Integrand size = 17, antiderivative size = 1 \[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 8.75 (sec) , antiderivative size = 382, normalized size of antiderivative = 382.00 \[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=\frac {6 i \arctan \left (\frac {(\cos (a)-i \sin (a)) \left (\cos (a) \cos \left (\frac {b x}{2}\right )-\sin (a) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (a) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (a)}\right ) \cos (a-c)}{\frac {3 b}{8}+\frac {1}{8} b \cos (4 a-4 c)-\frac {1}{2} b \cos (2 a-2 c)}-\frac {\csc ^2(a-c) \csc ^2\left (\frac {c}{2}+\frac {b x}{2}\right )}{8 b}+\frac {\csc ^3(a-c) \csc (a+b x)}{b}-\frac {3 (3+\cos (2 a-2 c)) \csc ^4(a-c) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{4 b}+\frac {3 (3+\cos (2 a-2 c)) \csc ^4(a-c) \log \left (\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{4 b}+\frac {\csc ^2(a-c) \sec ^2\left (\frac {c}{2}+\frac {b x}{2}\right )}{8 b}-\frac {\csc ^3(a-c) \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {b x}{2}\right ) \left (\sin \left (a-c-\frac {b x}{2}\right )-\sin \left (a-c+\frac {b x}{2}\right )\right )}{2 b}-\frac {\csc ^3(a-c) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {b x}{2}\right ) \left (-\sin \left (a-c-\frac {b x}{2}\right )+\sin \left (a-c+\frac {b x}{2}\right )\right )}{2 b} \] Input:
Integrate[Csc[a + b*x]^2*Csc[c + b*x]^3,x]
Output:
((6*I)*ArcTan[((Cos[a] - I*Sin[a])*(Cos[a]*Cos[(b*x)/2] - Sin[a]*Sin[(b*x) /2]))/(I*Cos[a]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[a])]*Cos[a - c])/((3*b)/8 + (b*Cos[4*a - 4*c])/8 - (b*Cos[2*a - 2*c])/2) - (Csc[a - c]^2*Csc[c/2 + ( b*x)/2]^2)/(8*b) + (Csc[a - c]^3*Csc[a + b*x])/b - (3*(3 + Cos[2*a - 2*c]) *Csc[a - c]^4*Log[Cos[c/2 + (b*x)/2]])/(4*b) + (3*(3 + Cos[2*a - 2*c])*Csc [a - c]^4*Log[Sin[c/2 + (b*x)/2]])/(4*b) + (Csc[a - c]^2*Sec[c/2 + (b*x)/2 ]^2)/(8*b) - (Csc[a - c]^3*Sec[c/2]*Sec[c/2 + (b*x)/2]*(Sin[a - c - (b*x)/ 2] - Sin[a - c + (b*x)/2]))/(2*b) - (Csc[a - c]^3*Csc[c/2]*Csc[c/2 + (b*x) /2]*(-Sin[a - c - (b*x)/2] + Sin[a - c + (b*x)/2]))/(2*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.
