Integrand size = 16, antiderivative size = 65 \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=-\frac {\text {arctanh}(\sin (c-b x)) \cos (a+c)}{2 b}-\frac {\sec ^3(c-b x) \sin (a+c)}{3 b}-\frac {\cos (a+c) \sec (c-b x) \tan (c-b x)}{2 b} \] Output:
1/2*arctanh(sin(b*x-c))*cos(a+c)/b-1/3*sec(b*x-c)^3*sin(a+c)/b+1/2*cos(a+c )*sec(b*x-c)*tan(b*x-c)/b
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=-\frac {12 \text {arctanh}\left (\sin (c)-\cos (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a+c)+\sec ^3(c-b x) (4 \sin (a+c)+3 \cos (a+c) \sin (2 (c-b x)))}{12 b} \] Input:
Integrate[Cos[a + b*x]*Sec[c - b*x]^4,x]
Output:
-1/12*(12*ArcTanh[Sin[c] - Cos[c]*Tan[(b*x)/2]]*Cos[a + c] + Sec[c - b*x]^ 3*(4*Sin[a + c] + 3*Cos[a + c]*Sin[2*(c - 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.
\(\displaystyle \int \cos (a+b x) \sec ^4(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos (a+b x) \sec ^4(c-b x)dx\) |
Input:
Int[Cos[a + b*x]*Sec[c - b*x]^4,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 10.38 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.80
method | result | size |
risch | \(\frac {i \left (3 \,{\mathrm e}^{i \left (b x +7 a +6 c \right )}+8 \,{\mathrm e}^{i \left (3 b x +7 a +4 c \right )}+3 \,{\mathrm e}^{i \left (b x +5 a +4 c \right )}-3 \,{\mathrm e}^{i \left (5 b x +7 a +2 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}-3 \,{\mathrm e}^{5 i \left (b x +a \right )}\right )}{6 b \left ({\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a +c \right )}\right ) \cos \left (a +c \right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a +c \right )}\right ) \cos \left (a +c \right )}{2 b}\) | \(182\) |
default | \(\text {Expression too large to display}\) | \(2374\) |
Input:
int(cos(b*x+a)*sec(b*x-c)^4,x,method=_RETURNVERBOSE)
Output:
1/6*I/b/(exp(2*I*(a+c))+exp(2*I*(b*x+a)))^3*(3*exp(I*(b*x+7*a+6*c))+8*exp( I*(3*b*x+7*a+4*c))+3*exp(I*(b*x+5*a+4*c))-3*exp(I*(5*b*x+7*a+2*c))-8*exp(I *(3*b*x+5*a+2*c))-3*exp(5*I*(b*x+a)))-1/2*ln(exp(I*(b*x+a))-I*exp(I*(a+c)) )/b*cos(a+c)+1/2*ln(exp(I*(b*x+a))+I*exp(I*(a+c)))/b*cos(a+c)
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (63) = 126\).
Time = 0.10 (sec) , antiderivative size = 433, normalized size of antiderivative = 6.66 \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=\frac {6 \, {\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 ) + 3 \, {\left ({\left (4 \, \cos \left (a + c\right )^{4} - 3 \, \cos \left (a + c\right )^{2}\right )} \cos \left (b x + a\right )^{3} + {\left ({\left (4 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{3} + \cos \left (a + c\right )\right )} \sin \left (b x + a\right ) \sin \left (a + c\right ) - 3 \, {\left (\cos \left (a + c\right )^{4} - \cos \left (a + c\right )^{2}\right )} \cos \left (b x + a\right )\right )} \log \left (\frac {2 \, {\left (\cos \left (a + c\right ) \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \sin \left (a + c\right ) + 1\right )}}{\cos \left (a + c\right ) + 1}\right ) - 3 \, {\left ({\left (4 \, \cos \left (a + c\right )^{4} - 3 \, \cos \left (a + c\right )^{2}\right )} \cos \left (b x + a\right )^{3} + {\left ({\left (4 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{3} + \cos \left (a + c\right )\right )} \sin \left (b x + a\right ) \sin \left (a + c\right ) - 3 \, {\left (\cos \left (a + c\right )^{4} - \cos \left (a + c\right )^{2}\right )} \cos \left (b x + a\right )\right )} \log \left (-\frac {2 \, {\left (\cos \left (a + c\right ) \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \sin \left (a + c\right ) - 1\right )}}{\cos \left (a + c\right ) + 1}\right ) - 2 \, {\left (6 \, \cos \left (b x + a\right )^{2} \cos \left (a + c\right )^{2} - 3 \, \cos \left (a + c\right )^{2} + 2\right )} \sin \left (a + c\right )}{12 \, {\left ({\left (4 \, b \cos \left (a + c\right )^{3} - 3 \, b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{3} + {\left ({\left (4 \, 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 )} \sin \left (b x + a\right ) \sin \left (a + c\right ) - 3 \, {\left (b \cos \left (a + c\right )^{3} - b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )\right )}} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^4,x, algorithm="fricas")
Output:
1/12*(6*(2*cos(a + c)^3 - cos(a + c))*cos(b*x + a)*sin(b*x + a) + 3*((4*co s(a + c)^4 - 3*cos(a + c)^2)*cos(b*x + a)^3 + ((4*cos(a + c)^3 - cos(a + c ))*cos(b*x + a)^2 - cos(a + c)^3 + cos(a + c))*sin(b*x + a)*sin(a + c) - 3 *(cos(a + c)^4 - cos(a + c)^2)*cos(b*x + a))*log(2*(cos(a + c)*sin(b*x + a ) - cos(b*x + a)*sin(a + c) + 1)/(cos(a + c) + 1)) - 3*((4*cos(a + c)^4 - 3*cos(a + c)^2)*cos(b*x + a)^3 + ((4*cos(a + c)^3 - cos(a + c))*cos(b*x + a)^2 - cos(a + c)^3 + cos(a + c))*sin(b*x + a)*sin(a + c) - 3*(cos(a + c)^ 4 - cos(a + c)^2)*cos(b*x + a))*log(-2*(cos(a + c)*sin(b*x + a) - cos(b*x + a)*sin(a + c) - 1)/(cos(a + c) + 1)) - 2*(6*cos(b*x + a)^2*cos(a + c)^2 - 3*cos(a + c)^2 + 2)*sin(a + c))/((4*b*cos(a + c)^3 - 3*b*cos(a + c))*cos (b*x + a)^3 + ((4*b*cos(a + c)^2 - b)*cos(b*x + a)^2 - b*cos(a + c)^2 + b) *sin(b*x + a)*sin(a + c) - 3*(b*cos(a + c)^3 - b*cos(a + c))*cos(b*x + a))
Timed out. \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (63) = 126\).
Time = 0.20 (sec) , antiderivative size = 1427, normalized size of antiderivative = 21.95 \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^4,x, algorithm="maxima")
Output:
1/12*(2*(3*sin(5*b*x) + 3*sin(5*b*x + 2*a + 2*c) - 8*sin(3*b*x + 2*a + 4*c ) + 8*sin(3*b*x + 2*c) - 3*sin(b*x + 2*a + 6*c) - 3*sin(b*x + 4*c))*cos(6* b*x + a) - 6*(3*sin(4*b*x + a + 2*c) + 3*sin(2*b*x + a + 4*c) + sin(a + 6* c))*cos(5*b*x + 2*a + 2*c) + 6*(3*sin(5*b*x) - 8*sin(3*b*x + 2*a + 4*c) + 8*sin(3*b*x + 2*c) - 3*sin(b*x + 2*a + 6*c) - 3*sin(b*x + 4*c))*cos(4*b*x + a + 2*c) + 16*(3*sin(2*b*x + a + 4*c) + sin(a + 6*c))*cos(3*b*x + 2*a + 4*c) - 16*(3*sin(2*b*x + a + 4*c) + sin(a + 6*c))*cos(3*b*x + 2*c) + 18*(s in(5*b*x) - sin(b*x + 2*a + 6*c) - sin(b*x + 4*c))*cos(2*b*x + a + 4*c) - 3*(cos(6*b*x + a)^2*cos(a + c) + 9*cos(4*b*x + a + 2*c)^2*cos(a + c) + 9*c os(2*b*x + a + 4*c)^2*cos(a + c) + 6*cos(2*b*x + a + 4*c)*cos(a + 6*c)*cos (a + c) + cos(a + 6*c)^2*cos(a + c) + cos(a + c)*sin(6*b*x + a)^2 + 9*cos( a + c)*sin(4*b*x + a + 2*c)^2 + 9*cos(a + c)*sin(2*b*x + a + 4*c)^2 + 6*co s(a + c)*sin(2*b*x + a + 4*c)*sin(a + 6*c) + cos(a + c)*sin(a + 6*c)^2 + 2 *(3*cos(4*b*x + a + 2*c)*cos(a + c) + 3*cos(2*b*x + a + 4*c)*cos(a + c) + cos(a + 6*c)*cos(a + c))*cos(6*b*x + a) + 6*(3*cos(2*b*x + a + 4*c)*cos(a + c) + cos(a + 6*c)*cos(a + c))*cos(4*b*x + a + 2*c) + 2*(3*cos(a + c)*sin (4*b*x + a + 2*c) + 3*cos(a + c)*sin(2*b*x + a + 4*c) + cos(a + c)*sin(a + 6*c))*sin(6*b*x + a) + 6*(3*cos(a + c)*sin(2*b*x + a + 4*c) + cos(a + c)* sin(a + 6*c))*sin(4*b*x + a + 2*c))*log((cos(b*x)^2 + cos(c)^2 - 2*cos(c)* sin(b*x) + sin(b*x)^2 + 2*cos(b*x)*sin(c) + sin(c)^2)/(cos(b*x)^2 + cos...
