Integrand size = 18, antiderivative size = 80 \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {\sec ^2(c-b x) \sin (2 (a+c))}{2 b}-\frac {\cos ^2(a+c) \tan (c-b x)}{b}+\frac {\cos (2 (a+c)) \tan (c-b x)}{b}-\frac {\cos ^2(a+c) \tan ^3(c-b x)}{3 b} \] Output:
1/2*sec(b*x-c)^2*sin(2*a+2*c)/b+cos(a+c)^2*tan(b*x-c)/b-cos(2*a+2*c)*tan(b *x-c)/b+1/3*cos(a+c)^2*tan(b*x-c)^3/b
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {\sec (c) \sec ^3(c-b x) (3 \sin (b x)-\sin (2 c-3 b x)+\sin (2 a+4 c-3 b x)+3 \sin (2 a+2 c-b x)+3 \sin (2 a+b x)-\sin (2 a+3 b x))}{12 b} \] Input:
Integrate[Sec[c - b*x]^4*Sin[a + b*x]^2,x]
Output:
(Sec[c]*Sec[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.
\(\displaystyle \int \sin ^2(a+b x) \sec ^4(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \sec ^4(c-b x)dx\) |
Input:
Int[Sec[c - b*x]^4*Sin[a + b*x]^2,x]
Output:
$Aborted
Time = 5.48 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(\frac {3 \sin \left (b x -c \right )+\sin \left (3 b x -3 c \right )-2 \sin \left (3 b x +2 a -c \right )}{3 b \left (\cos \left (3 b x -3 c \right )+3 \cos \left (b x -c \right )\right )}\) | \(62\) |
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 | \(\frac {-\frac {1}{\left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )}-\frac {-2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{2 \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {\left (\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )^{2}}{3 \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )^{3}}}{b}\) | \(186\) |
Input:
int(sec(b*x-c)^4*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/3*(3*sin(b*x-c)+sin(3*b*x-3*c)-2*sin(3*b*x+2*a-c))/b/(cos(3*b*x-3*c)+3*c os(b*x-c))
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (84) = 168\).
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.78 \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {{\left (4 \, \cos \left (a + c\right )^{5} - {\left (16 \, \cos \left (a + c\right )^{5} - 24 \, \cos \left (a + c\right )^{3} + 9 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2} - 9 \, \cos \left (a + c\right )^{3} + 6 \, \cos \left (a + c\right )\right )} \sin \left (b x + a\right ) + {\left ({\left (16 \, \cos \left (a + c\right )^{4} - 16 \, \cos \left (a + c\right )^{2} + 3\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (4 \, \cos \left (a + c\right )^{4} - 5 \, \cos \left (a + c\right )^{2} + 1\right )} \cos \left (b x + a\right )\right )} \sin \left (a + c\right )}{3 \, {\left ({\left (4 \, b \cos \left (a + c\right )^{3} - 3 \, b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{3} + {\left ({\left (4 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{2} - b \cos \left (a + c\right )^{2} + b\right )} \sin \left (b x + a\right ) \sin \left (a + c\right ) - 3 \, {\left (b \cos \left (a + c\right )^{3} - b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )\right )}} \] Input:
integrate(sec(b*x-c)^4*sin(b*x+a)^2,x, algorithm="fricas")
Output:
1/3*((4*cos(a + c)^5 - (16*cos(a + c)^5 - 24*cos(a + c)^3 + 9*cos(a + c))* cos(b*x + a)^2 - 9*cos(a + c)^3 + 6*cos(a + c))*sin(b*x + a) + ((16*cos(a + c)^4 - 16*cos(a + c)^2 + 3)*cos(b*x + a)^3 - 3*(4*cos(a + c)^4 - 5*cos(a + c)^2 + 1)*cos(b*x + a))*sin(a + c))/((4*b*cos(a + c)^3 - 3*b*cos(a + c) )*cos(b*x + a)^3 + ((4*b*cos(a + c)^2 - b)*cos(b*x + a)^2 - b*cos(a + c)^2 + b)*sin(b*x + a)*sin(a + c) - 3*(b*cos(a + c)^3 - b*cos(a + c))*cos(b*x + a))
Timed out. \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate(sec(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. 927 vs. \(2 (84) = 168\).
