\(\int \sec ^2(a+b x) \sec ^4(c+b x) \, dx\) [323]

Optimal result

Integrand size = 17, antiderivative size = 1 \[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=0 \] Output:

0
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 6.49 (sec) , antiderivative size = 966, normalized size of antiderivative = 966.00 \[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx =\text {Too large to display} \] Input:

Integrate[Sec[a + b*x]^2*Sec[c + b*x]^4,x]
 

Output:

((-4*I)*ArcTan[Tan[a + b*x]]*Cot[a - c]*Csc[a - c]^4)/b + ((4*I)*ArcTan[Ta 
n[c + b*x]]*Cot[a - c]*Csc[a - c]^4)/b + (2*Cot[a - c]*Csc[a - c]^4*Log[Co 
s[a + b*x]^2])/b - (2*Cot[a - c]*Csc[a - c]^4*Log[Cos[c + b*x]^2])/b + (Cs 
c[a - c]^4*Sec[a]*Sec[c]*Sec[a + b*x]*Sec[c + b*x]^3*(-15*Sin[2*a] + 3*Sin 
[2*a - 4*c] + 3*Sin[4*a - 4*c] + 9*Sin[2*a - 2*c] - 18*Sin[2*c] + 25*Sin[2 
*b*x] - Sin[2*a - 4*c - 4*b*x] + 2*Sin[2*a - 4*c - 2*b*x] - Sin[4*a - 4*c 
- 2*b*x] - 7*Sin[2*a - 2*c - 2*b*x] + 16*Sin[2*a + 2*b*x] + 3*Sin[2*a - 2* 
c + 2*b*x] + 3*Sin[4*a - 2*c + 2*b*x] + 7*Sin[2*c + 2*b*x] - 6*Sin[2*a + 2 
*c + 2*b*x] - 6*Sin[4*c + 2*b*x] + Sin[2*a + 4*b*x] + Sin[4*a + 4*b*x] + 1 
0*Sin[2*c + 4*b*x] + 7*Sin[2*a + 2*c + 4*b*x] + 4*Sin[4*c + 4*b*x]))/(48*b 
) + x*(((2*I)*Cos[a]*Cos[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + ((2*I)*Co 
s[c]*Sec[a])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + (4*Cos[c]*Sin[a])/(Cos[c] 
*Sin[a] - Cos[a]*Sin[c])^5 - (2*Cos[a]*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin 
[c])^5 + (2*Sec[a]*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + ((4*I)*Sin[ 
a]*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^5 + ((2*I)*Cos[a]*Cos[c])/(-(Co 
s[c]*Sin[a]) + Cos[a]*Sin[c])^5 + ((2*I)*Cos[a]*Sec[c])/(-(Cos[c]*Sin[a]) 
+ Cos[a]*Sin[c])^5 - (2*Cos[c]*Sin[a])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^ 
5 + (2*Sec[c]*Sin[a])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^5 + (4*Cos[a]*Sin 
[c])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^5 + ((4*I)*Sin[a]*Sin[c])/(-(Cos[c 
]*Sin[a]) + Cos[a]*Sin[c])^5 + ((-1 + Cos[2*c] + I*Sin[2*c])*(Cos[2*a] ...
 

Rubi [F]

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.

\(\displaystyle \int \sec ^2(a+b x) \sec ^4(b x+c) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sec ^2(a+b x) \sec ^4(b x+c)dx\)

Input:

Int[Sec[a + b*x]^2*Sec[c + b*x]^4,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 18.17 (sec) , antiderivative size = 506, normalized size of antiderivative = 506.00

method	result	size
default	\(\text {Expression too large to display}\)	\(506\)
risch	\(\frac {32 i \left (6 \,{\mathrm e}^{2 i \left (3 b x +6 a +4 c \right )}+6 \,{\mathrm e}^{2 i \left (3 b x +5 a +5 c \right )}+15 \,{\mathrm e}^{2 i \left (2 b x +6 a +3 c \right )}+18 \,{\mathrm e}^{2 i \left (2 b x +5 a +4 c \right )}+3 \,{\mathrm e}^{2 i \left (2 b x +4 a +5 c \right )}+{\mathrm e}^{2 i \left (b x +7 a +c \right )}+7 \,{\mathrm e}^{2 i \left (b x +6 a +2 c \right )}+25 \,{\mathrm e}^{2 i \left (b x +5 a +3 c \right )}+3 \,{\mathrm e}^{2 i \left (b x +4 a +4 c \right )}+{\mathrm e}^{2 i \left (6 a +c \right )}+10 \,{\mathrm e}^{2 i \left (5 a +2 c \right )}+{\mathrm e}^{2 i \left (4 a +3 c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3} \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )^{4} b}+\frac {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (3 a +2 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 {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (2 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 {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (3 a +2 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 {64 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (2 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}\)	\(617\)

