Integrand size = 17, antiderivative size = 1 \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 4.34 (sec) , antiderivative size = 186, normalized size of antiderivative = 186.00 \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\frac {96 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cot (a-c) \csc ^3(a-c)+\csc ^3(a-c) \left (-16 \csc (c+b x)+(-33-16 \cos (2 (a-c))+\cos (4 (a-c))) \csc (a-c) \left (\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )-2 \cot ^2(a-c) \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )+64 \cos \left (a+\frac {b x}{2}\right ) \cot (a-c) \csc (a) \csc ^2(a-c) \csc (a+b x) \sin \left (\frac {b x}{2}\right )}{16 b} \] Input:
Integrate[Cot[c + b*x]^2*Csc[a + b*x]^3,x]
Output:
(96*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cot[a - c]*Csc[a - c]^3 + Csc[a - c]^3*(-16*Csc[c + b*x] + (-33 - 16*Cos[2*(a - c)] + Cos[4*(a - c)])*Csc[ a - c]*(Log[Cos[(a + b*x)/2]] - Log[Sin[(a + b*x)/2]])) - 2*Cot[a - c]^2*( Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2) + 64*Cos[a + (b*x)/2]*Cot[a - c]* Csc[a]*Csc[a - c]^2*Csc[a + b*x]*Sin[(b*x)/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.
\(\displaystyle \int \csc ^3(a+b x) \cot ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \csc ^3(a+b x) \cot ^2(b x+c)dx\) |
Input:
Int[Cot[c + b*x]^2*Csc[a + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 1263, normalized size of antiderivative = 1263.00
Input:
int(cot(b*x+c)^2*csc(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/b/(exp(2*I*a)-exp(2*I*c))^3/(exp(2*I*(b*x+a))-1)^2/(-exp(2*I*(b*x+a+c)) +exp(2*I*a))*(-exp(I*(5*b*x+11*a+2*c))-33*exp(I*(5*b*x+9*a+4*c))-15*exp(I* (5*b*x+7*a+6*c))+exp(I*(5*b*x+5*a+8*c))+exp(I*(3*b*x+11*a))+16*exp(I*(3*b* x+9*a+2*c))+62*exp(I*(3*b*x+7*a+4*c))+16*exp(I*(3*b*x+5*a+6*c))+exp(I*(3*b *x+3*a+8*c))+exp(I*(b*x+9*a))-15*exp(I*(b*x+7*a+2*c))-33*exp(I*(b*x+5*a+4* c))-exp(I*(b*x+3*a+6*c)))+24*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b/(exp(8*I*a) -4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))*exp(I* (5*a+3*c))+24*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b/(exp(8*I*a)-4*exp(2*I*(3*a +c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))*exp(I*(3*a+5*c))-1/2/ (exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I *c))/b*ln(exp(I*(b*x+a))-1)*exp(8*I*a)+8/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6* exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(2 *I*(3*a+c))+33/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*( a+3*c))+exp(8*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(4*I*(a+c))+8/(exp(8*I*a)-4* exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*ln(exp( I*(b*x+a))-1)*exp(2*I*(a+3*c))-1/2/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4* I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(8*I*c)+ 1/2/(exp(8*I*a)-4*exp(2*I*(3*a+c))+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp (8*I*c))/b*ln(exp(I*(b*x+a))+1)*exp(8*I*a)-8/(exp(8*I*a)-4*exp(2*I*(3*a+c) )+6*exp(4*I*(a+c))-4*exp(2*I*(a+3*c))+exp(8*I*c))/b*ln(exp(I*(b*x+a))+1...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.15 (sec) , antiderivative size = 905, normalized size of antiderivative = 905.00 \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)^2*csc(b*x+a)^3,x, algorithm="fricas")