\(\displaystyle \int \csc ^2(a+b x) \csc ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \csc ^2(a+b x) \csc ^3(b x+c)dx\) |
Input:
Int[Csc[a + b*x]^2*Csc[c + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 2.25 (sec) , antiderivative size = 770, normalized size of antiderivative = 770.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(770\) |
risch | \(\text {Expression too large to display}\) | \(928\) |
Input:
int(csc(b*x+a)^2*csc(b*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(-4/(-sin(a)*cos(c)+cos(a)*sin(c))^4*(((-1/4*sin(c)^3*cos(a)^3-9/4*cos (c)*sin(c)^2*sin(a)*cos(a)^2-3/2*sin(c)^3*sin(a)^2*cos(a)+3/2*cos(c)*sin(c )^2*sin(a)^3+9/4*cos(c)^2*sin(c)*sin(a)^2*cos(a)-3/2*cos(c)^2*sin(c)*cos(a )^3+3/2*cos(c)^3*sin(a)*cos(a)^2+1/4*cos(c)^3*sin(a)^3)*tan(1/2*a+1/2*b*x) ^3+(-5/4*sin(c)^3*sin(a)*cos(a)^2-5/4*cos(c)*sin(c)^2*cos(a)^3+5/2*sin(c)^ 3*sin(a)^3+10*cos(c)*sin(c)^2*cos(a)*sin(a)^2+10*cos(c)^2*sin(c)*cos(a)^2* sin(a)+5/2*cos(c)^3*cos(a)^3-5/4*cos(c)^2*sin(c)*sin(a)^3-5/4*cos(c)^3*sin (a)^2*cos(a))*tan(1/2*a+1/2*b*x)^2+(-1/4*sin(c)^3*cos(a)^3+7/2*sin(c)^3*si n(a)^2*cos(a)+31/4*cos(c)*sin(c)^2*sin(a)*cos(a)^2-7/2*cos(c)*sin(c)^2*sin (a)^3+7/2*cos(c)^2*sin(c)*cos(a)^3-31/4*cos(c)^2*sin(c)*sin(a)^2*cos(a)-7/ 2*cos(c)^3*sin(a)*cos(a)^2+1/4*cos(c)^3*sin(a)^3)*tan(1/2*a+1/2*b*x)+5/4*s in(c)^3*sin(a)*cos(a)^2+5/4*cos(c)*sin(c)^2*cos(a)^3-5/2*cos(c)*sin(c)^2*c os(a)*sin(a)^2-5/2*cos(c)^2*sin(c)*cos(a)^2*sin(a)+5/4*cos(c)^2*sin(c)*sin (a)^3+5/4*cos(c)^3*sin(a)^2*cos(a))/(sin(c)*cos(a)*tan(1/2*a+1/2*b*x)^2-co s(c)*sin(a)*tan(1/2*a+1/2*b*x)^2-2*tan(1/2*a+1/2*b*x)*sin(a)*sin(c)-2*tan( 1/2*a+1/2*b*x)*cos(a)*cos(c)-cos(a)*sin(c)+sin(a)*cos(c))^2+3/4*(sin(c)^2* cos(a)^2+2*sin(a)^2*sin(c)^2+2*cos(a)*cos(c)*sin(a)*sin(c)+2*cos(a)^2*cos( c)^2+cos(c)^2*sin(a)^2)/(-cos(c)^2*sin(a)^2-cos(a)^2*cos(c)^2-sin(a)^2*sin (c)^2-sin(c)^2*cos(a)^2)^(1/2)*arctan(1/2*(2*(-sin(a)*cos(c)+cos(a)*sin(c) )*tan(1/2*a+1/2*b*x)-2*cos(a)*cos(c)-2*sin(a)*sin(c))/(-cos(c)^2*sin(a)...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.14 (sec) , antiderivative size = 678, normalized size of antiderivative = 678.00 \[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx =\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*csc(b*x+c)^3,x, algorithm="fricas")
Output:
-1/4*(6*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)*sin(b*x + c) + 6*((cos( b*x + c)^2*cos(-a + c)^2 - cos(-a + c)^2)*sin(b*x + c) - (cos(b*x + c)^3*c os(-a + c) - cos(b*x + c)*cos(-a + c))*sin(-a + c))*log((cos(b*x + c)*cos( -a + c) + sin(b*x + c)*sin(-a + c) + 1)/(cos(-a + c) + 1)) - 6*((cos(b*x + c)^2*cos(-a + c)^2 - cos(-a + c)^2)*sin(b*x + c) - (cos(b*x + c)^3*cos(-a + c) - cos(b*x + c)*cos(-a + c))*sin(-a + c))*log(-(cos(b*x + c)*cos(-a + c) + sin(b*x + c)*sin(-a + c) - 1)/(cos(-a + c) + 1)) - 3*(((cos(-a + c)^ 3 + cos(-a + c))*cos(b*x + c)^2 - cos(-a + c)^3 - cos(-a + c))*sin(b*x + c ) - ((cos(-a + c)^2 + 1)*cos(b*x + c)^3 - (cos(-a + c)^2 + 1)*cos(b*x + c) )*sin(-a + c))*log(1/2*cos(b*x + c) + 1/2) + 3*(((cos(-a + c)^3 + cos(-a + c))*cos(b*x + c)^2 - cos(-a + c)^3 - cos(-a + c))*sin(b*x + c) - ((cos(-a + c)^2 + 1)*cos(b*x + c)^3 - (cos(-a + c)^2 + 1)*cos(b*x + c))*sin(-a + c ))*log(-1/2*cos(b*x + c) + 1/2) - 2*(3*(cos(-a + c)^2 + 1)*cos(b*x + c)^2 - 4*cos(-a + c)^2 - 2)*sin(-a + c))/((b*cos(-a + c)^5 - 2*b*cos(-a + c)^3 - (b*cos(-a + c)^5 - 2*b*cos(-a + c)^3 + b*cos(-a + c))*cos(b*x + c)^2 + b *cos(-a + c))*sin(b*x + c) + ((b*cos(-a + c)^4 - 2*b*cos(-a + c)^2 + b)*co s(b*x + c)^3 - (b*cos(-a + c)^4 - 2*b*cos(-a + c)^2 + b)*cos(b*x + c))*sin (-a + c))
\[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=\int \csc ^{2}{\left (a + b x \right )} \csc ^{3}{\left (b x + c \right )}\, dx \] Input:
integrate(csc(b*x+a)**2*csc(b*x+c)**3,x)
Output:
Integral(csc(a + b*x)**2*csc(b*x + c)**3, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 32.84 (sec) , antiderivative size = 810260, normalized size of antiderivative = 810260.00 \[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*csc(b*x+c)^3,x, algorithm="maxima")
Output:
-(24*(((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*c) + (sin(8*a) - 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))*cos(3*a + 5*c) + 4*(sin (5*a + 3*c) + sin(3*a + 5*c))*cos(2*a + 6*c) - (cos(8*a) - 4*cos(6*a + 2*c ) + cos(8*c))*sin(5*a + 3*c) - 6*cos(4*a + 4*c)*sin(5*a + 3*c) + 6*cos(5*a + 3*c)*sin(4*a + 4*c) - (cos(8*a) - 4*cos(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*sin(3*a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c))*sin(2*a + 6*c))*cos(6*b*x + 8*a + 4*c)^2 + 9*((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c ))*cos(5*a + 3*c) + (sin(8*a) - 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin( 8*c))*cos(3*a + 5*c) + 4*(sin(5*a + 3*c) + sin(3*a + 5*c))*cos(2*a + 6*c) - (cos(8*a) - 4*cos(6*a + 2*c) + cos(8*c))*sin(5*a + 3*c) - 6*cos(4*a + 4* c)*sin(5*a + 3*c) + 6*cos(5*a + 3*c)*sin(4*a + 4*c) - (cos(8*a) - 4*cos(6* a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*sin(3*a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c))*sin(2*a + 6*c))*cos(6*b*x + 6*a + 6*c)^2 + 9*((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*c) + (sin(8*a) - 