Integrand size = 17, antiderivative size = 1 \[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 8.71 (sec) , antiderivative size = 491, normalized size of antiderivative = 491.00 \[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\frac {6 i \arctan \left (\frac {(i \cos (a)+\sin (a)) \left (\cos \left (\frac {b x}{2}\right ) \sin (a)+\cos (a) \sin \left (\frac {b x}{2}\right )\right )}{\cos (a) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (a)}\right ) \cos (a-c)}{\frac {3 b}{8}+\frac {1}{8} b \cos (4 a-4 c)-\frac {1}{2} b \cos (2 a-2 c)}-\frac {3 (3+\cos (2 a-2 c)) \csc ^4(a-c) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )-\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{4 b}+\frac {3 (3+\cos (2 a-2 c)) \csc ^4(a-c) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )+\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{4 b}+\frac {\csc ^3(a-c) \sec (a+b x)}{b}+\frac {\csc ^2(a-c)}{4 b \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )-\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )^2}-\frac {\csc ^3(a-c) \left (\sin \left (a-c-\frac {b x}{2}\right )-\sin \left (a-c+\frac {b x}{2}\right )\right )}{b \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )-\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}-\frac {\csc ^2(a-c)}{4 b \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )+\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )^2}-\frac {\csc ^3(a-c) \left (-\sin \left (a-c-\frac {b x}{2}\right )+\sin \left (a-c+\frac {b x}{2}\right )\right )}{b \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )+\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )} \] Input:
Integrate[Sec[a + b*x]^2*Sec[c + b*x]^3,x]
Output:
((6*I)*ArcTan[((I*Cos[a] + Sin[a])*(Cos[(b*x)/2]*Sin[a] + Cos[a]*Sin[(b*x) /2]))/(Cos[a]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[a])]*Cos[a - c])/((3*b)/8 + (b*Cos[4*a - 4*c])/8 - (b*Cos[2*a - 2*c])/2) - (3*(3 + Cos[2*a - 2*c])*C sc[a - c]^4*Log[Cos[c/2 + (b*x)/2] - Sin[c/2 + (b*x)/2]])/(4*b) + (3*(3 + Cos[2*a - 2*c])*Csc[a - c]^4*Log[Cos[c/2 + (b*x)/2] + Sin[c/2 + (b*x)/2]]) /(4*b) + (Csc[a - c]^3*Sec[a + b*x])/b + Csc[a - c]^2/(4*b*(Cos[c/2 + (b*x )/2] - Sin[c/2 + (b*x)/2])^2) - (Csc[a - c]^3*(Sin[a - c - (b*x)/2] - Sin[ a - c + (b*x)/2]))/(b*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (b*x)/2] - Sin[c/2 + (b*x)/2])) - Csc[a - c]^2/(4*b*(Cos[c/2 + (b*x)/2] + Sin[c/2 + (b*x)/2]) ^2) - (Csc[a - c]^3*(-Sin[a - c - (b*x)/2] + Sin[a - c + (b*x)/2]))/(b*(Co s[c/2] + Sin[c/2])*(Cos[c/2 + (b*x)/2] + Sin[c/2 + (b*x)/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.
