Integrand size = 17, antiderivative size = 166 \[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{4 d}-\frac {\cot (c+d x) \csc (c+d x)}{4 d}+\frac {2 e^{-2 i a-2 i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {3}{2}-\frac {b}{d},\frac {5}{2}-\frac {b}{d},e^{2 i (c+d x)}\right )}{2 b-3 d}-\frac {2 e^{2 i a+2 i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {3}{2}+\frac {b}{d},\frac {5}{2}+\frac {b}{d},e^{2 i (c+d x)}\right )}{2 b+3 d} \] Output:
-1/4*arctanh(cos(d*x+c))/d-1/4*cot(d*x+c)*csc(d*x+c)/d+2*exp(-2*I*a-2*I*b* x+3*I*(d*x+c))*hypergeom([3, 3/2-b/d],[5/2-b/d],exp(2*I*(d*x+c)))/(2*b-3*d )-2*exp(2*I*a+2*I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2+b/d],[5/2+b/d],exp(2* I*(d*x+c)))/(2*b+3*d)
Time = 1.87 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.24 \[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\frac {16 (2 b+d) e^{-i (2 a-c+2 b x-d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {b}{d},\frac {3}{2}-\frac {b}{d},e^{2 i (c+d x)}\right )-16 (2 b-d) e^{i (2 a+c+(2 b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {b}{d},\frac {3}{2}+\frac {b}{d},e^{2 i (c+d x)}\right )-16 d \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+16 d \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-16 \csc ^2(c+d x) \sin (a+b x) ((-2 b+d) \sin (a-c+b x-d x)+(2 b+d) \sin (a+c+(b+d) x))}{64 d^2} \] Input:
Integrate[Csc[c + d*x]^3*Sin[a + b*x]^2,x]
Output:
((16*(2*b + d)*Hypergeometric2F1[1, 1/2 - b/d, 3/2 - b/d, E^((2*I)*(c + d* x))])/E^(I*(2*a - c + 2*b*x - d*x)) - 16*(2*b - d)*E^(I*(2*a + c + (2*b + d)*x))*Hypergeometric2F1[1, 1/2 + b/d, 3/2 + b/d, E^((2*I)*(c + d*x))] - 1 6*d*Log[Cos[(c + d*x)/2]] + 16*d*Log[Sin[(c + d*x)/2]] - 16*Csc[c + d*x]^2 *Sin[a + b*x]*((-2*b + d)*Sin[a - c + b*x - d*x] + (2*b + d)*Sin[a + c + ( b + d)*x]))/(64*d^2)
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.
\(\displaystyle \int \sin ^2(a+b x) \csc ^3(c+d x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \csc ^3(c+d x)dx\) |
Input:
Int[Csc[c + d*x]^3*Sin[a + b*x]^2,x]
Output:
$Aborted
\[\int \csc \left (d x +c \right )^{3} \sin \left (b x +a \right )^{2}d x\]
Input:
int(csc(d*x+c)^3*sin(b*x+a)^2,x)
Output:
int(csc(d*x+c)^3*sin(b*x+a)^2,x)
\[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\int { \csc \left (d x + c\right )^{3} \sin \left (b x + a\right )^{2} \,d x } \] Input:
integrate(csc(d*x+c)^3*sin(b*x+a)^2,x, algorithm="fricas")
Output:
integral(-(cos(b*x + a)^2 - 1)*csc(d*x + c)^3, x)
Timed out. \[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**3*sin(b*x+a)**2,x)
Output:
Timed out
\[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\int { \csc \left (d x + c\right )^{3} \sin \left (b x + a\right )^{2} \,d x } \] Input:
integrate(csc(d*x+c)^3*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/4*((2*b - d)*cos((4*b + d)*x + 4*a + c)*cos(2*b*x + 2*a) + 2*d*cos((2*b + d)*x + 2*a + c)*cos(2*b*x + 2*a) + (2*b - d)*cos(2*b*x + 2*a)*cos(3*d*x + 3*c) - (2*b + d)*cos(2*b*x + 2*a)*cos(d*x + c) + (2*b - d)*sin((4*b + d) *x + 4*a + c)*sin(2*b*x + 2*a) + 2*d*sin((2*b + d)*x + 2*a + c)*sin(2*b*x + 2*a) + (2*b - d)*sin(2*b*x + 2*a)*sin(3*d*x + 3*c) - (2*b + d)*sin(2*b*x + 2*a)*sin(d*x + c) + (2*(2*b + d)*cos(2*(b + d)*x + 2*a + 2*c) - (2*b + d)*cos(2*b*x + 2*a))*cos((4*b + 3*d)*x + 4*a + 3*c) - 2*(2*d*cos(2*(b + d) *x + 2*a + 2*c) - d*cos(2*b*x + 2*a))*cos((2*b + 3*d)*x + 2*a + 3*c) - ((2 *b + d)*cos((4*b + 3*d)*x + 4*a + 3*c) - (2*b - d)*cos((4*b + d)*x + 4*a + c) - 2*d*cos((2*b + 3*d)*x + 2*a + 3*c) - 2*d*cos((2*b + d)*x + 2*a + c) - (2*b - d)*cos(3*d*x + 3*c) + (2*b + d)*cos(d*x + c))*cos(2*(b + 2*d)*x + 2*a + 4*c) - 2*((2*b - d)*cos((4*b + d)*x + 4*a + c) + 2*d*cos((2*b + d)* x + 2*a + c) + (2*b - d)*cos(3*d*x + 3*c) - (2*b + d)*cos(d*x + c))*cos(2* (b + d)*x + 2*a + 2*c) - 4*(d^2*cos(2*(b + 2*d)*x + 2*a + 4*c)^2 + 4*d^2*c os(2*(b + d)*x + 2*a + 2*c)^2 - 4*d^2*cos(2*(b + d)*x + 2*a + 2*c)*cos(2*b *x + 2*a) + d^2*cos(2*b*x + 2*a)^2 + d^2*sin(2*(b + 2*d)*x + 2*a + 4*c)^2 + 4*d^2*sin(2*(b + d)*x + 2*a + 2*c)^2 - 4*d^2*sin(2*(b + d)*x + 2*a + 2*c )*sin(2*b*x + 2*a) + d^2*sin(2*b*x + 2*a)^2 - 2*(2*d^2*cos(2*(b + d)*x + 2 *a + 2*c) - d^2*cos(2*b*x + 2*a))*cos(2*(b + 2*d)*x + 2*a + 4*c) - 2*(2*d^ 2*sin(2*(b + d)*x + 2*a + 2*c) - d^2*sin(2*b*x + 2*a))*sin(2*(b + 2*d)*...
\[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\int { \csc \left (d x + c\right )^{3} \sin \left (b x + a\right )^{2} \,d x } \] Input:
integrate(csc(d*x+c)^3*sin(b*x+a)^2,x, algorithm="giac")
Output:
integrate(csc(d*x + c)^3*sin(b*x + a)^2, x)
Timed out. \[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{{\sin \left (c+d\,x\right )}^3} \,d x \] Input:
int(sin(a + b*x)^2/sin(c + d*x)^3,x)
Output:
int(sin(a + b*x)^2/sin(c + d*x)^3, x)
\[ \int \csc ^3(c+d x) \sin ^2(a+b x) \, dx=\int \csc \left (d x +c \right )^{3} \sin \left (b x +a \right )^{2}d x \] Input:
int(csc(d*x+c)^3*sin(b*x+a)^2,x)
Output:
int(csc(c + d*x)**3*sin(a + b*x)**2,x)