Integrand size = 18, antiderivative size = 79 \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {\cos (2 (a+c)) \cot (c-b x)}{b}+\frac {\cot (c-b x) \sin ^2(a+c)}{b}+\frac {\cot ^3(c-b x) \sin ^2(a+c)}{3 b}-\frac {\csc ^2(c-b x) \sin (2 (a+c))}{2 b} \] Output:
-cos(2*a+2*c)*cot(b*x-c)/b-cot(b*x-c)*sin(a+c)^2/b-1/3*cot(b*x-c)^3*sin(a+ c)^2/b-1/2*csc(b*x-c)^2*sin(2*a+2*c)/b
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.03 \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {\csc (c) \csc ^3(c-b x) (3 \sin (b x)+\sin (2 c-3 b x)+\sin (2 a+4 c-3 b x)-3 \sin (2 a+2 c-b x)+3 \sin (2 a+b x)-\sin (2 a+3 b x))}{12 b} \] Input:
Integrate[Csc[c - b*x]^4*Sin[a + b*x]^2,x]
Output:
(Csc[c]*Csc[c - b*x]^3*(3*Sin[b*x] + Sin[2*c - 3*b*x] + Sin[2*a + 4*c - 3* b*x] - 3*Sin[2*a + 2*c - b*x] + 3*Sin[2*a + b*x] - Sin[2*a + 3*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.
\(\displaystyle \int \sin ^2(a+b x) \csc ^4(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \csc ^4(c-b x)dx\) |
Input:
Int[Csc[c - b*x]^4*Sin[a + b*x]^2,x]
Output:
$Aborted
Time = 3.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(\frac {\sec \left (\frac {b x}{2}-\frac {c}{2}\right )^{3} \csc \left (\frac {b x}{2}-\frac {c}{2}\right )^{3} \left (\cos \left (3 b x -3 c \right )-3 \cos \left (b x -c \right )+2 \cos \left (3 b x +2 a -c \right )\right )}{96 b}\) | \(62\) |
risch | \(\frac {2 i \left ({\mathrm e}^{8 i \left (a +c \right )}-3 \,{\mathrm e}^{2 i \left (b x +4 a +3 c \right )}+{\mathrm e}^{6 i \left (a +c \right )}+3 \,{\mathrm e}^{4 i \left (b x +2 a +c \right )}-3 \,{\mathrm e}^{2 i \left (b x +3 a +2 c \right )}+{\mathrm e}^{4 i \left (a +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (a +c \right )}-{\mathrm e}^{2 i \left (b x +a \right )}\right )^{3} b}\) | \(96\) |
default | \(\frac {\frac {2 \sin \left (a \right ) \cos \left (c \right )+2 \cos \left (a \right ) \sin \left (c \right )}{2 \left (-\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )+\sin \left (a \right ) \cos \left (c \right )\right )^{2}}-\frac {\left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}{3 \left (-\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )+\sin \left (a \right ) \cos \left (c \right )\right )^{3}}-\frac {1}{\left (-\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )+\sin \left (a \right ) \cos \left (c \right )\right )}}{b}\) | \(188\) |
Input:
int(csc(b*x-c)^4*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/96*sec(1/2*b*x-1/2*c)^3*csc(1/2*b*x-1/2*c)^3*(cos(3*b*x-3*c)-3*cos(b*x-c )+2*cos(3*b*x+2*a-c))/b
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (85) = 170\).