| \(\displaystyle \int \sec ^2(a+b x) \sec ^3(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sec ^2(a+b x) \sec ^3(b x+c)dx\) |
Input:
Int[Sec[a + b*x]^2*Sec[c + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 7.40 (sec) , antiderivative size = 943, normalized size of antiderivative = 943.00
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(943\) |
| default | \(\text {Expression too large to display}\) | \(1071\) |
Input:
int(sec(b*x+a)^2*sec(b*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-4*I/(exp(2*I*(b*x+a))+1)/(exp(2*I*(b*x+a+c))+exp(2*I*a))^2/(exp(2*I*a)-ex p(2*I*c))^3/b*(3*exp(5*I*(b*x+2*a+c))+9*exp(I*(5*b*x+8*a+7*c))+5*exp(I*(3* b*x+10*a+3*c))+14*exp(I*(3*b*x+8*a+5*c))+5*exp(I*(3*b*x+6*a+7*c))+9*exp(I* (b*x+8*a+3*c))+3*exp(I*(b*x+6*a+5*c)))-24*ln(exp(I*(b*x+a))+I)/(exp(8*I*a) -4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*exp( I*(5*a+3*c))-24*ln(exp(I*(b*x+a))+I)/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp( 4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*exp(I*(3*a+5*c))-6*ln(exp(I*(b *x+a))-I*exp(I*(a-c)))/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*e xp(2*I*(a+3*c))+exp(8*I*c))/b*exp(2*I*(3*a+c))-36*ln(exp(I*(b*x+a))-I*exp( I*(a-c)))/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c ))+exp(8*I*c))/b*exp(4*I*(a+c))-6*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/(exp(8 *I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b *exp(2*I*(a+3*c))+6*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))/(exp(8*I*a)-4*exp(2* I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*exp(2*I*(3*a+ c))+36*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6* exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*exp(4*I*(a+c))+6*ln(exp(I* (b*x+a))+I*exp(I*(a-c)))/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4 *exp(2*I*(a+3*c))+exp(8*I*c))/b*exp(2*I*(a+3*c))+24*ln(exp(I*(b*x+a))-I)/( exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I* c))/b*exp(I*(5*a+3*c))+24*ln(exp(I*(b*x+a))-I)/(exp(8*I*a)-4*exp(2*I*(3...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.15 (sec) , antiderivative size = 472, normalized size of antiderivative = 472.00 \[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\frac {6 \, {\left (\cos \left (-a + c\right )^{3} - \cos \left (-a + c\right )\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) - 6 \, {\left (\cos \left (b x + c\right )^{3} \cos \left (-a + c\right )^{2} + \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right )\right )} \log \left (\frac {2 \, {\left (\cos \left (-a + c\right ) \sin \left (b x + c\right ) - \cos \left (b x + c\right ) \sin \left (-a + c\right ) + 1\right )}}{\cos \left (-a + c\right ) + 1}\right ) + 6 \, {\left (\cos \left (b x + c\right )^{3} \cos \left (-a + c\right )^{2} + \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right )\right )} \log \left (-\frac {2 \, {\left (\cos \left (-a + c\right ) \sin \left (b x + c\right ) - \cos \left (b x + c\right ) \sin \left (-a + c\right ) - 1\right )}}{\cos \left (-a + c\right ) + 1}\right ) + 3 \, {\left ({\left (\cos \left (-a + c\right )^{2} + 1\right )} \cos \left (b x + c\right )^{2} \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (\cos \left (-a + c\right )^{3} + \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{3}\right )} \log \left (\sin \left (b x + c\right ) + 1\right ) - 3 \, {\left ({\left (\cos \left (-a + c\right )^{2} + 1\right )} \cos \left (b x + c\right )^{2} \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (\cos \left (-a + c\right )^{3} + \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{3}\right )} \log \left (-\sin \left (b x + c\right ) + 1\right ) - 2 \, {\left (3 \, {\left (\cos \left (-a + c\right )^{2} + 1\right )} \cos \left (b x + c\right )^{2} + \cos \left (-a + c\right )^{2} - 1\right )} \sin \left (-a + c\right )}{4 \, {\left ({\left (b \cos \left (-a + c\right )^{4} - 2 \, b \cos \left (-a + c\right )^{2} + b\right )} \cos \left (b x + c\right )^{2} \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (b \cos \left (-a + c\right )^{5} - 2 \, b \cos \left (-a + c\right )^{3} + b \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{3}\right )}} \] Input:
integrate(sec(b*x+a)^2*sec(b*x+c)^3,x, algorithm="fricas")
Output:
1/4*(6*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)*sin(b*x + c) - 6*(cos(b* x + c)^3*cos(-a + c)^2 + cos(b*x + c)^2*cos(-a + c)*sin(b*x + c)*sin(-a + c))*log(2*(cos(-a + c)*sin(b*x + c) - cos(b*x + c)*sin(-a + c) + 1)/(cos(- a + c) + 1)) + 6*(cos(b*x + c)^3*cos(-a + c)^2 + cos(b*x + c)^2*cos(-a + c )*sin(b*x + c)*sin(-a + c))*log(-2*(cos(-a + c)*sin(b*x + c) - cos(b*x + c )*sin(-a + c) - 1)/(cos(-a + c) + 1)) + 3*((cos(-a + c)^2 + 1)*cos(b*x + c )^2*sin(b*x + c)*sin(-a + c) + (cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^ 3)*log(sin(b*x + c) + 1) - 3*((cos(-a + c)^2 + 1)*cos(b*x + c)^2*sin(b*x + c)*sin(-a + c) + (cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^3)*log(-sin(b *x + c) + 1) - 2*(3*(cos(-a + c)^2 + 1)*cos(b*x + c)^2 + cos(-a + c)^2 - 1 )*sin(-a + c))/((b*cos(-a + c)^4 - 2*b*cos(-a + c)^2 + b)*cos(b*x + c)^2*s in(b*x + c)*sin(-a + c) + (b*cos(-a + c)^5 - 2*b*cos(-a + c)^3 + b*cos(-a + c))*cos(b*x + c)^3)
\[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\int \sec ^{2}{\left (a + b x \right )} \sec ^{3}{\left (b x + c \right )}\, dx \] Input:
integrate(sec(b*x+a)**2*sec(b*x+c)**3,x)
Output:
Integral(sec(a + b*x)**2*sec(b*x + c)**3, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 12.63 (sec) , antiderivative size = 504922, normalized size of antiderivative = 504922.00 \[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^2*sec(b*x+c)^3,x, algorithm="maxima")
Output:
-(24*(((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))*sin(3*a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c))*sin(2*a + 6*c))*cos(6*b*x + 8*a + 4*c)^2 + 9*((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))*sin(3*a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c))*sin(2*a + 6*c))*cos(6*b*x + 6*a + 6*c)^2 + 9*((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) + si n(3*a + 5*c))*cos(2*a + 6*c) - (cos(8*a) - 4*cos(6*a + 2*c) + cos(8*c))*si n(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(6*b*x + 4*a + 8*c)^2 + ((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 2.87 (sec) , antiderivative size = 15002, normalized size of antiderivative = 15002.00 \[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)^2*sec(b*x+c)^3,x, algorithm="giac")
