Integrand size = 16, antiderivative size = 37 \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=-\frac {\sec ^2(c-b x) \sin (a+c)}{2 b}-\frac {\cos (a+c) \tan (c-b x)}{b} \] Output:
-1/2*sec(b*x-c)^2*sin(a+c)/b+cos(a+c)*tan(b*x-c)/b
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=-\frac {\sec (c) \sec ^2(c-b x) (\sin (a)+\cos (a+c) \sin (c-2 b x))}{2 b} \] Input:
Integrate[Cos[a + b*x]*Sec[c - b*x]^3,x]
Output:
-1/2*(Sec[c]*Sec[c - b*x]^2*(Sin[a] + Cos[a + c]*Sin[c - 2*b*x]))/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.
\(\displaystyle \int \cos (a+b x) \sec ^3(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos (a+b x) \sec ^3(c-b x)dx\) |
Input:
Int[Cos[a + b*x]*Sec[c - b*x]^3,x]
Output:
$Aborted
Time = 1.93 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \(\frac {\sin \left (2 b x +a -c \right )}{b \left (1+\cos \left (2 b x -2 c \right )\right )}\) | \(28\) |
default | \(-\frac {1}{2 b \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )^{2}}\) | \(55\) |
risch | \(\frac {i \left ({\mathrm e}^{5 i \left (a +c \right )}+2 \,{\mathrm e}^{i \left (2 b x +5 a +3 c \right )}+{\mathrm e}^{3 i \left (a +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right )^{2} b}\) | \(58\) |
Input:
int(cos(b*x+a)*sec(b*x-c)^3,x,method=_RETURNVERBOSE)
Output:
1/b*sin(2*b*x+a-c)/(1+cos(2*b*x-2*c))
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (36) = 72\).
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.30 \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=\frac {2 \, {\left (2 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - {\left (4 \, \cos \left (b x + a\right )^{2} \cos \left (a + c\right )^{2} - 2 \, \cos \left (a + c\right )^{2} + 1\right )} \sin \left (a + c\right )}{2 \, {\left (2 \, b \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + {\left (2 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{2} - b \cos \left (a + c\right )^{2} + b\right )}} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^3,x, algorithm="fricas")
Output:
1/2*(2*(2*cos(a + c)^3 - cos(a + c))*cos(b*x + a)*sin(b*x + a) - (4*cos(b* x + a)^2*cos(a + c)^2 - 2*cos(a + c)^2 + 1)*sin(a + c))/(2*b*cos(b*x + a)* cos(a + c)*sin(b*x + a)*sin(a + c) + (2*b*cos(a + c)^2 - b)*cos(b*x + a)^2 - b*cos(a + c)^2 + b)
Exception generated. \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)**3,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (36) = 72\).
Time = 0.06 (sec) , antiderivative size = 419, normalized size of antiderivative = 11.32 \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=-\frac {{\left (2 \, \sin \left (2 \, b x + 2 \, a + 3 \, c\right ) + \sin \left (2 \, a + 5 \, c\right ) + \sin \left (3 \, c\right )\right )} \cos \left (4 \, b x + a\right ) - 2 \, {\left (2 \, \sin \left (2 \, b x + a + 2 \, c\right ) + \sin \left (a + 4 \, c\right )\right )} \cos \left (2 \, b x + 2 \, a + 3 \, c\right ) + 2 \, {\left (\sin \left (2 \, a + 5 \, c\right ) + \sin \left (3 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) - {\left (2 \, \cos \left (2 \, b x + 2 \, a + 3 \, c\right ) + \cos \left (2 \, a + 5 \, c\right ) + \cos \left (3 \, c\right )\right )} \sin \left (4 \, b x + a\right ) + 2 \, {\left (2 \, \cos \left (2 \, b x + a + 2 \, c\right ) + \cos \left (a + 4 \, c\right )\right )} \sin \left (2 \, b x + 2 \, a + 3 \, c\right ) - 2 \, {\left (\cos \left (2 \, a + 5 \, c\right ) + \cos \left (3 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) + \cos \left (a + 4 \, c\right ) \sin \left (2 \, a + 5 \, c\right ) - \cos \left (2 \, a + 5 \, c\right ) \sin \left (a + 4 \, c\right ) - \cos \left (3 \, c\right ) \sin \left (a + 4 \, c\right ) + \cos \left (a + 4 \, c\right ) \sin \left (3 \, c\right )}{b \cos \left (4 \, b x + a\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 2 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a + 4 \, c\right ) + b \cos \left (a + 4 \, c\right )^{2} + b \sin \left (4 \, b x + a\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a + 4 \, c\right ) + b \sin \left (a + 4 \, c\right )^{2} + 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) + b \cos \left (a + 4 \, c\right )\right )} \cos \left (4 \, b x + a\right ) + 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) + b \sin \left (a + 4 \, c\right )\right )} \sin \left (4 \, b x + a\right )} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^3,x, algorithm="maxima")
Output:
-((2*sin(2*b*x + 2*a + 3*c) + sin(2*a + 5*c) + sin(3*c))*cos(4*b*x + a) - 2*(2*sin(2*b*x + a + 2*c) + sin(a + 4*c))*cos(2*b*x + 2*a + 3*c) + 2*(sin( 2*a + 5*c) + sin(3*c))*cos(2*b*x + a + 2*c) - (2*cos(2*b*x + 2*a + 3*c) + cos(2*a + 5*c) + cos(3*c))*sin(4*b*x + a) + 2*(2*cos(2*b*x + a + 2*c) + co s(a + 4*c))*sin(2*b*x + 2*a + 3*c) - 2*(cos(2*a + 5*c) + cos(3*c))*sin(2*b *x + a + 2*c) + cos(a + 4*c)*sin(2*a + 5*c) - cos(2*a + 5*c)*sin(a + 4*c) - cos(3*c)*sin(a + 4*c) + cos(a + 4*c)*sin(3*c))/(b*cos(4*b*x + a)^2 + 4*b *cos(2*b*x + a + 2*c)^2 + 4*b*cos(2*b*x + a + 2*c)*cos(a + 4*c) + b*cos(a + 4*c)^2 + b*sin(4*b*x + a)^2 + 4*b*sin(2*b*x + a + 2*c)^2 + 4*b*sin(2*b*x + a + 2*c)*sin(a + 4*c) + b*sin(a + 4*c)^2 + 2*(2*b*cos(2*b*x + a + 2*c) + b*cos(a + 4*c))*cos(4*b*x + a) + 2*(2*b*sin(2*b*x + a + 2*c) + b*sin(a + 4*c))*sin(4*b*x + a))
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (36) = 72\).
Time = 0.15 (sec) , antiderivative size = 313, normalized size of antiderivative = 8.46 \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=\frac {\tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{6} + \tan \left (\frac {1}{2} \, a\right )^{6} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{4} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, c\right )^{2} + 1}{4 \, {\left (2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{2} - 1\right )}^{2} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} b} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^3,x, algorithm="giac")
Output:
1/4*(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)/((2*tan(b*x + a)*tan(1/2*a)^2*tan(1/2*c) + 2*tan(b*x + a)*tan(1/2*a)*t an(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^2 - 2*tan(b*x + a)*tan(1/2*a) + tan( 1/2*a)^2 - 2*tan(b*x + a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) + tan(1/2*c )^2 - 1)^2*(tan(1/2*a)^2*tan(1/2*c) + tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c))*b)
Timed out. \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=\text {Hanged} \] Input:
int(cos(a + b*x)/cos(c - b*x)^3,x)
Output:
\text{Hanged}
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int \cos (a+b x) \sec ^3(c-b x) \, dx=\frac {-\cos \left (b x -c \right ) \sin \left (b x +a \right )-\cos \left (b x +a \right ) \sin \left (b x -c \right )}{2 b \left (\sin \left (b x -c \right )^{2}-1\right )} \] Input:
int(cos(b*x+a)*sec(b*x-c)^3,x)
Output:
( - (cos(b*x - c)*sin(a + b*x) + cos(a + b*x)*sin(b*x - c)))/(2*b*(sin(b*x - c)**2 - 1))