Integrand size = 16, antiderivative size = 60 \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\frac {\cos (a+c) \csc ^3(c-b x)}{3 b}+\frac {1}{2} \left (\frac {\text {arctanh}(\cos (c-b x))}{b}+\frac {\cot (c-b x) \csc (c-b x)}{b}\right ) \sin (a+c) \] Output:
-1/3*cos(a+c)*csc(b*x-c)^3/b+1/2*(arctanh(cos(b*x-c))/b+cot(b*x-c)*csc(b*x -c)/b)*sin(a+c)
Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\frac {12 \text {arctanh}\left (\cos (c)+\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \sin (a+c)+\csc ^3(c-b x) (4 \cos (a+c)+3 \sin (a+c) \sin (2 (c-b x)))}{12 b} \] Input:
Integrate[Cos[a + b*x]*Csc[c - b*x]^4,x]
Output:
(12*ArcTanh[Cos[c] + Sin[c]*Tan[(b*x)/2]]*Sin[a + c] + Csc[c - b*x]^3*(4*C os[a + c] + 3*Sin[a + c]*Sin[2*(c - b*x)]))/(12*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.
| \(\displaystyle \int \cos (a+b x) \csc ^4(c-b x) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \cos (a+b x) \csc ^4(c-b x)dx\) |
Input:
Int[Cos[a + b*x]*Csc[c - b*x]^4,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 5.69 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.00
| method | result | size |
| risch | \(\frac {i \left (3 \,{\mathrm e}^{i \left (b x +7 a +6 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +7 a +4 c \right )}-3 \,{\mathrm e}^{i \left (b x +5 a +4 c \right )}-3 \,{\mathrm e}^{i \left (5 b x +7 a +2 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}+3 \,{\mathrm e}^{5 i \left (b x +a \right )}\right )}{6 b \left ({\mathrm e}^{2 i \left (a +c \right )}-{\mathrm e}^{2 i \left (b x +a \right )}\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \sin \left (a +c \right )}{2 b}-\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \sin \left (a +c \right )}{2 b}\) | \(180\) |
| default | \(\text {Expression too large to display}\) | \(2370\) |
Input:
int(cos(b*x+a)*csc(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*(a+c))+exp(I*(b*x+a)))/ b*sin(a+c)-1/2*ln(-exp(I*(a+c))+exp(I*(b*x+a)))/b*sin(a+c)
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (60) = 120\).
Time = 0.11 (sec) , antiderivative size = 420, normalized size of antiderivative = 7.00 \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\frac {6 \, {\left (2 \, \cos \left (a + c\right )^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + 12 \, {\left (\cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2} - 6 \, \cos \left (a + c\right )^{3} - 3 \, {\left ({\left (4 \, \cos \left (a + c\right )^{4} - 5 \, \cos \left (a + c\right )^{2} + 1\right )} \cos \left (b x + a\right )^{3} + {\left ({\left (4 \, \cos \left (a + c\right )^{3} - 3 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{3}\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 {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) + 1}{\cos \left (a + c\right ) + 1}\right ) + 3 \, {\left ({\left (4 \, \cos \left (a + c\right )^{4} - 5 \, \cos \left (a + c\right )^{2} + 1\right )} \cos \left (b x + a\right )^{3} + {\left ({\left (4 \, \cos \left (a + c\right )^{3} - 3 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{3}\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 {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) - 1}{\cos \left (a + c\right ) + 1}\right ) + 2 \, \cos \left (a + c\right )}{12 \, {\left ({\left (b \cos \left (a + c\right )^{3} - {\left (4 \, b \cos \left (a + c\right )^{3} - 3 \, b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) + {\left ({\left (4 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{3} - 3 \, b \cos \left (b x + a\right ) \cos \left (a + c\right )^{2}\right )} \sin \left (a + c\right )\right )}} \] Input:
integrate(cos(b*x+a)*csc(b*x-c)^4,x, algorithm="fricas")
Output:
1/12*(6*(2*cos(a + c)^2 - 1)*cos(b*x + a)*sin(b*x + a)*sin(a + c) + 12*(co s(a + c)^3 - cos(a + c))*cos(b*x + a)^2 - 6*cos(a + c)^3 - 3*((4*cos(a + c )^4 - 5*cos(a + c)^2 + 1)*cos(b*x + a)^3 + ((4*cos(a + c)^3 - 3*cos(a + c) )*cos(b*x + a)^2 - cos(a + c)^3)*sin(b*x + a)*sin(a + c) - 3*(cos(a + c)^4 - cos(a + c)^2)*cos(b*x + a))*log((cos(b*x + a)*cos(a + c) + sin(b*x + a) *sin(a + c) + 1)/(cos(a + c) + 1)) + 3*((4*cos(a + c)^4 - 5*cos(a + c)^2 + 1)*cos(b*x + a)^3 + ((4*cos(a + c)^3 - 3*cos(a + c))*cos(b*x + a)^2 - cos (a + c)^3)*sin(b*x + a)*sin(a + c) - 3*(cos(a + c)^4 - cos(a + c)^2)*cos(b *x + a))*log(-(cos(b*x + a)*cos(a + c) + sin(b*x + a)*sin(a + c) - 1)/(cos (a + c) + 1)) + 2*cos(a + c))/((b*cos(a + c)^3 - (4*b*cos(a + c)^3 - 3*b*c os(a + c))*cos(b*x + a)^2)*sin(b*x + a) + ((4*b*cos(a + c)^2 - b)*cos(b*x + a)^3 - 3*b*cos(b*x + a)*cos(a + c)^2)*sin(a + c))
Timed out. \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)*csc(b*x-c)**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1794 vs. \(2 (60) = 120\).
Time = 0.09 (sec) , antiderivative size = 1794, normalized size of antiderivative = 29.90 \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*csc(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*sin(a + c) + 9*cos(4*b*x + a + 2*c)^2*sin(a + c) + 9*c os(2*b*x + a + 4*c)^2*sin(a + c) - 6*cos(2*b*x + a + 4*c)*cos(a + 6*c)*sin (a + c) + cos(a + 6*c)^2*sin(a + c) + sin(6*b*x + a)^2*sin(a + c) + 9*sin( 4*b*x + a + 2*c)^2*sin(a + c) + 9*sin(2*b*x + a + 4*c)^2*sin(a + c) - 6*si n(2*b*x + a + 4*c)*sin(a + 6*c)*sin(a + c) + sin(a + 6*c)^2*sin(a + c) - 2 *(3*cos(4*b*x + a + 2*c)*sin(a + c) - 3*cos(2*b*x + a + 4*c)*sin(a + c) + cos(a + 6*c)*sin(a + c))*cos(6*b*x + a) - 6*(3*cos(2*b*x + a + 4*c)*sin(a + c) - cos(a + 6*c)*sin(a + c))*cos(4*b*x + a + 2*c) - 2*(3*sin(4*b*x + a + 2*c)*sin(a + c) - 3*sin(2*b*x + a + 4*c)*sin(a + c) + sin(a + 6*c)*sin(a + c))*sin(6*b*x + a) - 6*(3*sin(2*b*x + a + 4*c)*sin(a + c) - sin(a + 6*c )*sin(a + c))*sin(4*b*x + a + 2*c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + c os(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) - 3*(cos(6*b*x + a...
Leaf count of result is larger than twice the leaf count of optimal. 11496 vs. \(2 (60) = 120\).
