Integrand size = 17, antiderivative size = 89 \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \cos (2 (a-c))}{b}-\frac {\text {arctanh}(\cos (c+b x)) \sin ^2(a-c)}{2 b}-\frac {\cot (c+b x) \csc (c+b x) \sin ^2(a-c)}{2 b}-\frac {\csc (c+b x) \sin (2 (a-c))}{b} \] Output:
-arctanh(cos(b*x+c))*cos(2*a-2*c)/b-1/2*arctanh(cos(b*x+c))*sin(a-c)^2/b-1 /2*cot(b*x+c)*csc(b*x+c)*sin(a-c)^2/b-csc(b*x+c)*sin(2*a-2*c)/b
Leaf count is larger than twice the leaf count of optimal. \(247\) vs. \(2(89)=178\).
Time = 2.74 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=\frac {\left (\cos \left (2 a-2 c-\frac {b x}{2}\right )-\cos \left (2 a-2 c+\frac {b x}{2}\right )\right ) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {b x}{2}\right )}{4 b}+\frac {(-1+\cos (2 a-2 c)) \csc ^2\left (\frac {c}{2}+\frac {b x}{2}\right )}{16 b}+\frac {(-1-3 \cos (2 a-2 c)) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{4 b}+\frac {(1+3 \cos (2 a-2 c)) \log \left (\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{4 b}+\frac {\left (-\cos \left (2 a-2 c-\frac {b x}{2}\right )+\cos \left (2 a-2 c+\frac {b x}{2}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {b x}{2}\right )}{4 b}+\frac {(1-\cos (2 a-2 c)) \sec ^2\left (\frac {c}{2}+\frac {b x}{2}\right )}{16 b} \] Input:
Integrate[Csc[c + b*x]^3*Sin[a + b*x]^2,x]
Output:
((Cos[2*a - 2*c - (b*x)/2] - Cos[2*a - 2*c + (b*x)/2])*Csc[c/2]*Csc[c/2 + (b*x)/2])/(4*b) + ((-1 + Cos[2*a - 2*c])*Csc[c/2 + (b*x)/2]^2)/(16*b) + (( -1 - 3*Cos[2*a - 2*c])*Log[Cos[c/2 + (b*x)/2]])/(4*b) + ((1 + 3*Cos[2*a - 2*c])*Log[Sin[c/2 + (b*x)/2]])/(4*b) + ((-Cos[2*a - 2*c - (b*x)/2] + Cos[2 *a - 2*c + (b*x)/2])*Sec[c/2]*Sec[c/2 + (b*x)/2])/(4*b) + ((1 - Cos[2*a - 2*c])*Sec[c/2 + (b*x)/2]^2)/(16*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.
\(\displaystyle \int \sin ^2(a+b x) \csc ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \csc ^3(b x+c)dx\) |
Input:
Int[Csc[c + b*x]^3*Sin[a + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.00 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.73
method | result | size |
risch | \(\frac {-5 \,{\mathrm e}^{i \left (3 b x +6 a +c \right )}+2 \,{\mathrm e}^{i \left (3 b x +4 a +3 c \right )}+3 \,{\mathrm e}^{i \left (3 b x +2 a +5 c \right )}+3 \,{\mathrm e}^{i \left (b x +6 a -c \right )}+2 \,{\mathrm e}^{i \left (b x +4 a +c \right )}-5 \,{\mathrm e}^{i \left (b x +2 a +3 c \right )}}{4 \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{4 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right )}{4 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{4 b}\) | \(243\) |
default | \(\text {Expression too large to display}\) | \(1156\) |
Input:
int(csc(b*x+c)^3*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/4/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^2/b*(-5*exp(I*(3*b*x+6*a+c))+2*exp(I* (3*b*x+4*a+3*c))+3*exp(I*(3*b*x+2*a+5*c))+3*exp(I*(b*x+6*a-c))+2*exp(I*(b* x+4*a+c))-5*exp(I*(b*x+2*a+3*c)))-1/4/b*ln(exp(I*(b*x+a))+exp(I*(a-c)))-3/ 4/b*ln(exp(I*(b*x+a))+exp(I*(a-c)))*cos(2*a-2*c)+1/4/b*ln(exp(I*(b*x+a))-e xp(I*(a-c)))+3/4/b*ln(exp(I*(b*x+a))-exp(I*(a-c)))*cos(2*a-2*c)
Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.66 \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=-\frac {8 \, \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + 2 \, {\left (\cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right ) + {\left ({\left (3 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - 3 \, \cos \left (-a + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) - {\left ({\left (3 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - 3 \, \cos \left (-a + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )}} \] Input:
integrate(csc(b*x+c)^3*sin(b*x+a)^2,x, algorithm="fricas")
Output:
-1/4*(8*cos(-a + c)*sin(b*x + c)*sin(-a + c) + 2*(cos(-a + c)^2 - 1)*cos(b *x + c) + ((3*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - 3*cos(-a + c)^2 + 1)*log (1/2*cos(b*x + c) + 1/2) - ((3*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - 3*cos(- a + c)^2 + 1)*log(-1/2*cos(b*x + c) + 1/2))/(b*cos(b*x + c)^2 - b)
Timed out. \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+c)**3*sin(b*x+a)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1595 vs. \(2 (85) = 170\).