Time = 0.05 (sec) , antiderivative size = 927, normalized size of antiderivative = 11.59 \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(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. 786 vs. \(2 (84) = 168\).
Time = 0.15 (sec) , antiderivative size = 786, normalized size of antiderivative = 9.82 \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x-c)^4*sin(b*x+a)^2,x, algorithm="giac")
Output:
1/3*(tan(b*x - c)^3*tan(1/2*a)^4*tan(1/2*c)^4 - 2*tan(b*x - c)^3*tan(1/2*a )^4*tan(1/2*c)^2 - 8*tan(b*x - c)^3*tan(1/2*a)^3*tan(1/2*c)^3 - 6*tan(b*x - c)^2*tan(1/2*a)^4*tan(1/2*c)^3 - 2*tan(b*x - c)^3*tan(1/2*a)^2*tan(1/2*c )^4 - 6*tan(b*x - c)^2*tan(1/2*a)^3*tan(1/2*c)^4 + tan(b*x - c)^3*tan(1/2* a)^4 + 8*tan(b*x - c)^3*tan(1/2*a)^3*tan(1/2*c) + 6*tan(b*x - c)^2*tan(1/2 *a)^4*tan(1/2*c) + 20*tan(b*x - c)^3*tan(1/2*a)^2*tan(1/2*c)^2 + 36*tan(b* x - c)^2*tan(1/2*a)^3*tan(1/2*c)^2 + 12*tan(b*x - c)*tan(1/2*a)^4*tan(1/2* c)^2 + 8*tan(b*x - c)^3*tan(1/2*a)*tan(1/2*c)^3 + 36*tan(b*x - c)^2*tan(1/ 2*a)^2*tan(1/2*c)^3 + 24*tan(b*x - c)*tan(1/2*a)^3*tan(1/2*c)^3 + tan(b*x - c)^3*tan(1/2*c)^4 + 6*tan(b*x - c)^2*tan(1/2*a)*tan(1/2*c)^4 + 12*tan(b* x - c)*tan(1/2*a)^2*tan(1/2*c)^4 - 2*tan(b*x - c)^3*tan(1/2*a)^2 - 6*tan(b *x - c)^2*tan(1/2*a)^3 - 8*tan(b*x - c)^3*tan(1/2*a)*tan(1/2*c) - 36*tan(b *x - c)^2*tan(1/2*a)^2*tan(1/2*c) - 24*tan(b*x - c)*tan(1/2*a)^3*tan(1/2*c ) - 2*tan(b*x - c)^3*tan(1/2*c)^2 - 36*tan(b*x - c)^2*tan(1/2*a)*tan(1/2*c )^2 - 48*tan(b*x - c)*tan(1/2*a)^2*tan(1/2*c)^2 - 6*tan(b*x - c)^2*tan(1/2 *c)^3 - 24*tan(b*x - c)*tan(1/2*a)*tan(1/2*c)^3 + tan(b*x - c)^3 + 6*tan(b *x - c)^2*tan(1/2*a) + 12*tan(b*x - c)*tan(1/2*a)^2 + 6*tan(b*x - c)^2*tan (1/2*c) + 24*tan(b*x - c)*tan(1/2*a)*tan(1/2*c) + 12*tan(b*x - c)*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(...
Timed out. \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\text {Hanged} \] Input:
int(sin(a + b*x)^2/cos(c - b*x)^4,x)
Output:
\text{Hanged}
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int \sec ^4(c-b x) \sin ^2(a+b x) \, dx=\frac {\cos \left (b x -c \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )+\sin \left (b x -c \right )^{3}-\sin \left (b x -c \right ) \sin \left (b x +a \right )^{2}-\sin \left (b x -c \right )}{3 \cos \left (b x -c \right ) b \left (\sin \left (b x -c \right )^{2}-1\right )} \] Input:
int(sec(b*x-c)^4*sin(b*x+a)^2,x)
Output:
(cos(b*x - c)*cos(a + b*x)*sin(a + b*x) + sin(b*x - c)**3 - sin(b*x - c)*s in(a + b*x)**2 - sin(b*x - c))/(3*cos(b*x - c)*b*(sin(b*x - c)**2 - 1))