Output:
1/4*(2*(cos(-2*a + 2*c)^3 - 7*cos(-2*a + 2*c)^2 - cos(-2*a + 2*c) + 7)*cos (b*x + a)*sin(b*x + a) - 12*sqrt(2)*(((cos(-2*a + 2*c)^2 + 2*cos(-2*a + 2* c) + 1)*cos(b*x + a)^2 - cos(-2*a + 2*c)^2 - 2*cos(-2*a + 2*c) - 1)*sin(b* x + a) + ((cos(-2*a + 2*c) + 1)*cos(b*x + a)^3 - (cos(-2*a + 2*c) + 1)*cos (b*x + a))*sin(-2*a + 2*c))*log((2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos( b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) + 2*sqrt(2)*((cos(-2*a + 2*c) + 1)*c os(b*x + a) - sin(b*x + a)*sin(-2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - co s(-2*a + 2*c) + 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin( b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) - 1))/sqrt(cos(-2*a + 2*c) + 1) - (((cos(-2*a + 2*c)^3 - 7*cos(-2*a + 2*c)^2 - 25*cos(-2*a + 2*c) - 17)*c os(b*x + a)^2 - cos(-2*a + 2*c)^3 + 7*cos(-2*a + 2*c)^2 + 25*cos(-2*a + 2* c) + 17)*sin(b*x + a) + ((cos(-2*a + 2*c)^2 - 8*cos(-2*a + 2*c) - 17)*cos( b*x + a)^3 - (cos(-2*a + 2*c)^2 - 8*cos(-2*a + 2*c) - 17)*cos(b*x + a))*si n(-2*a + 2*c))*log(1/2*cos(b*x + a) + 1/2) + (((cos(-2*a + 2*c)^3 - 7*cos( -2*a + 2*c)^2 - 25*cos(-2*a + 2*c) - 17)*cos(b*x + a)^2 - cos(-2*a + 2*c)^ 3 + 7*cos(-2*a + 2*c)^2 + 25*cos(-2*a + 2*c) + 17)*sin(b*x + a) + ((cos(-2 *a + 2*c)^2 - 8*cos(-2*a + 2*c) - 17)*cos(b*x + a)^3 - (cos(-2*a + 2*c)^2 - 8*cos(-2*a + 2*c) - 17)*cos(b*x + a))*sin(-2*a + 2*c))*log(-1/2*cos(b*x + a) + 1/2) + 2*((cos(-2*a + 2*c)^2 - 8*cos(-2*a + 2*c) - 17)*cos(b*x + a) ^2 + 8*cos(-2*a + 2*c) + 16)*sin(-2*a + 2*c))/((b*cos(-2*a + 2*c)^3 - (...
Timed out. \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\text {Timed out} \] Input:
integrate(cot(b*x+c)**2*csc(b*x+a)**3,x)
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 52.89 (sec) , antiderivative size = 919799, normalized size of antiderivative = 919799.00 \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)^2*csc(b*x+a)^3,x, algorithm="maxima")
Output:
1/4*(24*(((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 6*(sin(8*a) - 5*sin(6*a + 2*c) + sin(8*c))*cos(4*a + 4*c) + (sin(8*a) + 30*sin(4*a + 4*c) + sin(8*c ))*cos(2*a + 6*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2*c) - 6*(cos(8*a) - 5 *cos(6*a + 2*c) + cos(8*c))*sin(4*a + 4*c) - (cos(8*a) + 30*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c))*cos(6*b*x + 10*a + 2*c)^2 + 9*((sin(8*a) + si n(8*c))*cos(6*a + 2*c) + 6*(sin(8*a) - 5*sin(6*a + 2*c) + sin(8*c))*cos(4* a + 4*c) + (sin(8*a) + 30*sin(4*a + 4*c) + sin(8*c))*cos(2*a + 6*c) - (cos (8*a) + cos(8*c))*sin(6*a + 2*c) - 6*(cos(8*a) - 5*cos(6*a + 2*c) + cos(8* c))*sin(4*a + 4*c) - (cos(8*a) + 30*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6 *c))*cos(6*b*x + 8*a + 4*c)^2 + 9*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 6*(sin(8*a) - 5*sin(6*a + 2*c) + sin(8*c))*cos(4*a + 4*c) + (sin(8*a) + 30 *sin(4*a + 4*c) + sin(8*c))*cos(2*a + 