Integrand size = 15, antiderivative size = 38 \[ \int \cos (a+b x) \csc ^3(c+b x) \, dx=-\frac {\cos (a-c) \csc ^2(c+b x)}{2 b}+\frac {\cot (c+b x) \sin (a-c)}{b} \]
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \cos (a+b x) \csc ^3(c+b x) \, dx=-\frac {\csc (c) \csc ^2(c+b x) (\sin (a)-\cos (c+2 b x) \sin (a-c))}{2 b} \]
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5092, 3042, 25, 3086, 15, 4254, 24}
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.
\(\displaystyle \int \cos (a+b x) \csc ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 5092 |
\(\displaystyle \cos (a-c) \int \cot (c+b x) \csc ^2(c+b x)dx-\sin (a-c) \int \csc ^2(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos (a-c) \int -\sec \left (c+b x-\frac {\pi }{2}\right )^2 \tan \left (c+b x-\frac {\pi }{2}\right )dx-\sin (a-c) \int \csc (c+b x)^2dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\sin (a-c) \int \csc (c+b x)^2dx-\cos (a-c) \int \sec \left (\frac {1}{2} (2 c-\pi )+b x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+b x\right )dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\cos (a-c) \int \csc (c+b x)d\csc (c+b x)}{b}-\sin (a-c) \int \csc (c+b x)^2dx\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\sin (a-c) \int \csc (c+b x)^2dx-\frac {\cos (a-c) \csc ^2(b x+c)}{2 b}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {\sin (a-c) \int 1d\cot (c+b x)}{b}-\frac {\cos (a-c) \csc ^2(b x+c)}{2 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sin (a-c) \cot (b x+c)}{b}-\frac {\cos (a-c) \csc ^2(b x+c)}{2 b}\) |
3.3.29.3.1 Defintions of rubi rules used
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_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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]
Time = 1.58 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(-\frac {\sec \left (\frac {x b}{2}+\frac {c}{2}\right )^{2} \csc \left (\frac {x b}{2}+\frac {c}{2}\right )^{2} \cos \left (2 x b +a +c \right )}{8 b}\) | \(36\) |
default | \(-\frac {1}{2 b \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}\) | \(55\) |
risch | \(-\frac {-2 \,{\mathrm e}^{i \left (2 x b +5 a +c \right )}+{\mathrm e}^{i \left (5 a -c \right )}-{\mathrm e}^{i \left (3 a +c \right )}}{\left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} b}\) | \(64\) |
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18 \[ \int \cos (a+b x) \csc ^3(c+b x) \, dx=\frac {2 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + \cos \left (-a + c\right )}{2 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )}} \]
Timed out. \[ \int \cos (a+b x) \csc ^3(c+b x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (36) = 72\).
Time = 0.23 (sec) , antiderivative size = 395, normalized size of antiderivative = 10.39 \[ \int \cos (a+b x) \csc ^3(c+b x) \, dx=\frac {{\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) - {\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) - {\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right ) + 2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} - 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) - b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) - b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \]
((2*cos(2*b*x + 2*a + 2*c) - cos(2*a) + cos(2*c))*cos(4*b*x + a + 5*c) - 2 *(2*cos(2*b*x + 2*a + 2*c) - cos(2*a) + cos(2*c))*cos(2*b*x + a + 3*c) - ( cos(2*a) - cos(2*c))*cos(a + c) + 2*cos(2*b*x + 2*a + 2*c)*cos(a + c) + (2 *sin(2*b*x + 2*a + 2*c) - sin(2*a) + sin(2*c))*sin(4*b*x + a + 5*c) - 2*(2 *sin(2*b*x + 2*a + 2*c) - sin(2*a) + sin(2*c))*sin(2*b*x + a + 3*c) - (sin (2*a) - sin(2*c))*sin(a + c) + 2*sin(2*b*x + 2*a + 2*c)*sin(a + c))/(b*cos (4*b*x + a + 5*c)^2 + 4*b*cos(2*b*x + a + 3*c)^2 - 4*b*cos(2*b*x + a + 3*c )*cos(a + c) + b*cos(a + c)^2 + b*sin(4*b*x + a + 5*c)^2 + 4*b*sin(2*b*x + a + 3*c)^2 - 4*b*sin(2*b*x + a + 3*c)*sin(a + c) + b*sin(a + c)^2 - 2*(2* b*cos(2*b*x + a + 3*c) - b*cos(a + c))*cos(4*b*x + a + 5*c) - 2*(2*b*sin(2 *b*x + a + 3*c) - b*sin(a + c))*sin(4*b*x + a + 5*c))
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (36) = 72\).
Time = 0.33 (sec) , antiderivative size = 327, normalized size of antiderivative = 8.61 \[ \int \cos (a+b x) \csc ^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}{2 \, {\left (\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 )^{2} + 4 \, \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 ) - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right ) + 2 \, \tan \left (\frac {1}{2} \, c\right )\right )}^{2} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 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 )} b} \]
-1/2*(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)/((tan(b*x + a)*tan(1/2*a)^2*tan(1/2*c)^2 - tan(b*x + a)*tan(1/2*a)^2 + 4*tan(b*x + a)*tan(1/2*a)*tan(1/2*c) - 2*tan(1/2*a)^2*tan(1/2*c) - tan(b *x + a)*tan(1/2*c)^2 + 2*tan(1/2*a)*tan(1/2*c)^2 + tan(b*x + a) - 2*tan(1/ 2*a) + 2*tan(1/2*c))^2*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1 /2*a)*tan(1/2*c) - tan(1/2*c)^2 + 1)*b)
Timed out. \[ \int \cos (a+b x) \csc ^3(c+b x) \, dx=\text {Hanged} \]