Time = 1.06 (sec) , antiderivative size = 11496, normalized size of antiderivative = 191.60 \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*csc(b*x-c)^4,x, algorithm="giac")
Output:
1/24*(24*(tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^2*tan(1/2*c)^3 - 2*tan(1/ 2*a)^2*tan(1/2*c) - 2*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) + tan(1/2*c))*l og(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*c)))/(tan(1/2*a)^3*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)^2 + tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2 - 1) - 24*(tan(1/2*a)^3*tan(1/2*c) + 2*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^2 - 2*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2)*log(abs(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) + 1))/(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*a)^2*tan(1/2*c) + tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*c)^3 + tan(1/2*a) + tan(1/2*c)) + (3*tan(1/2*b*x + 1/2*a)^5*tan(1/2*a)^12*tan(1/2*c)^10 + 6*ta n(1/2*b*x + 1/2*a)^5*tan(1/2*a)^11*tan(1/2*c)^11 - 3*tan(1/2*b*x + 1/2*a)^ 4*tan(1/2*a)^12*tan(1/2*c)^11 + 3*tan(1/2*b*x + 1/2*a)^5*tan(1/2*a)^10*tan (1/2*c)^12 - 3*tan(1/2*b*x + 1/2*a)^4*tan(1/2*a)^11*tan(1/2*c)^12 + tan(1/ 2*b*x + 1/2*a)^3*tan(1/2*a)^12*tan(1/2*c)^12 + 12*tan(1/2*b*x + 1/2*a)^5*t an(1/2*a)^12*tan(1/2*c)^8 + 18*tan(1/2*b*x + 1/2*a)^5*tan(1/2*a)^11*tan(1/ 2*c)^9 - 9*tan(1/2*b*x + 1/2*a)^4*tan(1/2*a)^12*tan(1/2*c)^9 + 12*tan(1/2* b*x + 1/2*a)^5*tan(1/2*a)^10*tan(1/2*c)^10 + 6*tan(1/2*b*x + 1/2*a)^4*tan( 1/2*a)^11*tan(1/2*c)^10 - 2*tan(1/2*b*x + 1/2*a)^3*tan(1/2*a)^12*tan(1/...
Timed out. \[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\text {Hanged} \] Input:
int(cos(a + b*x)/sin(c - b*x)^4,x)
Output:
\text{Hanged}
\[ \int \cos (a+b x) \csc ^4(c-b x) \, dx=\frac {-24 \cos \left (b x -c \right ) \cos \left (b x +a \right )+20 \cos \left (b x -c \right ) \sin \left (b x -c \right )^{2}+4 \cos \left (b x -c \right ) \sin \left (b x -c \right ) \sin \left (b x +a \right )-8 \cos \left (b x -c \right )+4 \cos \left (b x +a \right ) \sin \left (b x -c \right )^{2}-8 \cos \left (b x +a \right )-12 \left (\int \frac {\tan \left (\frac {b x}{2}-\frac {c}{2}\right )^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}d x \right ) \sin \left (b x -c \right )^{3} b -6 \left (\int \frac {1}{\tan \left (\frac {b x}{2}-\frac {c}{2}\right )^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+\tan \left (\frac {b x}{2}-\frac {c}{2}\right )^{4}}d x \right ) \sin \left (b x -c \right )^{3} b +9 \sin \left (b x -c \right )^{3} \sin \left (b x +a \right )+24 \sin \left (b x -c \right )^{3} \tan \left (\frac {b x}{2}-\frac {c}{2}\right )-3 \sin \left (b x -c \right )^{3} b x +12 \sin \left (b x -c \right ) \sin \left (b x +a \right )-8}{48 \sin \left (b x -c \right )^{3} b} \] Input:
int(cos(b*x+a)*csc(b*x-c)^4,x)
Output:
( - 24*cos(b*x - c)*cos(a + b*x) + 20*cos(b*x - c)*sin(b*x - c)**2 + 4*cos (b*x - c)*sin(b*x - c)*sin(a + b*x) - 8*cos(b*x - c) + 4*cos(a + b*x)*sin( b*x - c)**2 - 8*cos(a + b*x) - 12*int((tan((b*x - c)/2)**2*tan((a + b*x)/2 )**2)/(tan((a + b*x)/2)**2 + 1),x)*sin(b*x - c)**3*b - 6*int(1/(tan((b*x - c)/2)**4*tan((a + b*x)/2)**2 + tan((b*x - c)/2)**4),x)*sin(b*x - c)**3*b + 9*sin(b*x - c)**3*sin(a + b*x) + 24*sin(b*x - c)**3*tan((b*x - c)/2) - 3 *sin(b*x - c)**3*b*x + 12*sin(b*x - c)*sin(a + b*x) - 8)/(48*sin(b*x - c)* *3*b)