2*sin(b*x + c)**2*sin(a + b*x) + 9102*cos(b*x + c)*c os(a + b*x)**2*sin(b*x + c) + 13566*cos(b*x + c)*cos(a + b*x)**2*sin(a + b *x) - 8580*cos(b*x + c)*cos(a + b*x)*int(cos(b*x + c)/(cos(a + b*x)*sin(b* x + c)**4*sin(a + b*x)**2 - cos(a + b*x)*sin(b*x + c)**4 - 2*cos(a + b*x)* sin(b*x + c)**2*sin(a + b*x)**2 + 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)**4*sin(a + b*x)**2*b + 8580*cos(b*x + c)*cos(a + b*x)*int(cos(b*x + c)/(cos(a + b*x)*sin(b*x + c)**4*sin(a + b*x)**2 - cos(a + b*x)*sin(b*x + c)**4 - 2*cos(a + b*x)*...