Input:

int(sec(b*x+a)^2*sec(b*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

1/b*(tan(b*x+a)/(sin(a)*cos(c)-cos(a)*sin(c))^4-(2*cos(c)^2*sin(a)^2+6*cos 
(a)^2*cos(c)^2+8*cos(a)*cos(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/(tan(b*x+a)*sin(a)*cos(c)-tan( 
b*x+a)*cos(a)*sin(c)+cos(a)*cos(c)+sin(a)*sin(c))-(4*cos(a)*cos(c)+4*sin(a 
)*sin(c))/(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))-1/3/(sin(a)*cos(c)-cos(a 
)*sin(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*c 
os(a)^4*cos(c)^2*sin(c)^2+sin(a)^4*sin(c)^4+2*cos(a)^2*sin(a)^2*sin(c)^4+c 
os(a)^4*sin(c)^4)/(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*(4*cos(c)^3*sin(a)^2*cos(a)+4*cos(c)^3*cos(a 
)^3+4*cos(c)^2*sin(c)*sin(a)^3+4*cos(c)^2*sin(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)-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)
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.11 (sec) , antiderivative size = 429, normalized size of antiderivative = 429.00 \[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=\frac {4 \, {\left (\cos \left (-a + c\right )^{4} + \cos \left (-a + c\right )^{2} - 2\right )} \cos \left (b x + c\right )^{4} + \cos \left (-a + c\right )^{4} - 2 \, {\left (\cos \left (-a + c\right )^{4} + \cos \left (-a + c\right )^{2} - 2\right )} \cos \left (b x + c\right )^{2} + 2 \, {\left (2 \, {\left (\cos \left (-a + c\right )^{3} + 2 \, \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{3} + {\left (\cos \left (-a + c\right )^{3} - \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )\right )} \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 2 \, \cos \left (-a + c\right )^{2} + 6 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right )^{2} + \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x + c\right )^{2}\right ) - 6 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right )^{2} + \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right )\right )} \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right ) + 1}{3 \, {\left ({\left (b \cos \left (-a + c\right )^{5} - 2 \, b \cos \left (-a + c\right )^{3} + b \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{4} \sin \left (-a + c\right ) - {\left (b \cos \left (-a + c\right )^{6} - 3 \, b \cos \left (-a + c\right )^{4} + 3 \, b \cos \left (-a + c\right )^{2} - b\right )} \cos \left (b x + c\right )^{3} \sin \left (b x + c\right )\right )}} \] Input:

integrate(sec(b*x+a)^2*sec(b*x+c)^4,x, algorithm="fricas")
 

Output:

1/3*(4*(cos(-a + c)^4 + cos(-a + c)^2 - 2)*cos(b*x + c)^4 + cos(-a + c)^4 
- 2*(cos(-a + c)^4 + cos(-a + c)^2 - 2)*cos(b*x + c)^2 + 2*(2*(cos(-a + c) 
^3 + 2*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 + 6*(cos(b*x + c)^4*cos( 
-a + c)^2 + cos(b*x + c)^3*cos(-a + c)*sin(b*x + c)*sin(-a + c))*log(cos(b 
*x + c)^2) - 6*(cos(b*x + c)^4*cos(-a + c)^2 + cos(b*x + c)^3*cos(-a + c)* 
sin(b*x + c)*sin(-a + c))*log(4*(2*cos(b*x + c)*cos(-a + c)*sin(b*x + c)*s 
in(-a + c) + (2*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)/(co 
s(-a + c)^2 + 2*cos(-a + c) + 1)) + 1)/((b*cos(-a + c)^5 - 2*b*cos(-a + c) 
^3 + b*cos(-a + c))*cos(b*x + c)^4*sin(-a + c) - (b*cos(-a + c)^6 - 3*b*co 
s(-a + c)^4 + 3*b*cos(-a + c)^2 - b)*cos(b*x + c)^3*sin(b*x + c))
 

Sympy [F]

\[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=\int \sec ^{2}{\left (a + b x \right )} \sec ^{4}{\left (b x + c \right )}\, dx \] Input:

integrate(sec(b*x+a)**2*sec(b*x+c)**4,x)
 

Output:

Integral(sec(a + b*x)**2*sec(b*x + c)**4, x)
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 34.92 (sec) , antiderivative size = 1342628, normalized size of antiderivative = 1342628.00 \[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=\text {Too large to display} \] Input:

integrate(sec(b*x+a)^2*sec(b*x+c)^4,x, algorithm="maxima")
 

Output:

32/3*(25*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) - (cos(8*a) + cos(8*c))*sin 
(6*a + 2*c))*cos(8*a + 2*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) 
+ 2*(5*sin(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) - (cos(8* 
a) + cos(8*c))*sin(6*a + 2*c) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos( 
8*c))*sin(4*a + 4*c))*cos(6*a + 4*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6* 
a + 2*c) + 2*(5*sin(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) 
+ (sin(8*a) + 46*sin(4*a + 4*c) + sin(8*c))*cos(2*a + 6*c) - (cos(8*a) + c 
os(8*c))*sin(6*a + 2*c) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos(8*c))* 
sin(4*a + 4*c) - (cos(8*a) + 46*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c)) 
*cos(4*a + 6*c)^2 + 25*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 2*(5*sin(8* 
a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) + (sin(8*a) + 46*sin(4 
*a + 4*c) + sin(8*c))*cos(2*a + 6*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2*c 
) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos(8*c))*sin(4*a + 4*c) - (cos( 
8*a) + 46*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c))*cos(2*a + 8*c)^2 + 25 
*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2 
*c))*sin(8*a + 2*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 2*(5*s 
in(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) - (cos(8*a) + cos 
(8*c))*sin(6*a + 2*c) - 2*(5*cos(8*a) - 23*cos(6*a + 2*c) + 5*cos(8*c))*si 
n(4*a + 4*c))*sin(6*a + 4*c)^2 + 100*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) 
 + 2*(5*sin(8*a) - 23*sin(6*a + 2*c) + 5*sin(8*c))*cos(4*a + 4*c) + (si...
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.43 (sec) , antiderivative size = 13226, normalized size of antiderivative = 13226.00 \[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=\text {Too large to display} \] Input:

integrate(sec(b*x+a)^2*sec(b*x+c)^4,x, algorithm="giac")
 

Output:

-1/96*(12*(tan(1/2*a)^10*tan(1/2*c)^10 + 3*tan(1/2*a)^10*tan(1/2*c)^8 + 4* 
tan(1/2*a)^9*tan(1/2*c)^9 + 3*tan(1/2*a)^8*tan(1/2*c)^10 + 2*tan(1/2*a)^10 
*tan(1/2*c)^6 + 16*tan(1/2*a)^9*tan(1/2*c)^7 + 9*tan(1/2*a)^8*tan(1/2*c)^8 
 + 16*tan(1/2*a)^7*tan(1/2*c)^9 + 2*tan(1/2*a)^6*tan(1/2*c)^10 - 2*tan(1/2 
*a)^10*tan(1/2*c)^4 + 24*tan(1/2*a)^9*tan(1/2*c)^5 + 6*tan(1/2*a)^8*tan(1/ 
2*c)^6 + 64*tan(1/2*a)^7*tan(1/2*c)^7 + 6*tan(1/2*a)^6*tan(1/2*c)^8 + 24*t 
an(1/2*a)^5*tan(1/2*c)^9 - 2*tan(1/2*a)^4*tan(1/2*c)^10 - 3*tan(1/2*a)^10* 
tan(1/2*c)^2 + 16*tan(1/2*a)^9*tan(1/2*c)^3 - 6*tan(1/2*a)^8*tan(1/2*c)^4 
+ 96*tan(1/2*a)^7*tan(1/2*c)^5 + 4*tan(1/2*a)^6*tan(1/2*c)^6 + 96*tan(1/2* 
a)^5*tan(1/2*c)^7 - 6*tan(1/2*a)^4*tan(1/2*c)^8 + 16*tan(1/2*a)^3*tan(1/2* 
c)^9 - 3*tan(1/2*a)^2*tan(1/2*c)^10 - tan(1/2*a)^10 + 4*tan(1/2*a)^9*tan(1 
/2*c) - 9*tan(1/2*a)^8*tan(1/2*c)^2 + 64*tan(1/2*a)^7*tan(1/2*c)^3 - 4*tan 
(1/2*a)^6*tan(1/2*c)^4 + 144*tan(1/2*a)^5*tan(1/2*c)^5 - 4*tan(1/2*a)^4*ta 
n(1/2*c)^6 + 64*tan(1/2*a)^3*tan(1/2*c)^7 - 9*tan(1/2*a)^2*tan(1/2*c)^8 + 
4*tan(1/2*a)*tan(1/2*c)^9 - tan(1/2*c)^10 - 3*tan(1/2*a)^8 + 16*tan(1/2*a) 
^7*tan(1/2*c) - 6*tan(1/2*a)^6*tan(1/2*c)^2 + 96*tan(1/2*a)^5*tan(1/2*c)^3 
 + 4*tan(1/2*a)^4*tan(1/2*c)^4 + 96*tan(1/2*a)^3*tan(1/2*c)^5 - 6*tan(1/2* 
a)^2*tan(1/2*c)^6 + 16*tan(1/2*a)*tan(1/2*c)^7 - 3*tan(1/2*c)^8 - 2*tan(1/ 
2*a)^6 + 24*tan(1/2*a)^5*tan(1/2*c) + 6*tan(1/2*a)^4*tan(1/2*c)^2 + 64*tan 
(1/2*a)^3*tan(1/2*c)^3 + 6*tan(1/2*a)^2*tan(1/2*c)^4 + 24*tan(1/2*a)*ta...
 

Mupad [F(-1)]

Timed out. \[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=\text {Hanged} \] Input:

int(1/(cos(a + b*x)^2*cos(c + b*x)^4),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \sec ^2(a+b x) \sec ^4(c+b x) \, dx=\int \sec \left (b x +a \right )^{2} \sec \left (b x +c \right )^{4}d x \] Input:

int(sec(b*x+a)^2*sec(b*x+c)^4,x)
 

Output:

int(sec(b*x+a)^2*sec(b*x+c)^4,x)