Time = 0.09 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.59 \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {{\left (16 \, \cos \left (a + c\right )^{5} - 16 \, \cos \left (a + c\right )^{3} + 3 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (a + c\right )^{4} - {\left (16 \, \cos \left (a + c\right )^{4} - 8 \, \cos \left (a + c\right )^{2} + 1\right )} \cos \left (b x + a\right )^{2} + \cos \left (a + c\right )^{2} + 1\right )} \sin \left (b x + a\right ) \sin \left (a + c\right ) - 3 \, {\left (4 \, \cos \left (a + c\right )^{5} - 3 \, \cos \left (a + c\right )^{3}\right )} \cos \left (b x + a\right )}{3 \, {\left ({\left (b \cos \left (a + c\right )^{3} - {\left (4 \, b \cos \left (a + c\right )^{3} - 3 \, b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) + {\left ({\left (4 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{3} - 3 \, b \cos \left (b x + a\right ) \cos \left (a + c\right )^{2}\right )} \sin \left (a + c\right )\right )}} \] Input:
integrate(csc(b*x-c)^4*sin(b*x+a)^2,x, algorithm="fricas")
Output:
1/3*((16*cos(a + c)^5 - 16*cos(a + c)^3 + 3*cos(a + c))*cos(b*x + a)^3 - ( 4*cos(a + c)^4 - (16*cos(a + c)^4 - 8*cos(a + c)^2 + 1)*cos(b*x + a)^2 + c os(a + c)^2 + 1)*sin(b*x + a)*sin(a + c) - 3*(4*cos(a + c)^5 - 3*cos(a + c )^3)*cos(b*x + a))/((b*cos(a + c)^3 - (4*b*cos(a + c)^3 - 3*b*cos(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. \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x-c)**4*sin(b*x+a)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (85) = 170\).
Time = 0.05 (sec) , antiderivative size = 937, normalized size of antiderivative = 11.86 \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x-c)^4*sin(b*x+a)^2,x, algorithm="maxima")
Output:
2/3*((3*sin(4*b*x + 4*a + 4*c) - 3*sin(2*b*x + 4*a + 6*c) - 3*sin(2*b*x + 2*a + 4*c) + sin(4*a + 8*c) + sin(2*a + 6*c) + sin(4*c))*cos(6*b*x + 2*a) + 3*(3*sin(4*b*x + 2*a + 2*c) - 3*sin(2*b*x + 2*a + 4*c) + sin(2*a + 6*c)) *cos(4*b*x + 4*a + 4*c) + 3*(3*sin(2*b*x + 4*a + 6*c) + 3*sin(2*b*x + 2*a + 4*c) - sin(4*a + 8*c) - sin(2*a + 6*c) - sin(4*c))*cos(4*b*x + 2*a + 2*c ) + 3*(3*sin(2*b*x + 2*a + 4*c) - sin(2*a + 6*c))*cos(2*b*x + 4*a + 6*c) + 3*(sin(4*a + 8*c) + sin(4*c))*cos(2*b*x + 2*a + 4*c) - (3*cos(4*b*x + 4*a + 4*c) - 3*cos(2*b*x + 4*a + 6*c) - 3*cos(2*b*x + 2*a + 4*c) + cos(4*a + 8*c) + cos(2*a + 6*c) + cos(4*c))*sin(6*b*x + 2*a) - 3*(3*cos(4*b*x + 2*a + 2*c) - 3*cos(2*b*x + 2*a + 4*c) + cos(2*a + 6*c))*sin(4*b*x + 4*a + 4*c) - 3*(3*cos(2*b*x + 4*a + 6*c) + 3*cos(2*b*x + 2*a + 4*c) - cos(4*a + 8*c) - cos(2*a + 6*c) - cos(4*c))*sin(4*b*x + 2*a + 2*c) - 3*(3*cos(2*b*x + 2* a + 4*c) - cos(2*a + 6*c))*sin(2*b*x + 4*a + 6*c) - 3*(cos(4*a + 8*c) + co s(4*c))*sin(2*b*x + 2*a + 4*c) - cos(2*a + 6*c)*sin(4*a + 8*c) + cos(4*a + 8*c)*sin(2*a + 6*c) + cos(4*c)*sin(2*a + 6*c) - cos(2*a + 6*c)*sin(4*c))/ (b*cos(6*b*x + 2*a)^2 + 9*b*cos(4*b*x + 2*a + 2*c)^2 + 9*b*cos(2*b*x + 2*a + 4*c)^2 - 6*b*cos(2*b*x + 2*a + 4*c)*cos(2*a + 6*c) + b*cos(2*a + 6*c)^2 + b*sin(6*b*x + 2*a)^2 + 9*b*sin(4*b*x + 2*a + 2*c)^2 + 9*b*sin(2*b*x + 2 *a + 4*c)^2 - 6*b*sin(2*b*x + 2*a + 4*c)*sin(2*a + 6*c) + b*sin(2*a + 6*c) ^2 - 2*(3*b*cos(4*b*x + 2*a + 2*c) - 3*b*cos(2*b*x + 2*a + 4*c) + b*cos...
Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (85) = 170\).
Time = 0.15 (sec) , antiderivative size = 705, normalized size of antiderivative = 8.92 \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x-c)^4*sin(b*x+a)^2,x, algorithm="giac")
Output:
-1/3*(3*tan(b*x - c)^2*tan(1/2*a)^4*tan(1/2*c)^4 - 6*tan(b*x - c)^2*tan(1/ 2*a)^4*tan(1/2*c)^2 - 24*tan(b*x - c)^2*tan(1/2*a)^3*tan(1/2*c)^3 - 6*tan( b*x - c)*tan(1/2*a)^4*tan(1/2*c)^3 - 6*tan(b*x - c)^2*tan(1/2*a)^2*tan(1/2 *c)^4 - 6*tan(b*x - c)*tan(1/2*a)^3*tan(1/2*c)^4 + 3*tan(b*x - c)^2*tan(1/ 2*a)^4 + 24*tan(b*x - c)^2*tan(1/2*a)^3*tan(1/2*c) + 6*tan(b*x - c)*tan(1/ 2*a)^4*tan(1/2*c) + 60*tan(b*x - c)^2*tan(1/2*a)^2*tan(1/2*c)^2 + 36*tan(b *x - c)*tan(1/2*a)^3*tan(1/2*c)^2 + 4*tan(1/2*a)^4*tan(1/2*c)^2 + 24*tan(b *x - c)^2*tan(1/2*a)*tan(1/2*c)^3 + 36*tan(b*x - c)*tan(1/2*a)^2*tan(1/2*c )^3 + 8*tan(1/2*a)^3*tan(1/2*c)^3 + 3*tan(b*x - c)^2*tan(1/2*c)^4 + 6*tan( b*x - c)*tan(1/2*a)*tan(1/2*c)^4 + 4*tan(1/2*a)^2*tan(1/2*c)^4 - 6*tan(b*x - c)^2*tan(1/2*a)^2 - 6*tan(b*x - c)*tan(1/2*a)^3 - 24*tan(b*x - c)^2*tan (1/2*a)*tan(1/2*c) - 36*tan(b*x - c)*tan(1/2*a)^2*tan(1/2*c) - 8*tan(1/2*a )^3*tan(1/2*c) - 6*tan(b*x - c)^2*tan(1/2*c)^2 - 36*tan(b*x - c)*tan(1/2*a )*tan(1/2*c)^2 - 16*tan(1/2*a)^2*tan(1/2*c)^2 - 6*tan(b*x - c)*tan(1/2*c)^ 3 - 8*tan(1/2*a)*tan(1/2*c)^3 + 3*tan(b*x - c)^2 + 6*tan(b*x - c)*tan(1/2* a) + 4*tan(1/2*a)^2 + 6*tan(b*x - c)*tan(1/2*c) + 8*tan(1/2*a)*tan(1/2*c) + 4*tan(1/2*c)^2)/((tan(1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^ 2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c) ^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*tan(1/2*c)^2 + 1)*b*tan(b*x - c)^3)
Timed out. \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Hanged} \] Input:
int(sin(a + b*x)^2/sin(c - b*x)^4,x)
Output:
\text{Hanged}
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \csc ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {-\cos \left (b x -c \right ) \sin \left (b x -c \right )^{2}-\cos \left (b x -c \right ) \sin \left (b x +a \right )^{2}-\cos \left (b x +a \right ) \sin \left (b x -c \right ) \sin \left (b x +a \right )}{3 \sin \left (b x -c \right )^{3} b} \] Input:
int(csc(b*x-c)^4*sin(b*x+a)^2,x)
Output:
( - (cos(b*x - c)*sin(b*x - c)**2 + cos(b*x - c)*sin(a + b*x)**2 + cos(a + b*x)*sin(b*x - c)*sin(a + b*x)))/(3*sin(b*x - c)**3*b)