Integrand size = 17, antiderivative size = 1 \[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 14.86 (sec) , antiderivative size = 243, normalized size of antiderivative = 243.00 \[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=-\frac {\csc ^5(a-c) \left (240 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) (15 \cos (a-c)+\cos (3 (a-c))) \csc (a-c)+240 (5+3 \cos (2 (a-c))) \csc (a-c) \left (\log \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )+(349+338 \cos (2 (a-c))+33 \cos (4 (a-c))+120 \cos (2 (a-2 c-b x))+400 \cos (2 (a+b x))+15 \cos (4 (a+b x))+400 \cos (2 (c+b x))+75 \cos (4 (c+b x))+40 \cos (4 a-2 c+2 b x)+150 \cos (2 (a+c+2 b x))) \sec ^2(a+b x) \sec ^3(c+b x)\right )}{192 b} \] Input:
Integrate[Sec[a + b*x]^3*Sec[c + b*x]^4,x]
Output:
-1/192*(Csc[a - c]^5*(240*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*(15*Cos[a - c] + Cos[3*(a - c)])*Csc[a - c] + 240*(5 + 3*Cos[2*(a - c)])*Csc[a - c]* (Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]]) + (349 + 338*Cos[2*(a - c)] + 33*Cos[4*(a - c)] + 120*Cos[2*(a - 2*c - b*x)] + 400*Cos[2*(a + b*x)] + 15*Cos[4*(a + b*x)] + 400*Cos[2*(c + b*x)] + 75*Cos[4*(c + b*x)] + 40*Cos[4*a - 2*c + 2*b*x] + 150*Cos[2*(a + c + 2*b*x)])*Sec[a + b*x]^2*Sec[c + b*x]^3))/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.
\(\displaystyle \int \sec ^3(a+b x) \sec ^4(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sec ^3(a+b x) \sec ^4(b x+c)dx\) |
Input:
Int[Sec[a + b*x]^3*Sec[c + b*x]^4,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 35.21 (sec) , antiderivative size = 1761, normalized size of antiderivative = 1761.00
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1761\) |
default | \(\text {Expression too large to display}\) | \(3107\) |
Input:
int(sec(b*x+a)^3*sec(b*x+c)^4,x,method=_RETURNVERBOSE)
Output:
-8/3*I/(exp(2*I*(b*x+a))+1)^2/(exp(2*I*(b*x+a+c))+exp(2*I*a))^3/(exp(2*I*a )-exp(2*I*c))^5/b*(15*exp(I*(9*b*x+17*a+8*c))+150*exp(I*(9*b*x+15*a+10*c)) +75*exp(I*(9*b*x+13*a+12*c))+40*exp(I*(7*b*x+17*a+6*c))+400*exp(I*(7*b*x+1 5*a+8*c))+400*exp(I*(7*b*x+13*a+10*c))+120*exp(I*(7*b*x+11*a+12*c))+33*exp (I*(5*b*x+17*a+4*c))+338*exp(I*(5*b*x+15*a+6*c))+698*exp(I*(5*b*x+13*a+8*c ))+338*exp(I*(5*b*x+11*a+10*c))+33*exp(I*(5*b*x+9*a+12*c))+120*exp(I*(3*b* x+15*a+4*c))+400*exp(I*(3*b*x+13*a+6*c))+400*exp(I*(3*b*x+11*a+8*c))+40*ex p(I*(3*b*x+9*a+10*c))+75*exp(I*(b*x+13*a+4*c))+150*exp(I*(b*x+11*a+6*c))+1 5*exp(I*(b*x+9*a+8*c)))-120*ln(exp(I*(b*x+a))+I)/(exp(12*I*a)-6*exp(2*I*(5 *a+c))+15*exp(4*I*(2*a+c))-20*exp(6*I*(a+c))+15*exp(4*I*(a+2*c))-6*exp(2*I *(a+5*c))+exp(12*I*c))/b*exp(4*I*(2*a+c))-400*ln(exp(I*(b*x+a))+I)/(exp(12 *I*a)-6*exp(2*I*(5*a+c))+15*exp(4*I*(2*a+c))-20*exp(6*I*(a+c))+15*exp(4*I* (a+2*c))-6*exp(2*I*(a+5*c))+exp(12*I*c))/b*exp(6*I*(a+c))-120*ln(exp(I*(b* x+a))+I)/(exp(12*I*a)-6*exp(2*I*(5*a+c))+15*exp(4*I*(2*a+c))-20*exp(6*I*(a +c))+15*exp(4*I*(a+2*c))-6*exp(2*I*(a+5*c))+exp(12*I*c))/b*exp(4*I*(a+2*c) )-20*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/(exp(12*I*a)-6*exp(2*I*(5*a+c))+15* exp(4*I*(2*a+c))-20*exp(6*I*(a+c))+15*exp(4*I*(a+2*c))-6*exp(2*I*(a+5*c))+ exp(12*I*c))/b*exp(3*I*(3*a+c))-300*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/(exp (12*I*a)-6*exp(2*I*(5*a+c))+15*exp(4*I*(2*a+c))-20*exp(6*I*(a+c))+15*exp(4 *I*(a+2*c))-6*exp(2*I*(a+5*c))+exp(12*I*c))/b*exp(I*(7*a+5*c))-300*ln(e...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.24 (sec) , antiderivative size = 890, normalized size of antiderivative = 890.00 \[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^4,x, algorithm="fricas")
Output:
-1/12*(15*(2*(3*cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^4*sin(b*x + c)*s in(-a + c) + (6*cos(-a + c)^4 - cos(-a + c)^2 - 1)*cos(b*x + c)^5 - (3*cos (-a + c)^4 - 2*cos(-a + c)^2 - 1)*cos(b*x + c)^3)*log(2*(cos(-a + c)*sin(b *x + c) - cos(b*x + c)*sin(-a + c) + 1)/(cos(-a + c) + 1)) - 15*(2*(3*cos( -a + c)^3 + cos(-a + c))*cos(b*x + c)^4*sin(b*x + c)*sin(-a + c) + (6*cos( -a + c)^4 - cos(-a + c)^2 - 1)*cos(b*x + c)^5 - (3*cos(-a + c)^4 - 2*cos(- a + c)^2 - 1)*cos(b*x + c)^3)*log(-2*(cos(-a + c)*sin(b*x + c) - cos(b*x + c)*sin(-a + c) - 1)/(cos(-a + c) + 1)) - 15*(2*(cos(-a + c)^4 + 3*cos(-a + c)^2)*cos(b*x + c)^4*sin(b*x + c)*sin(-a + c) + (2*cos(-a + c)^5 + 5*cos (-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^5 - (cos(-a + c)^5 + 2*cos(-a + c )^3 - 3*cos(-a + c))*cos(b*x + c)^3)*log(sin(b*x + c) + 1) + 15*(2*(cos(-a + c)^4 + 3*cos(-a + c)^2)*cos(b*x + c)^4*sin(b*x + c)*sin(-a + c) + (2*co s(-a + c)^5 + 5*cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^5 - (cos(-a + c)^5 + 2*cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^3)*log(-sin(b*x + c) + 1) - 10*(6*(cos(-a + c)^5 + cos(-a + c)^3 - 2*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) + 2*(15*(2*cos(-a + c)^4 + 3*cos(-a + c)^2 - 1)*cos(b*x + c)^4 + 2*co s(-a + c)^4 - 10*(cos(-a + c)^4 - 1)*cos(b*x + c)^2 - 4*cos(-a + c)^2 + 2) *sin(-a + c))/(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)^4*sin(b*x + c)*sin(-a + c) + (2*b*cos(-a ...
\[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=\int \sec ^{3}{\left (a + b x \right )} \sec ^{4}{\left (b x + c \right )}\, dx \] Input:
integrate(sec(b*x+a)**3*sec(b*x+c)**4,x)
Output:
Integral(sec(a + b*x)**3*sec(b*x + c)**4, x)
Timed out. \[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Timed out} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^4,x, algorithm="maxima")
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 41.48 (sec) , antiderivative size = 53926, normalized size of antiderivative = 53926.00 \[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^3*sec(b*x+c)^4,x, algorithm="giac")
Output:
1/96*(15*(tan(1/2*a)^13*tan(1/2*c)^13 - tan(1/2*a)^13*tan(1/2*c)^12 + tan( 1/2*a)^12*tan(1/2*c)^13 + 3*tan(1/2*a)^13*tan(1/2*c)^11 + 7*tan(1/2*a)^12* tan(1/2*c)^12 + 3*tan(1/2*a)^11*tan(1/2*c)^13 - 3*tan(1/2*a)^13*tan(1/2*c) ^10 - 3*tan(1/2*a)^12*tan(1/2*c)^11 + 3*tan(1/2*a)^11*tan(1/2*c)^12 + 3*ta n(1/2*a)^10*tan(1/2*c)^13 + 3*tan(1/2*a)^13*tan(1/2*c)^9 + 21*tan(1/2*a)^1 2*tan(1/2*c)^10 + 30*tan(1/2*a)^11*tan(1/2*c)^11 + 21*tan(1/2*a)^10*tan(1/ 2*c)^12 + 3*tan(1/2*a)^9*tan(1/2*c)^13 - 3*tan(1/2*a)^13*tan(1/2*c)^8 - 15 *tan(1/2*a)^12*tan(1/2*c)^9 - 6*tan(1/2*a)^11*tan(1/2*c)^10 + 6*tan(1/2*a) ^10*tan(1/2*c)^11 + 15*tan(1/2*a)^9*tan(1/2*c)^12 + 3*tan(1/2*a)^8*tan(1/2 *c)^13 + 27*tan(1/2*a)^12*tan(1/2*c)^8 + 57*tan(1/2*a)^11*tan(1/2*c)^9 + 1 18*tan(1/2*a)^10*tan(1/2*c)^10 + 57*tan(1/2*a)^9*tan(1/2*c)^11 + 27*tan(1/ 2*a)^8*tan(1/2*c)^12 - 24*tan(1/2*a)^12*tan(1/2*c)^7 - 15*tan(1/2*a)^11*ta n(1/2*c)^8 - 55*tan(1/2*a)^10*tan(1/2*c)^9 + 55*tan(1/2*a)^9*tan(1/2*c)^10 + 15*tan(1/2*a)^8*tan(1/2*c)^11 + 24*tan(1/2*a)^7*tan(1/2*c)^12 - 3*tan(1 /2*a)^13*tan(1/2*c)^5 + 24*tan(1/2*a)^12*tan(1/2*c)^6 + 24*tan(1/2*a)^11*t an(1/2*c)^7 + 231*tan(1/2*a)^10*tan(1/2*c)^8 + 163*tan(1/2*a)^9*tan(1/2*c) ^9 + 231*tan(1/2*a)^8*tan(1/2*c)^10 + 24*tan(1/2*a)^7*tan(1/2*c)^11 + 24*t an(1/2*a)^6*tan(1/2*c)^12 - 3*tan(1/2*a)^5*tan(1/2*c)^13 + 3*tan(1/2*a)^13 *tan(1/2*c)^4 - 27*tan(1/2*a)^12*tan(1/2*c)^5 + 24*tan(1/2*a)^11*tan(1/2*c )^6 - 192*tan(1/2*a)^10*tan(1/2*c)^7 + 123*tan(1/2*a)^9*tan(1/2*c)^8 - ...
Timed out. \[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^3*cos(c + b*x)^4),x)
Output:
\text{Hanged}
\[ \int \sec ^3(a+b x) \sec ^4(c+b x) \, dx=\text {too large to display} \] Input:
int(sec(b*x+a)^3*sec(b*x+c)^4,x)
Output:
(60*cos(b*x + c)**2*cos(a + b*x)**2*sin(b*x + c) - 30*cos(b*x + c)**2*cos( a + b*x)*sin(b*x + c)*sin(a + b*x)**2 + 60*cos(b*x + c)**2*cos(a + b*x)*si n(b*x + c) + 24*cos(b*x + c)**2*sin(a + b*x)**3 - 24*cos(b*x + c)**2*sin(a + b*x) - 84*cos(b*x + c)*cos(a + b*x)**2*sin(b*x + c)**2*sin(a + b*x) + 1 20*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) + 24*cos(b*x + c)*cos(a + b*x)*int(cos(a + b*x)/(cos(b*x + c)*sin(b*x + c)**2*sin(a + b*x)**2 - cos(b*x + c)*sin(b*x + c)**2 - cos (b*x + c)*sin(a + b*x)**2 + cos(b*x + c)),x)*sin(b*x + c)**2*sin(a + b*x)* *2*b - 24*cos(b*x + c)*cos(a + b*x)*int(cos(a + b*x)/(cos(b*x + c)*sin(b*x + c)**2*sin(a + b*x)**2 - cos(b*x + c)*sin(b*x + c)**2 - cos(b*x + c)*sin (a + b*x)**2 + cos(b*x + c)),x)*sin(b*x + c)**2*b - 24*cos(b*x + c)*cos(a + b*x)*int(cos(a + b*x)/(cos(b*x + c)*sin(b*x + c)**2*sin(a + b*x)**2 - co s(b*x + c)*sin(b*x + c)**2 - cos(b*x + c)*sin(a + b*x)**2 + cos(b*x + c)), x)*sin(a + b*x)**2*b + 24*cos(b*x + c)*cos(a + b*x)*int(cos(a + b*x)/(cos( b*x + c)*sin(b*x + c)**2*sin(a + b*x)**2 - cos(b*x + c)*sin(b*x + c)**2 - cos(b*x + c)*sin(a + b*x)**2 + cos(b*x + c)),x)*b + 24*cos(b*x + c)*cos(a + b*x)*int(cos(a + b*x)/(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 - 24*cos(b*x + c)*cos(a + b*x)*int(cos(a + b*x)/(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 - 24*cos(b*x + ...