Integrand size = 17, antiderivative size = 1 \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.55 (sec) , antiderivative size = 105, normalized size of antiderivative = 105.00 \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\frac {12 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) (3 \cos (a-c)+5 \cos (3 (a-c)))-(32+40 \cos (2 (a-c))+15 \cos (2 (a-2 c-b x))+33 \cos (2 (a+b x))+24 \cos (2 (c+b x))) \sec ^3(c+b x) \sin (a-c)}{48 b} \] Input:
Integrate[Cos[a + b*x]^3*Sec[c + b*x]^4,x]
Output:
(12*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*(3*Cos[a - c] + 5*Cos[3*(a - c)] ) - (32 + 40*Cos[2*(a - c)] + 15*Cos[2*(a - 2*c - b*x)] + 33*Cos[2*(a + b* x)] + 24*Cos[2*(c + b*x)])*Sec[c + b*x]^3*Sin[a - c])/(48*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.
\(\displaystyle \int \cos ^3(a+b x) \sec ^4(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^3(a+b x) \sec ^4(b x+c)dx\) |
Input:
Int[Cos[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 = 40.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 357.00
method | result | size |
risch | \(\frac {i \left (33 \,{\mathrm e}^{i \left (5 b x +9 a +2 c \right )}-9 \,{\mathrm e}^{i \left (5 b x +7 a +4 c \right )}-9 \,{\mathrm e}^{i \left (5 b x +5 a +6 c \right )}-15 \,{\mathrm e}^{i \left (5 b x +3 a +8 c \right )}+40 \,{\mathrm e}^{3 i \left (b x +3 a \right )}+24 \,{\mathrm e}^{i \left (3 b x +7 a +2 c \right )}-24 \,{\mathrm e}^{i \left (3 b x +5 a +4 c \right )}-40 \,{\mathrm e}^{3 i \left (b x +a +2 c \right )}+15 \,{\mathrm e}^{i \left (b x +9 a -2 c \right )}+9 \,{\mathrm e}^{i \left (b x +7 a \right )}+9 \,{\mathrm e}^{i \left (b x +5 a +2 c \right )}-33 \,{\mathrm e}^{i \left (b x +3 a +4 c \right )}\right )}{24 \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3} b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (3 a -3 c \right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}-\frac {5 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (3 a -3 c \right )}{8 b}\) | \(357\) |
default | \(\text {Expression too large to display}\) | \(4421\) |
Input:
int(cos(b*x+a)^3*sec(b*x+c)^4,x,method=_RETURNVERBOSE)
Output:
1/24*I/(exp(2*I*(b*x+a+c))+exp(2*I*a))^3/b*(33*exp(I*(5*b*x+9*a+2*c))-9*ex p(I*(5*b*x+7*a+4*c))-9*exp(I*(5*b*x+5*a+6*c))-15*exp(I*(5*b*x+3*a+8*c))+40 *exp(3*I*(b*x+3*a))+24*exp(I*(3*b*x+7*a+2*c))-24*exp(I*(3*b*x+5*a+4*c))-40 *exp(3*I*(b*x+a+2*c))+15*exp(I*(b*x+9*a-2*c))+9*exp(I*(b*x+7*a))+9*exp(I*( b*x+5*a+2*c))-33*exp(I*(b*x+3*a+4*c)))+3/8*ln(exp(I*(b*x+a))+I*exp(I*(a-c) ))/b*cos(a-c)+5/8*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))/b*cos(3*a-3*c)-3/8*ln( exp(I*(b*x+a))-I*exp(I*(a-c)))/b*cos(a-c)-5/8*ln(exp(I*(b*x+a))-I*exp(I*(a -c)))/b*cos(3*a-3*c)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 165.00 \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\frac {3 \, {\left (5 \, \cos \left (-a + c\right )^{3} - 3 \, \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{3} \log \left (\sin \left (b x + c\right ) + 1\right ) - 3 \, {\left (5 \, \cos \left (-a + c\right )^{3} - 3 \, \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{3} \log \left (-\sin \left (b x + c\right ) + 1\right ) - 18 \, {\left (\cos \left (-a + c\right )^{3} - \cos \left (-a + c\right )\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) + 4 \, {\left (3 \, {\left (4 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )} \sin \left (-a + c\right )}{12 \, b \cos \left (b x + c\right )^{3}} \] Input:
integrate(cos(b*x+a)^3*sec(b*x+c)^4,x, algorithm="fricas")
Output:
1/12*(3*(5*cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^3*log(sin(b*x + c) + 1) - 3*(5*cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^3*log(-sin(b*x + c ) + 1) - 18*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)*sin(b*x + c) + 4*(3 *(4*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*sin(-a + c))/(b *cos(b*x + c)^3)
Timed out. \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**3*sec(b*x+c)**4,x)
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.31 (sec) , antiderivative size = 2778, normalized size of antiderivative = 2778.00 \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^3*sec(b*x+c)^4,x, algorithm="maxima")
Output:
-1/48*(2*(33*sin(5*b*x + 6*a + 4*c) - 9*sin(5*b*x + 4*a + 6*c) - 9*sin(5*b *x + 2*a + 8*c) - 15*sin(5*b*x + 10*c) + 40*sin(3*b*x + 6*a + 2*c) + 24*si n(3*b*x + 4*a + 4*c) - 24*sin(3*b*x + 2*a + 6*c) - 40*sin(3*b*x + 8*c) + 1 5*sin(b*x + 6*a) + 9*sin(b*x + 4*a + 2*c) + 9*sin(b*x + 2*a + 4*c) - 33*si n(b*x + 6*c))*cos(6*b*x + 3*a + 8*c) - 66*(3*sin(4*b*x + 3*a + 6*c) + 3*si n(2*b*x + 3*a + 4*c) + sin(3*a + 2*c))*cos(5*b*x + 6*a + 4*c) + 18*(3*sin( 4*b*x + 3*a + 6*c) + 3*sin(2*b*x + 3*a + 4*c) + sin(3*a + 2*c))*cos(5*b*x + 4*a + 6*c) + 18*(3*sin(4*b*x + 3*a + 6*c) + 3*sin(2*b*x + 3*a + 4*c) + s in(3*a + 2*c))*cos(5*b*x + 2*a + 8*c) + 30*(3*sin(4*b*x + 3*a + 6*c) + 3*s in(2*b*x + 3*a + 4*c) + sin(3*a + 2*c))*cos(5*b*x + 10*c) + 6*(40*sin(3*b* x + 6*a + 2*c) + 24*sin(3*b*x + 4*a + 4*c) - 24*sin(3*b*x + 2*a + 6*c) - 4 0*sin(3*b*x + 8*c) + 15*sin(b*x + 6*a) + 9*sin(b*x + 4*a + 2*c) + 9*sin(b* x + 2*a + 4*c) - 33*sin(b*x + 6*c))*cos(4*b*x + 3*a + 6*c) - 80*(3*sin(2*b *x + 3*a + 4*c) + sin(3*a + 2*c))*cos(3*b*x + 6*a + 2*c) - 48*(3*sin(2*b*x + 3*a + 4*c) + sin(3*a + 2*c))*cos(3*b*x + 4*a + 4*c) + 48*(3*sin(2*b*x + 3*a + 4*c) + sin(3*a + 2*c))*cos(3*b*x + 2*a + 6*c) + 80*(3*sin(2*b*x + 3 *a + 4*c) + sin(3*a + 2*c))*cos(3*b*x + 8*c) + 18*(5*sin(b*x + 6*a) + 3*si n(b*x + 4*a + 2*c) + 3*sin(b*x + 2*a + 4*c) - 11*sin(b*x + 6*c))*cos(2*b*x + 3*a + 4*c) + 3*((3*cos(-a + c) + 5*cos(-3*a + 3*c))*cos(6*b*x + 3*a + 8 *c)^2 + 9*(3*cos(-a + c) + 5*cos(-3*a + 3*c))*cos(4*b*x + 3*a + 6*c)^2 ...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.07 (sec) , antiderivative size = 22836, normalized size of antiderivative = 22836.00 \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^3*sec(b*x+c)^4,x, algorithm="giac")
Output:
-1/3*(3*(tan(1/2*a)^7*tan(1/2*c)^7 - tan(1/2*a)^7*tan(1/2*c)^6 + tan(1/2*a )^6*tan(1/2*c)^7 - 9*tan(1/2*a)^7*tan(1/2*c)^5 + 25*tan(1/2*a)^6*tan(1/2*c )^6 - 9*tan(1/2*a)^5*tan(1/2*c)^7 + 9*tan(1/2*a)^7*tan(1/2*c)^4 - 33*tan(1 /2*a)^6*tan(1/2*c)^5 + 33*tan(1/2*a)^5*tan(1/2*c)^6 - 9*tan(1/2*a)^4*tan(1 /2*c)^7 + 9*tan(1/2*a)^7*tan(1/2*c)^3 - 81*tan(1/2*a)^6*tan(1/2*c)^4 + 165 *tan(1/2*a)^5*tan(1/2*c)^5 - 81*tan(1/2*a)^4*tan(1/2*c)^6 + 9*tan(1/2*a)^3 *tan(1/2*c)^7 - 9*tan(1/2*a)^7*tan(1/2*c)^2 + 81*tan(1/2*a)^6*tan(1/2*c)^3 - 213*tan(1/2*a)^5*tan(1/2*c)^4 + 213*tan(1/2*a)^4*tan(1/2*c)^5 - 81*tan( 1/2*a)^3*tan(1/2*c)^6 + 9*tan(1/2*a)^2*tan(1/2*c)^7 - tan(1/2*a)^7*tan(1/2 *c) + 33*tan(1/2*a)^6*tan(1/2*c)^2 - 213*tan(1/2*a)^5*tan(1/2*c)^3 + 397*t an(1/2*a)^4*tan(1/2*c)^4 - 213*tan(1/2*a)^3*tan(1/2*c)^5 + 33*tan(1/2*a)^2 *tan(1/2*c)^6 - tan(1/2*a)*tan(1/2*c)^7 + tan(1/2*a)^7 - 25*tan(1/2*a)^6*t an(1/2*c) + 165*tan(1/2*a)^5*tan(1/2*c)^2 - 397*tan(1/2*a)^4*tan(1/2*c)^3 + 397*tan(1/2*a)^3*tan(1/2*c)^4 - 165*tan(1/2*a)^2*tan(1/2*c)^5 + 25*tan(1 /2*a)*tan(1/2*c)^6 - tan(1/2*c)^7 - tan(1/2*a)^6 + 33*tan(1/2*a)^5*tan(1/2 *c) - 213*tan(1/2*a)^4*tan(1/2*c)^2 + 397*tan(1/2*a)^3*tan(1/2*c)^3 - 213* tan(1/2*a)^2*tan(1/2*c)^4 + 33*tan(1/2*a)*tan(1/2*c)^5 - tan(1/2*c)^6 - 9* tan(1/2*a)^5 + 81*tan(1/2*a)^4*tan(1/2*c) - 213*tan(1/2*a)^3*tan(1/2*c)^2 + 213*tan(1/2*a)^2*tan(1/2*c)^3 - 81*tan(1/2*a)*tan(1/2*c)^4 + 9*tan(1/2*c )^5 + 9*tan(1/2*a)^4 - 81*tan(1/2*a)^3*tan(1/2*c) + 165*tan(1/2*a)^2*ta...
Timed out. \[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\text {Hanged} \] Input:
int(cos(a + b*x)^3/cos(c + b*x)^4,x)
Output:
\text{Hanged}
\[ \int \cos ^3(a+b x) \sec ^4(c+b x) \, dx=\text {too large to display} \] Input:
int(cos(b*x+a)^3*sec(b*x+c)^4,x)
Output:
( - 63*cos(b*x + c)*cos(a + b*x)*sin(b*x + c)**2*sin(a + b*x) + 192*cos(b* x + c)*cos(a + b*x)*sin(b*x + c)*sin(a + b*x)**2 - 768*cos(b*x + c)*cos(a + b*x)*sin(b*x + c) + 309*cos(b*x + c)*cos(a + b*x)*sin(a + b*x) - 4992*co s(b*x + c)*int(tan((b*x + c)/2)**6/(tan((b*x + c)/2)**8*tan((a + b*x)/2)** 6 + 3*tan((b*x + c)/2)**8*tan((a + b*x)/2)**4 + 3*tan((b*x + c)/2)**8*tan( (a + b*x)/2)**2 + tan((b*x + c)/2)**8 - 4*tan((b*x + c)/2)**6*tan((a + b*x )/2)**6 - 12*tan((b*x + c)/2)**6*tan((a + b*x)/2)**4 - 12*tan((b*x + c)/2) **6*tan((a + b*x)/2)**2 - 4*tan((b*x + c)/2)**6 + 6*tan((b*x + c)/2)**4*ta n((a + b*x)/2)**6 + 18*tan((b*x + c)/2)**4*tan((a + b*x)/2)**4 + 18*tan((b *x + c)/2)**4*tan((a + b*x)/2)**2 + 6*tan((b*x + c)/2)**4 - 4*tan((b*x + c )/2)**2*tan((a + b*x)/2)**6 - 12*tan((b*x + c)/2)**2*tan((a + b*x)/2)**4 - 12*tan((b*x + c)/2)**2*tan((a + b*x)/2)**2 - 4*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)*si n(b*x + c)**2*b + 4992*cos(b*x + c)*int(tan((b*x + c)/2)**6/(tan((b*x + c) /2)**8*tan((a + b*x)/2)**6 + 3*tan((b*x + c)/2)**8*tan((a + b*x)/2)**4 + 3 *tan((b*x + c)/2)**8*tan((a + b*x)/2)**2 + tan((b*x + c)/2)**8 - 4*tan((b* x + c)/2)**6*tan((a + b*x)/2)**6 - 12*tan((b*x + c)/2)**6*tan((a + b*x)/2) **4 - 12*tan((b*x + c)/2)**6*tan((a + b*x)/2)**2 - 4*tan((b*x + c)/2)**6 + 6*tan((b*x + c)/2)**4*tan((a + b*x)/2)**6 + 18*tan((b*x + c)/2)**4*tan((a + b*x)/2)**4 + 18*tan((b*x + c)/2)**4*tan((a + b*x)/2)**2 + 6*tan((b*x...