Time = 0.09 (sec) , antiderivative size = 1595, normalized size of antiderivative = 17.92 \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sin(b*x+a)^2,x, algorithm="maxima")
Output:
-1/8*(2*(5*cos(3*b*x + 4*a + 2*c) - 2*cos(3*b*x + 2*a + 4*c) - 3*cos(3*b*x + 6*c) - 3*cos(b*x + 4*a) - 2*cos(b*x + 2*a + 2*c) + 5*cos(b*x + 4*c))*co s(4*b*x + 2*a + 5*c) - 10*(2*cos(2*b*x + 2*a + 3*c) - cos(2*a + c))*cos(3* b*x + 4*a + 2*c) + 4*(2*cos(2*b*x + 2*a + 3*c) - cos(2*a + c))*cos(3*b*x + 2*a + 4*c) + 6*(2*cos(2*b*x + 2*a + 3*c) - cos(2*a + c))*cos(3*b*x + 6*c) + 4*(3*cos(b*x + 4*a) + 2*cos(b*x + 2*a + 2*c) - 5*cos(b*x + 4*c))*cos(2* b*x + 2*a + 3*c) - 6*cos(b*x + 4*a)*cos(2*a + c) - 4*cos(b*x + 2*a + 2*c)* cos(2*a + c) + 10*cos(b*x + 4*c)*cos(2*a + c) + ((3*cos(-2*a + 2*c) + 1)*c os(4*b*x + 2*a + 5*c)^2 + 4*(3*cos(-2*a + 2*c) + 1)*cos(2*b*x + 2*a + 3*c) ^2 + (3*cos(-2*a + 2*c) + 1)*sin(4*b*x + 2*a + 5*c)^2 + 4*(3*cos(-2*a + 2* c) + 1)*sin(2*b*x + 2*a + 3*c)^2 - 2*(2*(3*cos(-2*a + 2*c) + 1)*cos(2*b*x + 2*a + 3*c) - 3*cos(2*a + c)*cos(-2*a + 2*c) - cos(2*a + c))*cos(4*b*x + 2*a + 5*c) - 4*(3*cos(2*a + c)*cos(-2*a + 2*c) + cos(2*a + c))*cos(2*b*x + 2*a + 3*c) + cos(2*a + c)^2 + 3*(cos(2*a + c)^2 + sin(2*a + c)^2)*cos(-2* a + 2*c) - 2*(2*(3*cos(-2*a + 2*c) + 1)*sin(2*b*x + 2*a + 3*c) - 3*cos(-2* a + 2*c)*sin(2*a + c) - sin(2*a + c))*sin(4*b*x + 2*a + 5*c) - 4*(3*cos(-2 *a + 2*c)*sin(2*a + c) + sin(2*a + c))*sin(2*b*x + 2*a + 3*c) + sin(2*a + c)^2)*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b *x)*sin(c) + sin(c)^2) - ((3*cos(-2*a + 2*c) + 1)*cos(4*b*x + 2*a + 5*c)^2 + 4*(3*cos(-2*a + 2*c) + 1)*cos(2*b*x + 2*a + 3*c)^2 + (3*cos(-2*a + 2...
Leaf count of result is larger than twice the leaf count of optimal. 2867 vs. \(2 (85) = 170\).
Time = 0.18 (sec) , antiderivative size = 2867, normalized size of antiderivative = 32.21 \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sin(b*x+a)^2,x, algorithm="giac")
Output:
1/2*(2*(tan(1/2*a)^4*tan(1/2*c)^4 - 4*tan(1/2*a)^4*tan(1/2*c)^2 + 12*tan(1 /2*a)^3*tan(1/2*c)^3 - 4*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 - 12*tan (1/2*a)^3*tan(1/2*c) + 28*tan(1/2*a)^2*tan(1/2*c)^2 - 12*tan(1/2*a)*tan(1/ 2*c)^3 + tan(1/2*c)^4 - 4*tan(1/2*a)^2 + 12*tan(1/2*a)*tan(1/2*c) - 4*tan( 1/2*c)^2 + 1)*log(abs(tan(1/2*b*x + 1/2*c)))/(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) + (tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^6 - 2*tan(1/2*b *x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^7 - 4*tan(1/2*b*x + 1/2*c)*tan(1/2*a )^8*tan(1/2*c)^7 + tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^8 + 4*ta n(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^8 + 2*tan(1/2*b*x + 1/2*c)^2*ta n(1/2*a)^8*tan(1/2*c)^4 - 2*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c) ^5 - 4*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^8*tan(1/2*c)^5 - 16*tan(1/2*b*x + 1 /2*c)*tan(1/2*a)^7*tan(1/2*c)^6 - 2*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*ta n(1/2*c)^7 + 16*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c)^7 + 2*tan(1/2 *b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)^8 + 4*tan(1/2*b*x + 1/2*c)*tan(1/2 *a)^5*tan(1/2*c)^8 + tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^2 + 2* tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^3 + 4*tan(1/2*b*x + 1/2*c)* tan(1/2*a)^8*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2* c)^4 - 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^4 - 2*tan(1/2*b*...
Timed out. \[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Hanged} \] Input:
int(sin(a + b*x)^2/sin(c + b*x)^3,x)
Output:
\text{Hanged}
\[ \int \csc ^3(c+b x) \sin ^2(a+b x) \, dx=\int \csc \left (b x +c \right )^{3} \sin \left (b x +a \right )^{2}d x \] Input:
int(csc(b*x+c)^3*sin(b*x+a)^2,x)
Output:
int(csc(b*x + c)**3*sin(a + b*x)**2,x)