Integrand size = 17, antiderivative size = 1 \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.60 (sec) , antiderivative size = 291, normalized size of antiderivative = 291.00 \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=\frac {\sec (c) \sec ^2(c+b x) (4 b x \cos (3 a-4 c)+4 b x \cos (3 a-2 c)+2 b x \cos (3 a-6 c-2 b x)+2 b x \cos (3 a-4 c-2 b x)+2 b x \cos (3 a+2 b x)+2 b x \cos (3 a-2 c+2 b x)-6 \sin (a)-2 \sin (3 a-4 c)+4 \log (\cos (c+b x)) \sin (3 a-4 c)+4 \sin (3 a-2 c)+4 \log (\cos (c+b x)) \sin (3 a-2 c)+2 \log (\cos (c+b x)) \sin (3 a-6 c-2 b x)+3 \sin (3 a-4 c-2 b x)+2 \log (\cos (c+b x)) \sin (3 a-4 c-2 b x)+3 \sin (a+2 b x)+2 \log (\cos (c+b x)) \sin (3 a+2 b x)-3 \sin (3 a-2 c+2 b x)+2 \log (\cos (c+b x)) \sin (3 a-2 c+2 b x)-3 \sin (a-2 (c+b x)))}{16 b} \] Input:
Integrate[Cos[a + b*x]^3*Sec[c + b*x]^3,x]
Output:
(Sec[c]*Sec[c + b*x]^2*(4*b*x*Cos[3*a - 4*c] + 4*b*x*Cos[3*a - 2*c] + 2*b* x*Cos[3*a - 6*c - 2*b*x] + 2*b*x*Cos[3*a - 4*c - 2*b*x] + 2*b*x*Cos[3*a + 2*b*x] + 2*b*x*Cos[3*a - 2*c + 2*b*x] - 6*Sin[a] - 2*Sin[3*a - 4*c] + 4*Lo g[Cos[c + b*x]]*Sin[3*a - 4*c] + 4*Sin[3*a - 2*c] + 4*Log[Cos[c + b*x]]*Si n[3*a - 2*c] + 2*Log[Cos[c + b*x]]*Sin[3*a - 6*c - 2*b*x] + 3*Sin[3*a - 4* c - 2*b*x] + 2*Log[Cos[c + b*x]]*Sin[3*a - 4*c - 2*b*x] + 3*Sin[a + 2*b*x] + 2*Log[Cos[c + b*x]]*Sin[3*a + 2*b*x] - 3*Sin[3*a - 2*c + 2*b*x] + 2*Log [Cos[c + b*x]]*Sin[3*a - 2*c + 2*b*x] - 3*Sin[a - 2*(c + b*x)]))/(16*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.
\(\displaystyle \int \cos ^3(a+b x) \sec ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^3(a+b x) \sec ^3(b x+c)dx\) |
Input:
Int[Cos[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 = 15.97 (sec) , antiderivative size = 192, normalized size of antiderivative = 192.00
method | result | size |
risch | \(x \,{\mathrm e}^{3 i \left (a -c \right )}-2 i \sin \left (3 a -3 c \right ) x -\frac {2 i \sin \left (3 a -3 c \right ) a}{b}-\frac {i \left (4 \,{\mathrm e}^{i \left (2 b x +7 a -c \right )}-6 \,{\mathrm e}^{i \left (2 b x +5 a +c \right )}+2 \,{\mathrm e}^{i \left (2 b x +a +5 c \right )}+3 \,{\mathrm e}^{i \left (7 a -3 c \right )}-3 \,{\mathrm e}^{i \left (5 a -c \right )}-3 \,{\mathrm e}^{i \left (3 a +c \right )}+3 \,{\mathrm e}^{i \left (a +3 c \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) \sin \left (3 a -3 c \right )}{b}\) | \(192\) |
default | \(\text {Expression too large to display}\) | \(720\) |
Input:
int(cos(b*x+a)^3*sec(b*x+c)^3,x,method=_RETURNVERBOSE)
Output:
x*exp(3*I*(a-c))-2*I*sin(3*a-3*c)*x-2*I/b*sin(3*a-3*c)*a-1/4*I/(exp(2*I*(b *x+a+c))+exp(2*I*a))^2/b*(4*exp(I*(2*b*x+7*a-c))-6*exp(I*(2*b*x+5*a+c))+2* exp(I*(2*b*x+a+5*c))+3*exp(I*(7*a-3*c))-3*exp(I*(5*a-c))-3*exp(I*(3*a+c))+ 3*exp(I*(a+3*c)))+1/b*ln(exp(2*I*(b*x+a))+exp(2*I*(a-c)))*sin(3*a-3*c)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 132.00 \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=-\frac {2 \, {\left (4 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} \log \left (-\cos \left (b x + c\right )\right ) \sin \left (-a + c\right ) - 2 \, {\left (4 \, b x \cos \left (-a + c\right )^{3} - 3 \, b x \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{2} + 6 \, {\left (\cos \left (-a + c\right )^{3} - \cos \left (-a + c\right )\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) + {\left (\cos \left (-a + c\right )^{2} - 1\right )} \sin \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )^{2}} \] Input:
integrate(cos(b*x+a)^3*sec(b*x+c)^3,x, algorithm="fricas")
Output:
-1/2*(2*(4*cos(-a + c)^2 - 1)*cos(b*x + c)^2*log(-cos(b*x + c))*sin(-a + c ) - 2*(4*b*x*cos(-a + c)^3 - 3*b*x*cos(-a + c))*cos(b*x + c)^2 + 6*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)*sin(b*x + c) + (cos(-a + c)^2 - 1)*sin (-a + c))/(b*cos(b*x + c)^2)
Timed out. \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**3*sec(b*x+c)**3,x)
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 1401, normalized size of antiderivative = 1401.00 \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^3*sec(b*x+c)^3,x, algorithm="maxima")
Output:
1/4*(4*(b*cos(3*a + 3*c)*cos(6*c) + b*sin(3*a + 3*c)*sin(6*c))*x + (8*b*x* cos(2*b*x + 8*c) + 4*b*x*cos(6*c) + 4*sin(2*b*x + 6*a + 2*c) - 6*sin(2*b*x + 4*a + 4*c) + 2*sin(2*b*x + 8*c) + 3*sin(6*a) - 3*sin(4*a + 2*c) - 3*sin (2*a + 4*c) + 3*sin(6*c))*cos(4*b*x + 3*a + 7*c) + 4*(b*x*cos(4*b*x + 3*a + 7*c) + 2*b*x*cos(2*b*x + 3*a + 5*c) + b*x*cos(3*a + 3*c))*cos(4*b*x + 10 *c) + 2*(4*b*x*cos(6*c) + 4*sin(2*b*x + 6*a + 2*c) - 6*sin(2*b*x + 4*a + 4 *c) + 3*sin(6*a) - 3*sin(4*a + 2*c) - 3*sin(2*a + 4*c) + 3*sin(6*c))*cos(2 *b*x + 3*a + 5*c) + 2*(8*b*x*cos(2*b*x + 3*a + 5*c) + 4*b*x*cos(3*a + 3*c) - 2*sin(2*b*x + 3*a + 5*c) - sin(3*a + 3*c))*cos(2*b*x + 8*c) + 3*(sin(6* a) - sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - 2*(cos(4*b*x + 3*a + 7*c) ^2*sin(-3*a + 3*c) + 4*cos(2*b*x + 3*a + 5*c)^2*sin(-3*a + 3*c) + 4*cos(2* b*x + 3*a + 5*c)*cos(3*a + 3*c)*sin(-3*a + 3*c) + cos(3*a + 3*c)^2*sin(-3* a + 3*c) + sin(4*b*x + 3*a + 7*c)^2*sin(-3*a + 3*c) + 4*sin(2*b*x + 3*a + 5*c)^2*sin(-3*a + 3*c) + 4*sin(2*b*x + 3*a + 5*c)*sin(3*a + 3*c)*sin(-3*a + 3*c) + sin(3*a + 3*c)^2*sin(-3*a + 3*c) + 2*(2*cos(2*b*x + 3*a + 5*c)*si n(-3*a + 3*c) + cos(3*a + 3*c)*sin(-3*a + 3*c))*cos(4*b*x + 3*a + 7*c) + 2 *(2*sin(2*b*x + 3*a + 5*c)*sin(-3*a + 3*c) + sin(3*a + 3*c)*sin(-3*a + 3*c ))*sin(4*b*x + 3*a + 7*c))*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*c) + cos( 2*c)^2 + sin(2*b*x)^2 - 2*sin(2*b*x)*sin(2*c) + sin(2*c)^2) + (8*b*x*sin(2 *b*x + 8*c) + 4*b*x*sin(6*c) - 4*cos(2*b*x + 6*a + 2*c) + 6*cos(2*b*x +...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 5096, normalized size of antiderivative = 5096.00 \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^3*sec(b*x+c)^3,x, algorithm="giac")
Output:
((tan(1/2*a)^6*tan(1/2*c)^6 - 15*tan(1/2*a)^6*tan(1/2*c)^4 + 36*tan(1/2*a) ^5*tan(1/2*c)^5 - 15*tan(1/2*a)^4*tan(1/2*c)^6 + 15*tan(1/2*a)^6*tan(1/2*c )^2 - 120*tan(1/2*a)^5*tan(1/2*c)^3 + 225*tan(1/2*a)^4*tan(1/2*c)^4 - 120* tan(1/2*a)^3*tan(1/2*c)^5 + 15*tan(1/2*a)^2*tan(1/2*c)^6 - tan(1/2*a)^6 + 36*tan(1/2*a)^5*tan(1/2*c) - 225*tan(1/2*a)^4*tan(1/2*c)^2 + 400*tan(1/2*a )^3*tan(1/2*c)^3 - 225*tan(1/2*a)^2*tan(1/2*c)^4 + 36*tan(1/2*a)*tan(1/2*c )^5 - tan(1/2*c)^6 + 15*tan(1/2*a)^4 - 120*tan(1/2*a)^3*tan(1/2*c) + 225*t an(1/2*a)^2*tan(1/2*c)^2 - 120*tan(1/2*a)*tan(1/2*c)^3 + 15*tan(1/2*c)^4 - 15*tan(1/2*a)^2 + 36*tan(1/2*a)*tan(1/2*c) - 15*tan(1/2*c)^2 + 1)*(b*x + a)/(tan(1/2*a)^6*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 3*tan(1/2*a) ^4*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^2 + 9*tan(1/2*a)^4*tan(1/2*c)^ 4 + 3*tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2*a)^6 + 9*tan(1/2*a)^4*tan(1/2*c) ^2 + 9*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*c)^6 + 3*tan(1/2*a)^4 + 9*tan(1 /2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*c)^4 + 3*tan(1/2*a)^2 + 3*tan(1/2*c)^2 + 1) - (3*tan(1/2*a)^6*tan(1/2*c)^5 - 3*tan(1/2*a)^5*tan(1/2*c)^6 - 10*tan(1 /2*a)^6*tan(1/2*c)^3 + 45*tan(1/2*a)^5*tan(1/2*c)^4 - 45*tan(1/2*a)^4*tan( 1/2*c)^5 + 10*tan(1/2*a)^3*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c) - 45*t an(1/2*a)^5*tan(1/2*c)^2 + 150*tan(1/2*a)^4*tan(1/2*c)^3 - 150*tan(1/2*a)^ 3*tan(1/2*c)^4 + 45*tan(1/2*a)^2*tan(1/2*c)^5 - 3*tan(1/2*a)*tan(1/2*c)^6 + 3*tan(1/2*a)^5 - 45*tan(1/2*a)^4*tan(1/2*c) + 150*tan(1/2*a)^3*tan(1/...
Timed out. \[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=\text {Hanged} \] Input:
int(cos(a + b*x)^3/cos(c + b*x)^3,x)
Output:
\text{Hanged}
\[ \int \cos ^3(a+b x) \sec ^3(c+b x) \, dx=\text {too large to display} \] Input:
int(cos(b*x+a)^3*sec(b*x+c)^3,x)
Output:
(36*cos(b*x + c)*cos(a + b*x)*sin(b*x + c)*sin(a + b*x)**2 - 144*cos(b*x + c)*cos(a + b*x)*sin(b*x + c) + 108*cos(b*x + c)*cos(a + b*x)*sin(a + b*x) + 108*cos(b*x + c)*sin(b*x + c)*sin(a + b*x)**2 - 144*cos(b*x + c)*sin(b* x + c) - 54*cos(b*x + c)*sin(a + b*x)**3 + 63*cos(b*x + c)*sin(a + b*x) + 27*cos(a + b*x)*sin(b*x + c)**2*sin(a + b*x) + 54*cos(a + b*x)*sin(b*x + c )*sin(a + b*x)**2 - 261*cos(a + b*x)*sin(b*x + c) + 81*cos(a + b*x)*sin(a + b*x) + 4320*int(tan((b*x + c)/2)**4/(tan((b*x + c)/2)**6*tan((a + b*x)/2 )**6 + 3*tan((b*x + c)/2)**6*tan((a + b*x)/2)**4 + 3*tan((b*x + c)/2)**6*t an((a + b*x)/2)**2 + tan((b*x + c)/2)**6 - 3*tan((b*x + c)/2)**4*tan((a + b*x)/2)**6 - 9*tan((b*x + c)/2)**4*tan((a + b*x)/2)**4 - 9*tan((b*x + c)/2 )**4*tan((a + b*x)/2)**2 - 3*tan((b*x + c)/2)**4 + 3*tan((b*x + c)/2)**2*t an((a + b*x)/2)**6 + 9*tan((b*x + c)/2)**2*tan((a + b*x)/2)**4 + 9*tan((b* x + c)/2)**2*tan((a + b*x)/2)**2 + 3*tan((b*x + c)/2)**2 - tan((a + b*x)/2 )**6 - 3*tan((a + b*x)/2)**4 - 3*tan((a + b*x)/2)**2 - 1),x)*sin(b*x + c)* *2*b - 4320*int(tan((b*x + c)/2)**4/(tan((b*x + c)/2)**6*tan((a + b*x)/2)* *6 + 3*tan((b*x + c)/2)**6*tan((a + b*x)/2)**4 + 3*tan((b*x + c)/2)**6*tan ((a + b*x)/2)**2 + tan((b*x + c)/2)**6 - 3*tan((b*x + c)/2)**4*tan((a + b* x)/2)**6 - 9*tan((b*x + c)/2)**4*tan((a + b*x)/2)**4 - 9*tan((b*x + c)/2)* *4*tan((a + b*x)/2)**2 - 3*tan((b*x + c)/2)**4 + 3*tan((b*x + c)/2)**2*tan ((a + b*x)/2)**6 + 9*tan((b*x + c)/2)**2*tan((a + b*x)/2)**4 + 9*tan((b...