Integrand size = 16, antiderivative size = 37 \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=-\frac {\cos (a+c) \cot (c-b x)}{b}+\frac {\csc ^2(c-b x) \sin (a+c)}{2 b} \] Output:
cos(a+c)*cot(b*x-c)/b+1/2*csc(b*x-c)^2*sin(a+c)/b
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=\frac {(\cos (a)-\cos (a+c) \cos (c-2 b x)) \csc (c) \csc ^2(c-b x)}{2 b} \] Input:
Integrate[Csc[c - b*x]^3*Sin[a + b*x],x]
Output:
((Cos[a] - Cos[a + c]*Cos[c - 2*b*x])*Csc[c]*Csc[c - b*x]^2)/(2*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.
\(\displaystyle \int \sin (a+b x) \csc ^3(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin (a+b x) \csc ^3(c-b x)dx\) |
Input:
Int[Csc[c - b*x]^3*Sin[a + b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62
method | result | size |
risch | \(-\frac {i \left ({\mathrm e}^{5 i \left (a +c \right )}-2 \,{\mathrm e}^{i \left (2 b x +5 a +3 c \right )}+{\mathrm e}^{3 i \left (a +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (a +c \right )}-{\mathrm e}^{2 i \left (b x +a \right )}\right )^{2} b}\) | \(60\) |
parallelrisch | \(\frac {\csc \left (\frac {b x}{2}-\frac {c}{2}\right ) \left (\sin \left (b x +a \right ) \left (-\frac {\sec \left (\frac {b x}{2}-\frac {c}{2}\right )^{2}}{2}+1\right ) \csc \left (\frac {b x}{2}-\frac {c}{2}\right )+\sec \left (\frac {b x}{2}-\frac {c}{2}\right ) \cos \left (b x +a \right )\right )}{4 b}\) | \(63\) |
default | \(-\frac {\frac {1}{\left (-\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )+\sin \left (a \right ) \cos \left (c \right )\right )}+\frac {-\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )}{2 \left (-\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )+\sin \left (a \right ) \cos \left (c \right )\right )^{2}}}{b}\) | \(123\) |
Input:
int(-csc(b*x-c)^3*sin(b*x+a),x,method=_RETURNVERBOSE)
Output:
-I/(exp(2*I*(a+c))-exp(2*I*(b*x+a)))^2/b*(exp(5*I*(a+c))-2*exp(I*(2*b*x+5* a+3*c))+exp(3*I*(a+c)))
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (36) = 72\).
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.27 \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=-\frac {2 \, {\left (2 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - {\left (4 \, \cos \left (b x + a\right )^{2} \cos \left (a + c\right )^{2} - 2 \, \cos \left (a + c\right )^{2} - 1\right )} \sin \left (a + c\right )}{2 \, {\left (2 \, b \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + {\left (2 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{2} - b \cos \left (a + c\right )^{2}\right )}} \] Input:
integrate(-csc(b*x-c)^3*sin(b*x+a),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)
Timed out. \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=\text {Timed out} \] Input:
integrate(-csc(b*x-c)**3*sin(b*x+a),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (36) = 72\).
Time = 0.04 (sec) , antiderivative size = 433, normalized size of antiderivative = 11.70 \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=-\frac {{\left (2 \, \sin \left (2 \, b x + 2 \, a + 3 \, c\right ) - \sin \left (2 \, a + 5 \, c\right ) - \sin \left (3 \, c\right )\right )} \cos \left (4 \, b x + a\right ) + 2 \, {\left (2 \, \sin \left (2 \, b x + a + 2 \, c\right ) - \sin \left (a + 4 \, c\right )\right )} \cos \left (2 \, b x + 2 \, a + 3 \, c\right ) + 2 \, {\left (\sin \left (2 \, a + 5 \, c\right ) + \sin \left (3 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) - {\left (2 \, \cos \left (2 \, b x + 2 \, a + 3 \, c\right ) - \cos \left (2 \, a + 5 \, c\right ) - \cos \left (3 \, c\right )\right )} \sin \left (4 \, b x + a\right ) - 2 \, {\left (2 \, \cos \left (2 \, b x + a + 2 \, c\right ) - \cos \left (a + 4 \, c\right )\right )} \sin \left (2 \, b x + 2 \, a + 3 \, c\right ) - 2 \, {\left (\cos \left (2 \, a + 5 \, c\right ) + \cos \left (3 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) - \cos \left (a + 4 \, c\right ) \sin \left (2 \, a + 5 \, c\right ) + \cos \left (2 \, a + 5 \, c\right ) \sin \left (a + 4 \, c\right ) + \cos \left (3 \, c\right ) \sin \left (a + 4 \, c\right ) - \cos \left (a + 4 \, c\right ) \sin \left (3 \, c\right )}{b \cos \left (4 \, b x + a\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 2 \, c\right )^{2} - 4 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a + 4 \, c\right ) + b \cos \left (a + 4 \, c\right )^{2} + b \sin \left (4 \, b x + a\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 4 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a + 4 \, c\right ) + b \sin \left (a + 4 \, c\right )^{2} - 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) - b \cos \left (a + 4 \, c\right )\right )} \cos \left (4 \, b x + a\right ) - 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) - b \sin \left (a + 4 \, c\right )\right )} \sin \left (4 \, b x + a\right )} \] Input:
integrate(-csc(b*x-c)^3*sin(b*x+a),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. 155 vs. \(2 (36) = 72\).
Time = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 4.19 \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=\frac {\tan \left (b x - c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x - c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 4 \, \tan \left (b x - c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x - c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x - c\right ) + \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b \tan \left (b x - c\right )^{2}} \] Input:
integrate(-csc(b*x-c)^3*sin(b*x+a),x, algorithm="giac")
Output:
(tan(b*x - c)*tan(1/2*a)^2*tan(1/2*c)^2 - tan(b*x - c)*tan(1/2*a)^2 - 4*ta n(b*x - c)*tan(1/2*a)*tan(1/2*c) - tan(1/2*a)^2*tan(1/2*c) - tan(b*x - c)* tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^2 + tan(b*x - c) + tan(1/2*a) + tan(1 /2*c))/((tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*b*ta n(b*x - c)^2)
Timed out. \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=\text {Hanged} \] Input:
int(sin(a + b*x)/sin(c - b*x)^3,x)
Output:
\text{Hanged}
Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.24 \[ \int \csc ^3(c-b x) \sin (a+b x) \, dx=\frac {\cos \left (b x -c \right ) \sin \left (b x +a \right )+\cos \left (b x +a \right ) \sin \left (b x -c \right )}{2 \sin \left (b x -c \right )^{2} b} \] Input:
int(-csc(b*x-c)^3*sin(b*x+a),x)
Output:
(cos(b*x - c)*sin(a + b*x) + cos(a + b*x)*sin(b*x - c))/(2*sin(b*x - c)**2 *b)