4*sin(6*a + 2 *c) + 6*sin(4*a + 4*c) + sin(8*c))*cos(3*a + 5*c) + 4*(sin(5*a + 3*c) + si n(3*a + 5*c))*cos(2*a + 6*c) - (cos(8*a) - 4*cos(6*a + 2*c) + cos(8*c))*si n(5*a + 3*c) - 6*cos(4*a + 4*c)*sin(5*a + 3*c) + 6*cos(5*a + 3*c)*sin(4*a + 4*c) - (cos(8*a) - 4*cos(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*sin(3 *a + 5*c) - 4*(cos(5*a + 3*c) + cos(3*a + 5*c))*sin(2*a + 6*c))*cos(6*b*x + 4*a + 8*c)^2 + ((sin(8*a) - 4*sin(6*a + 2*c) + sin(8*c))*cos(5*a + 3*...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 2.32 (sec) , antiderivative size = 10037, normalized size of antiderivative = 10037.00 \[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*csc(b*x+c)^3,x, algorithm="giac")
Output:
-1/32*(6*(tan(1/2*a)^8*tan(1/2*c)^8 + 2*tan(1/2*a)^8*tan(1/2*c)^6 + 4*tan( 1/2*a)^7*tan(1/2*c)^7 + 2*tan(1/2*a)^6*tan(1/2*c)^8 + 12*tan(1/2*a)^7*tan( 1/2*c)^5 + 4*tan(1/2*a)^6*tan(1/2*c)^6 + 12*tan(1/2*a)^5*tan(1/2*c)^7 - 2* tan(1/2*a)^8*tan(1/2*c)^2 + 12*tan(1/2*a)^7*tan(1/2*c)^3 + 36*tan(1/2*a)^5 *tan(1/2*c)^5 + 12*tan(1/2*a)^3*tan(1/2*c)^7 - 2*tan(1/2*a)^2*tan(1/2*c)^8 - tan(1/2*a)^8 + 4*tan(1/2*a)^7*tan(1/2*c) - 4*tan(1/2*a)^6*tan(1/2*c)^2 + 36*tan(1/2*a)^5*tan(1/2*c)^3 + 36*tan(1/2*a)^3*tan(1/2*c)^5 - 4*tan(1/2* a)^2*tan(1/2*c)^6 + 4*tan(1/2*a)*tan(1/2*c)^7 - tan(1/2*c)^8 - 2*tan(1/2*a )^6 + 12*tan(1/2*a)^5*tan(1/2*c) + 36*tan(1/2*a)^3*tan(1/2*c)^3 + 12*tan(1 /2*a)*tan(1/2*c)^5 - 2*tan(1/2*c)^6 + 12*tan(1/2*a)^3*tan(1/2*c) + 4*tan(1 /2*a)^2*tan(1/2*c)^2 + 12*tan(1/2*a)*tan(1/2*c)^3 + 2*tan(1/2*a)^2 + 4*tan (1/2*a)*tan(1/2*c) + 2*tan(1/2*c)^2 + 1)*log(abs(tan(1/2*b*x + 1/2*a)))/(t an(1/2*a)^8*tan(1/2*c)^4 - 4*tan(1/2*a)^7*tan(1/2*c)^5 + 6*tan(1/2*a)^6*ta n(1/2*c)^6 - 4*tan(1/2*a)^5*tan(1/2*c)^7 + tan(1/2*a)^4*tan(1/2*c)^8 + 4*t an(1/2*a)^7*tan(1/2*c)^3 - 16*tan(1/2*a)^6*tan(1/2*c)^4 + 24*tan(1/2*a)^5* tan(1/2*c)^5 - 16*tan(1/2*a)^4*tan(1/2*c)^6 + 4*tan(1/2*a)^3*tan(1/2*c)^7 + 6*tan(1/2*a)^6*tan(1/2*c)^2 - 24*tan(1/2*a)^5*tan(1/2*c)^3 + 36*tan(1/2* a)^4*tan(1/2*c)^4 - 24*tan(1/2*a)^3*tan(1/2*c)^5 + 6*tan(1/2*a)^2*tan(1/2* c)^6 + 4*tan(1/2*a)^5*tan(1/2*c) - 16*tan(1/2*a)^4*tan(1/2*c)^2 + 24*tan(1 /2*a)^3*tan(1/2*c)^3 - 16*tan(1/2*a)^2*tan(1/2*c)^4 + 4*tan(1/2*a)*tan(...
Timed out. \[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(sin(a + b*x)^2*sin(c + b*x)^3),x)
Output:
\text{Hanged}
\[ \int \csc ^2(a+b x) \csc ^3(c+b x) \, dx=\int \csc \left (b x +c \right )^{3} \csc \left (b x +a \right )^{2}d x \] Input:
int(csc(b*x+a)^2*csc(b*x+c)^3,x)
Output:
int(csc(b*x + c)**3*csc(a + b*x)**2,x)