Integrand size = 15, 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 = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \cos (a+b x) \csc ^4(c+b x) \, dx=\frac {12 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \sin (a-c)+\csc ^3(c+b x) (-4 \cos (a-c)+3 \sin (a-c) \sin (2 (c+b x)))}{12 b} \] 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* Cos[a - c] + 3*Sin[a - c]*Sin[2*(c + b*x)]))/(12*b)
Time = 0.37 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5092, 3042, 25, 3086, 15, 4255, 3042, 4257}
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.
\(\displaystyle \int \cos (a+b x) \csc ^4(b x+c) \, dx\) |
\(\Big \downarrow \) 5092 |
\(\displaystyle \cos (a-c) \int \cot (c+b x) \csc ^3(c+b x)dx-\sin (a-c) \int \csc ^3(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos (a-c) \int -\sec \left (c+b x-\frac {\pi }{2}\right )^3 \tan \left (c+b x-\frac {\pi }{2}\right )dx-\sin (a-c) \int \csc (c+b x)^3dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\sin (a-c) \int \csc (c+b x)^3dx-\cos (a-c) \int \sec \left (\frac {1}{2} (2 c-\pi )+b x\right )^3 \tan \left (\frac {1}{2} (2 c-\pi )+b x\right )dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\cos (a-c) \int \csc ^2(c+b x)d\csc (c+b x)}{b}-\sin (a-c) \int \csc (c+b x)^3dx\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\sin (a-c) \int \csc (c+b x)^3dx-\frac {\cos (a-c) \csc ^3(b x+c)}{3 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -\sin (a-c) \left (\frac {1}{2} \int \csc (c+b x)dx-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc ^3(b x+c)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\sin (a-c) \left (\frac {1}{2} \int \csc (c+b x)dx-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc ^3(b x+c)}{3 b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\sin (a-c) \left (-\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cos (a-c) \csc ^3(b x+c)}{3 b}\) |
Input:
Int[Cos[a + b*x]*Csc[c + b*x]^4,x]
Output:
-1/3*(Cos[a - c]*Csc[c + b*x]^3)/b - (-1/2*ArcTanh[Cos[c + b*x]]/b - (Cot[ c + b*x]*Csc[c + b*x])/(2*b))*Sin[a - c]
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cos[v_]*Csc[w_]^(n_.), x_Symbol] :> Simp[Cos[v - w] Int[Cot[w]*Csc[w] ^(n - 1), x], x] - Simp[Sin[v - w] Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0 ] && FreeQ[v - w, x] && NeQ[w, v]
Result contains complex when optimal does not.
Time = 5.49 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.17
method | result | size |
risch | \(\frac {i \left (-3 \,{\mathrm e}^{i \left (5 b x +7 a +4 c \right )}+3 \,{\mathrm e}^{i \left (5 b x +5 a +6 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +7 a +2 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +5 a +4 c \right )}+3 \,{\mathrm e}^{i \left (b x +7 a \right )}-3 \,{\mathrm e}^{i \left (b x +5 a +2 c \right )}\right )}{6 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}\) | \(190\) |
default | \(\text {Expression too large to display}\) | \(2375\) |
Input:
int(cos(b*x+a)*csc(b*x+c)^4,x,method=_RETURNVERBOSE)
Output:
1/6*I/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^3*(-3*exp(I*(5*b*x+7*a+4*c))+3*ex p(I*(5*b*x+5*a+6*c))-8*exp(I*(3*b*x+7*a+2*c))-8*exp(I*(3*b*x+5*a+4*c))+3*e xp(I*(b*x+7*a))-3*exp(I*(b*x+5*a+2*c)))-1/2*ln(exp(I*(b*x+a))-exp(I*(a-c)) )/b*sin(a-c)+1/2*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*sin(a-c)
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (56) = 112\).
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.08 \[ \int \cos (a+b x) \csc ^4(c+b x) \, dx=-\frac {3 \, {\left (\cos \left (b x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 3 \, {\left (\cos \left (b x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 6 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 4 \, \cos \left (-a + c\right )}{12 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )} \sin \left (b x + c\right )} \] Input:
integrate(cos(b*x+a)*csc(b*x+c)^4,x, algorithm="fricas")
Output:
-1/12*(3*(cos(b*x + c)^2 - 1)*log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin (-a + c) - 3*(cos(b*x + c)^2 - 1)*log(-1/2*cos(b*x + c) + 1/2)*sin(b*x + c )*sin(-a + c) - 6*cos(b*x + c)*sin(b*x + c)*sin(-a + c) - 4*cos(-a + c))/( (b*cos(b*x + c)^2 - b)*sin(b*x + 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. 1777 vs. \(2 (56) = 112\).