Output:
-1/16*(3*(tan(1/2*a)^9*tan(1/2*c)^9 - tan(1/2*a)^9*tan(1/2*c)^8 + tan(1/2* a)^8*tan(1/2*c)^9 + 2*tan(1/2*a)^9*tan(1/2*c)^7 + 5*tan(1/2*a)^8*tan(1/2*c )^8 + 2*tan(1/2*a)^7*tan(1/2*c)^9 - 2*tan(1/2*a)^9*tan(1/2*c)^6 - 2*tan(1/ 2*a)^8*tan(1/2*c)^7 + 2*tan(1/2*a)^7*tan(1/2*c)^8 + 2*tan(1/2*a)^6*tan(1/2 *c)^9 + 2*tan(1/2*a)^9*tan(1/2*c)^5 + 6*tan(1/2*a)^8*tan(1/2*c)^6 + 20*tan (1/2*a)^7*tan(1/2*c)^7 + 6*tan(1/2*a)^6*tan(1/2*c)^8 + 2*tan(1/2*a)^5*tan( 1/2*c)^9 - 2*tan(1/2*a)^9*tan(1/2*c)^4 - 2*tan(1/2*a)^8*tan(1/2*c)^5 - 12* tan(1/2*a)^7*tan(1/2*c)^6 + 12*tan(1/2*a)^6*tan(1/2*c)^7 + 2*tan(1/2*a)^5* tan(1/2*c)^8 + 2*tan(1/2*a)^4*tan(1/2*c)^9 + 2*tan(1/2*a)^9*tan(1/2*c)^3 - 2*tan(1/2*a)^8*tan(1/2*c)^4 + 32*tan(1/2*a)^7*tan(1/2*c)^5 + 20*tan(1/2*a )^6*tan(1/2*c)^6 + 32*tan(1/2*a)^5*tan(1/2*c)^7 - 2*tan(1/2*a)^4*tan(1/2*c )^8 + 2*tan(1/2*a)^3*tan(1/2*c)^9 - 2*tan(1/2*a)^9*tan(1/2*c)^2 + 6*tan(1/ 2*a)^8*tan(1/2*c)^3 - 32*tan(1/2*a)^7*tan(1/2*c)^4 + 24*tan(1/2*a)^6*tan(1 /2*c)^5 - 24*tan(1/2*a)^5*tan(1/2*c)^6 + 32*tan(1/2*a)^4*tan(1/2*c)^7 - 6* tan(1/2*a)^3*tan(1/2*c)^8 + 2*tan(1/2*a)^2*tan(1/2*c)^9 + tan(1/2*a)^9*tan (1/2*c) - 2*tan(1/2*a)^8*tan(1/2*c)^2 + 12*tan(1/2*a)^7*tan(1/2*c)^3 + 24* tan(1/2*a)^6*tan(1/2*c)^4 + 56*tan(1/2*a)^5*tan(1/2*c)^5 + 24*tan(1/2*a)^4 *tan(1/2*c)^6 + 12*tan(1/2*a)^3*tan(1/2*c)^7 - 2*tan(1/2*a)^2*tan(1/2*c)^8 + tan(1/2*a)*tan(1/2*c)^9 - tan(1/2*a)^9 + 5*tan(1/2*a)^8*tan(1/2*c) - 20 *tan(1/2*a)^7*tan(1/2*c)^2 + 20*tan(1/2*a)^6*tan(1/2*c)^3 - 56*tan(1/2*...
Timed out. \[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^2*cos(c + b*x)^3),x)
Output:
\text{Hanged}
\[ \int \sec ^2(a+b x) \sec ^3(c+b x) \, dx=\text {too large to display} \] Input:
int(sec(b*x+a)^2*sec(b*x+c)^3,x)
Output:
(32*cos(b*x + c)*sin(a + b*x) + 20*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 - 20*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 + 20*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 - 20*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)*b + 12*cos(a + b*x)*int(sin(b*x + c)**2/(sin(b*x + c)**2 - 1),x)*sin(b*x + c)**2*b - 12*cos(a + b*x)*int(sin(b*x + c)**2/(si n(b*x + c)**2 - 1),x)*b - 12*cos(a + b*x)*int(sin(b*x + c)**2/(cos(b*x + c )*cos(a + b*x)*sin(b*x + c)**2 - cos(b*x + c)*cos(a + b*x)),x)*sin(b*x + c )**2*b + 12*cos(a + b*x)*int(sin(b*x + c)**2/(cos(b*x + c)*cos(a + b*x)*si n(b*x + c)**2 - cos(b*x + c)*cos(a + b*x)),x)*b + 20*cos(a + b*x)*int(sin( b*x + c)**2/(cos(b*x + c)*sin(b*x + c)**2*sin(a + b*x)**2 - cos(b*x + c)*s in(b*x + c)**2 - cos(b*x + c)*sin(a + b*x)**2 + cos(b*x + c)),x)*sin(b*x + c)**2*b - 20*cos(a + b*x)*int(sin(b*x + c)**2/(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 - 12*cos(a + b*x)*int(sin(b*x + c)**2/(cos(b*x + c)*sin(b*x + c)**2 - cos(b*x + c)),x)*sin(b*x + c)**2*b + 12*cos(a +...