Leaf count of result is larger than twice the leaf count of optimal. 12160 vs. \(2 (63) = 126\).
Time = 0.95 (sec) , antiderivative size = 12160, normalized size of antiderivative = 187.08 \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^4,x, algorithm="giac")
Output:
1/6*(3*(tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a) ^2*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c) - 5*tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^3 + 5*tan(1/2*a)^2*tan(1/2*c) + 5*tan (1/2*a)*tan(1/2*c)^2 + tan(1/2*c)^3 + tan(1/2*a)^2 + 5*tan(1/2*a)*tan(1/2* c) + tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c) - 1)*log(abs(-tan(1/2*b*x + 1/ 2*a)*tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a)*tan(1/2*a) + tan(1/2*b*x + 1/2*a)*tan(1/2*c) - tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a) - tan( 1/2*a) - tan(1/2*c) + 1))/(tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/ 2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c) - tan(1/2*a)^ 2*tan(1/2*c)^2 + tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^3 - tan(1/2*a)^2*tan (1/2*c) - tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*c)^3 - tan(1/2*a)^2 + tan(1/2* a)*tan(1/2*c) - tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c) - 1) - 3*(tan(1/2*a )^3*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c) - 5*tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/ 2*c)^3 - tan(1/2*a)^3 - 5*tan(1/2*a)^2*tan(1/2*c) - 5*tan(1/2*a)*tan(1/2*c )^2 - tan(1/2*c)^3 + tan(1/2*a)^2 + 5*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 + tan(1/2*a) + tan(1/2*c) - 1)*log(abs(-tan(1/2*b*x + 1/2*a)*tan(1/2*a)*t an(1/2*c) - tan(1/2*b*x + 1/2*a)*tan(1/2*a) - tan(1/2*b*x + 1/2*a)*tan(1/2 *c) + tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a) - tan(1/2*a) - tan(1/2* c) - 1))/(tan(1/2*a)^3*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^2 + tan(1...
Timed out. \[ \int \cos (a+b x) \sec ^4(c-b x) \, dx=\text {Hanged} \] Input:
int(cos(a + b*x)/cos(c - b*x)^4,x)
Output:
\text{Hanged}
\[ \int \cos (a+b x) \sec ^4(c-b x) \, dx =\text {Too large to display} \] Input:
int(cos(b*x+a)*sec(b*x-c)^4,x)
Output:
( - 4*cos(b*x - c)**2*sin(a + b*x) + 8*cos(b*x - c)*int(cos(a + b*x)/(sin( b*x - c)**4 - 2*sin(b*x - c)**2 + 1),x)*sin(b*x - c)**2*b - 8*cos(b*x - c) *int(cos(a + b*x)/(sin(b*x - c)**4 - 2*sin(b*x - c)**2 + 1),x)*b - cos(b*x - c)*int((cos(a + b*x)*sin(b*x - c)**4)/(sin(b*x - c)**4 - 2*sin(b*x - c) **2 + 1),x)*sin(b*x - c)**2*b + cos(b*x - c)*int((cos(a + b*x)*sin(b*x - c )**4)/(sin(b*x - c)**4 - 2*sin(b*x - c)**2 + 1),x)*b - 4*cos(b*x - c)*int( (cos(a + b*x)*sin(b*x - c)**2)/(sin(b*x - c)**4 - 2*sin(b*x - c)**2 + 1),x )*sin(b*x - c)**2*b + 4*cos(b*x - c)*int((cos(a + b*x)*sin(b*x - c)**2)/(s in(b*x - c)**4 - 2*sin(b*x - c)**2 + 1),x)*b - 2*cos(b*x - c)*log(sin(b*x - c) - 1)*sin(b*x - c)**2 + 2*cos(b*x - c)*log(sin(b*x - c) - 1) + 2*cos(b *x - c)*log(sin(b*x - c) + 1)*sin(b*x - c)**2 - 2*cos(b*x - c)*log(sin(b*x - c) + 1) + 4*cos(b*x - c)*log(tan((b*x - c)/2) - 1)*sin(b*x - c)**2 - 4* cos(b*x - c)*log(tan((b*x - c)/2) - 1) - 4*cos(b*x - c)*log(tan((b*x - c)/ 2) + 1)*sin(b*x - c)**2 + 4*cos(b*x - c)*log(tan((b*x - c)/2) + 1) + cos(b *x - c)*sin(b*x - c)**2*sin(a + b*x) + cos(b*x - c)*sin(b*x - c)**2*a - 3* cos(b*x - c)*sin(a + b*x) - cos(b*x - c)*a - 4*cos(a + b*x)*sin(b*x - c) - 4*sin(b*x - c)**2*sin(a + b*x) + 4*sin(a + b*x))/(15*cos(b*x - c)*b*(sin( b*x - c)**2 - 1))