Integrand size = 17, antiderivative size = 54 \[ \int \csc ^4(c+b x) \sin ^2(a+b x) \, dx=-\frac {\cot (c+b x)}{2 b}-\frac {\cot ^3(c+b x)}{6 b}+\frac {\cos (2 a+c+3 b x) \csc ^3(c+b x)}{6 b} \] Output:
-1/2*cot(b*x+c)/b-1/6*cot(b*x+c)^3/b+1/6*cos(3*b*x+2*a+c)*csc(b*x+c)^3/b
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.48 \[ \int \csc ^4(c+b x) \sin ^2(a+b x) \, dx=-\frac {\csc (c) \csc ^3(c+b x) (-3 \sin (b x)-\sin (2 a-4 c-3 b x)+3 \sin (2 a-2 c-b x)-3 \sin (2 a+b x)+\sin (2 a+3 b x)+\sin (2 c+3 b x))}{12 b} \] Input:
Integrate[Csc[c + b*x]^4*Sin[a + b*x]^2,x]
Output:
-1/12*(Csc[c]*Csc[c + b*x]^3*(-3*Sin[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] + Sin[2*c + 3*b*x] ))/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.
\(\displaystyle \int \sin ^2(a+b x) \csc ^4(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \csc ^4(b x+c)dx\) |
Input:
Int[Csc[c + b*x]^4*Sin[a + b*x]^2,x]
Output:
$Aborted
Time = 3.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {\sec \left (\frac {b x}{2}+\frac {c}{2}\right )^{3} \csc \left (\frac {b x}{2}+\frac {c}{2}\right )^{3} \left (-3 \cos \left (b x +c \right )+2 \cos \left (3 b x +2 a +c \right )+\cos \left (3 b x +3 c \right )\right )}{96 b}\) | \(58\) |
risch | \(\frac {2 i \left (3 \,{\mathrm e}^{2 i \left (2 b x +4 a +c \right )}-3 \,{\mathrm e}^{2 i \left (b x +4 a \right )}-3 \,{\mathrm e}^{2 i \left (b x +3 a +c \right )}+{\mathrm e}^{2 i \left (4 a -c \right )}+{\mathrm e}^{6 i a}+{\mathrm e}^{2 i \left (2 a +c \right )}\right )}{3 \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3} b}\) | \(95\) |
default | \(\frac {-\frac {-2 \cos \left (a \right ) \sin \left (c \right )+2 \sin \left (a \right ) \cos \left (c \right )}{2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {\left (-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}{3 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{3}}-\frac {1}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}}{b}\) | \(186\) |
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*(-3*cos(b*x+c)+2*cos(3*b*x+ 2*a+c)+cos(3*b*x+3*c))/b
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \csc ^4(c+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (4 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{3} - 3 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )^{2} + 3 \, \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right )}{3 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )} \sin \left (b x + c\right )} \] Input:
integrate(csc(b*x+c)^4*sin(b*x+a)^2,x, algorithm="fricas")
Output:
-1/3*((4*cos(-a + c)^2 - 1)*cos(b*x + c)^3 - 3*cos(b*x + c)*cos(-a + c)^2 + 3*cos(-a + c)*sin(b*x + c)*sin(-a + c))/((b*cos(b*x + c)^2 - b)*sin(b*x + 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. 900 vs. \(2 (48) = 96\).
Time = 0.05 (sec) , antiderivative size = 900, normalized size of antiderivative = 16.67 \[ \int \csc ^4(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] 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 + 2*c) - 3*sin(2*b*x + 2*a + 4*c) + sin(4*a) + sin(2*a + 2*c) + sin(4*c))*cos(6*b*x + 2*a + 8*c) - 3*(3*sin(2*b*x + 2*a + 4*c) - sin(2*a + 2*c))*cos(4*b*x + 4*a + 4*c) - 3 *(3*sin(4*b*x + 4*a + 4*c) - 3*sin(2*b*x + 4*a + 2*c) - 3*sin(2*b*x + 2*a + 4*c) + sin(4*a) + sin(2*a + 2*c) + sin(4*c))*cos(4*b*x + 2*a + 6*c) - 3* (3*sin(2*b*x + 4*a + 2*c) - sin(4*a) - sin(4*c))*cos(2*b*x + 2*a + 4*c) - (sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (3*cos(4*b*x + 4*a + 4*c) - 3*cos(2 *b*x + 4*a + 2*c) - 3*cos(2*b*x + 2*a + 4*c) + cos(4*a) + cos(2*a + 2*c) + cos(4*c))*sin(6*b*x + 2*a + 8*c) + 3*(3*cos(2*b*x + 2*a + 4*c) - cos(2*a + 2*c))*sin(4*b*x + 4*a + 4*c) + 3*(3*cos(4*b*x + 4*a + 4*c) - 3*cos(2*b*x + 4*a + 2*c) - 3*cos(2*b*x + 2*a + 4*c) + cos(4*a) + cos(2*a + 2*c) + cos (4*c))*sin(4*b*x + 2*a + 6*c) + 3*cos(2*a + 2*c)*sin(2*b*x + 4*a + 2*c) + 3*(3*cos(2*b*x + 4*a + 2*c) - cos(4*a) - cos(4*c))*sin(2*b*x + 2*a + 4*c) + (cos(4*a) + cos(4*c))*sin(2*a + 2*c) - 3*cos(2*b*x + 4*a + 2*c)*sin(2*a + 2*c))/(b*cos(6*b*x + 2*a + 8*c)^2 + 9*b*cos(4*b*x + 2*a + 6*c)^2 + 9*b*c os(2*b*x + 2*a + 4*c)^2 - 6*b*cos(2*b*x + 2*a + 4*c)*cos(2*a + 2*c) + b*co s(2*a + 2*c)^2 + b*sin(6*b*x + 2*a + 8*c)^2 + 9*b*sin(4*b*x + 2*a + 6*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 + 2*c ) + b*sin(2*a + 2*c)^2 - 2*(3*b*cos(4*b*x + 2*a + 6*c) - 3*b*cos(2*b*x + 2 *a + 4*c) + b*cos(2*a + 2*c))*cos(6*b*x + 2*a + 8*c) - 6*(3*b*cos(2*b*x...
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (48) = 96\).
Time = 0.16 (sec) , antiderivative size = 653, normalized size of antiderivative = 12.09 \[ \int \csc ^4(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] 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 = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \csc ^4(c+b x) \sin ^2(a+b x) \, dx=\frac {-\cos \left (b x +c \right ) \sin \left (b x +c \right )^{2}-\cos \left (b x +c \right ) \sin \left (b x +a \right )^{2}-\cos \left (b x +a \right ) \sin \left (b x +c \right ) \sin \left (b x +a \right )}{3 \sin \left (b x +c \right )^{3} b} \] 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)