Time = 0.10 (sec) , antiderivative size = 1777, normalized size of antiderivative = 29.62 \[ \int \cos (a+b x) \csc ^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*csc(b*x+c)^4,x, algorithm="maxima")
Output:
-1/12*(2*(3*sin(5*b*x + 2*a + 4*c) - 3*sin(5*b*x + 6*c) + 8*sin(3*b*x + 2* a + 2*c) + 8*sin(3*b*x + 4*c) - 3*sin(b*x + 2*a) + 3*sin(b*x + 2*c))*cos(6 *b*x + a + 6*c) + 6*(3*sin(4*b*x + a + 4*c) - 3*sin(2*b*x + a + 2*c) + sin (a))*cos(5*b*x + 2*a + 4*c) - 6*(3*sin(4*b*x + a + 4*c) - 3*sin(2*b*x + a + 2*c) + sin(a))*cos(5*b*x + 6*c) - 6*(8*sin(3*b*x + 2*a + 2*c) + 8*sin(3* b*x + 4*c) - 3*sin(b*x + 2*a) + 3*sin(b*x + 2*c))*cos(4*b*x + a + 4*c) - 1 6*(3*sin(2*b*x + a + 2*c) - sin(a))*cos(3*b*x + 2*a + 2*c) - 16*(3*sin(2*b *x + a + 2*c) - sin(a))*cos(3*b*x + 4*c) - 18*(sin(b*x + 2*a) - sin(b*x + 2*c))*cos(2*b*x + a + 2*c) + 3*(cos(6*b*x + a + 6*c)^2*sin(-a + c) + 9*cos (4*b*x + a + 4*c)^2*sin(-a + c) + 9*cos(2*b*x + a + 2*c)^2*sin(-a + c) - 6 *cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(6*b*x + a + 6*c)^2*sin(-a + c) + 9*sin(4*b*x + a + 4*c)^2*sin(-a + c) + 9*sin(2*b*x + a + 2*c)^2*sin( -a + c) - 6*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) - 2*(3*cos(4*b*x + a + 4*c)*sin(-a + c) - 3*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c ))*cos(6*b*x + a + 6*c) - 6*(3*cos(2*b*x + a + 2*c)*sin(-a + c) - cos(a)*s in(-a + c))*cos(4*b*x + a + 4*c) - 2*(3*sin(4*b*x + a + 4*c)*sin(-a + c) - 3*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin(6*b*x + a + 6*c) - 6*(3*sin(2*b*x + a + 2*c)*sin(-a + c) - sin(a)*sin(-a + c))*sin(4*b *x + a + 4*c) + (cos(a)^2 + sin(a)^2)*sin(-a + c))*log(cos(b*x)^2 + 2*cos( b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + sin(c)^2) - 3...
Leaf count of result is larger than twice the leaf count of optimal. 11407 vs. \(2 (56) = 112\).
Time = 1.12 (sec) , antiderivative size = 11407, normalized size of antiderivative = 190.12 \[ \int \cos (a+b x) \csc ^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*csc(b*x+c)^4,x, algorithm="giac")
Output:
-1/24*(24*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c) - 2*t an(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*b*x + 1/2*a)*tan(1 /2*a) - 2*tan(1/2*a)^2 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c) + 4*tan(1/2*a)* tan(1/2*c) - 2*tan(1/2*c)^2)/abs(2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1 /2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*a)^2*ta n(1/2*c)^2 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a) - 2*tan(1/2*b*x + 1/2*a)*ta n(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) + 2))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan( 1/2*a)^2 + tan(1/2*c)^2 + 1) - (3*tan(1/2*b*x + 1/2*a)^5*tan(1/2*a)^12*tan (1/2*c)^10 - 6*tan(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*tan(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(1/2*c)^10 - 18*tan(1/2*b*x + 1/2*a)^5*tan(1/2*a)^9*tan(1/2*c)^ 11 - 6*tan(1/2*b*x + 1/2*a)^4*tan(1/2*a)^10*tan(1/2*c)^11 + 16*tan(1/2*b*x + 1/2*a)^3*tan(1/2*a)^11*tan(1/2*c)^11 - 3*tan(1/2*b*x + 1/2*a)^2*tan(...
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)