6*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2*c) - 6*(cos(8*a) - 5*cos(6*a + 2*c) + cos(8*c))*sin(4*a + 4*c) - (cos (8*a) + 30*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c))*cos(6*b*x + 6*a + 6* c)^2 + ((sin(8*a) + sin(8*c))*cos(6*a + 2*c) + 6*(sin(8*a) - 5*sin(6*a + 2 *c) + sin(8*c))*cos(4*a + 4*c) + (sin(8*a) + 30*sin(4*a + 4*c) + sin(8*c)) *cos(2*a + 6*c) - (cos(8*a) + cos(8*c))*sin(6*a + 2*c) - 6*(cos(8*a) - 5*c os(6*a + 2*c) + cos(8*c))*sin(4*a + 4*c) - (cos(8*a) + 30*cos(4*a + 4*c) + cos(8*c))*sin(2*a + 6*c))*cos(6*b*x + 4*a + 8*c)^2 + ((sin(8*a) + sin(8*c ))*cos(6*a + 2*c) + 6*(sin(8*a) - 5*sin(6*a + 2*c) + sin(8*c))*cos(4*a ...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.82 (sec) , antiderivative size = 7181, normalized size of antiderivative = 7181.00 \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)^2*csc(b*x+a)^3,x, algorithm="giac")
Output:
-1/32*(2*(3*tan(1/2*a)^9*tan(1/2*c)^8 + 6*tan(1/2*a)^9*tan(1/2*c)^6 + 12*t an(1/2*a)^8*tan(1/2*c)^7 + 6*tan(1/2*a)^7*tan(1/2*c)^8 - 2*tan(1/2*a)^9*ta n(1/2*c)^4 + 44*tan(1/2*a)^8*tan(1/2*c)^5 + 44*tan(1/2*a)^6*tan(1/2*c)^7 - 2*tan(1/2*a)^5*tan(1/2*c)^8 + 6*tan(1/2*a)^9*tan(1/2*c)^2 - 44*tan(1/2*a) ^8*tan(1/2*c)^3 + 212*tan(1/2*a)^7*tan(1/2*c)^4 - 180*tan(1/2*a)^6*tan(1/2 *c)^5 + 212*tan(1/2*a)^5*tan(1/2*c)^6 - 44*tan(1/2*a)^4*tan(1/2*c)^7 + 6*t an(1/2*a)^3*tan(1/2*c)^8 + 3*tan(1/2*a)^9 - 12*tan(1/2*a)^8*tan(1/2*c) + 1 80*tan(1/2*a)^6*tan(1/2*c)^3 - 132*tan(1/2*a)^5*tan(1/2*c)^4 + 180*tan(1/2 *a)^4*tan(1/2*c)^5 - 12*tan(1/2*a)^2*tan(1/2*c)^7 + 3*tan(1/2*a)*tan(1/2*c )^8 + 6*tan(1/2*a)^7 - 44*tan(1/2*a)^6*tan(1/2*c) + 212*tan(1/2*a)^5*tan(1 /2*c)^2 - 180*tan(1/2*a)^4*tan(1/2*c)^3 + 212*tan(1/2*a)^3*tan(1/2*c)^4 - 44*tan(1/2*a)^2*tan(1/2*c)^5 + 6*tan(1/2*a)*tan(1/2*c)^6 - 2*tan(1/2*a)^5 + 44*tan(1/2*a)^4*tan(1/2*c) + 44*tan(1/2*a)^2*tan(1/2*c)^3 - 2*tan(1/2*a) *tan(1/2*c)^4 + 6*tan(1/2*a)^3 + 12*tan(1/2*a)^2*tan(1/2*c) + 6*tan(1/2*a) *tan(1/2*c)^2 + 3*tan(1/2*a))*log(abs(tan(1/2*b*x)*tan(1/2*a) - 1))/(tan(1 /2*a)^9*tan(1/2*c)^4 - 4*tan(1/2*a)^8*tan(1/2*c)^5 + 6*tan(1/2*a)^7*tan(1/ 2*c)^6 - 4*tan(1/2*a)^6*tan(1/2*c)^7 + tan(1/2*a)^5*tan(1/2*c)^8 + 4*tan(1 /2*a)^8*tan(1/2*c)^3 - 16*tan(1/2*a)^7*tan(1/2*c)^4 + 24*tan(1/2*a)^6*tan( 1/2*c)^5 - 16*tan(1/2*a)^5*tan(1/2*c)^6 + 4*tan(1/2*a)^4*tan(1/2*c)^7 + 6* tan(1/2*a)^7*tan(1/2*c)^2 - 24*tan(1/2*a)^6*tan(1/2*c)^3 + 36*tan(1/2*a...
Time = 49.98 (sec) , antiderivative size = 74632, normalized size of antiderivative = 74632.00 \[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\text {Too large to display} \] Input:
int(cot(c + b*x)^2/sin(a + b*x)^3,x)
Output:
((6*sin(a)^4 - cos(a)^4*cot(c)^2 + 5*cos(a)^2*sin(a)^2 + 8*cot(c)^2*sin(a) ^4 + 2*cot(c)^4*sin(a)^4 + 5*cos(a)*cot(c)*sin(a)^3 + 2*cos(a)^3*cot(c)*si n(a) + 4*cos(a)*cot(c)^3*sin(a)^3 + 5*cos(a)^3*cot(c)^3*sin(a) + 10*cos(a) ^2*cot(c)^2*sin(a)^2 + 2*cos(a)^2*cot(c)^4*sin(a)^2)/((cos(a)^2 + sin(a)^2 )*(cos(a)^3 - cot(c)^3*sin(a)^3 - 3*cos(a)^2*cot(c)*sin(a) + 3*cos(a)*cot( c)^2*sin(a)^2)) - (tan((b*x)/2)^4*(3*cos(a)^2*sin(a)^4 - 2*cos(a)^6*cot(c) ^2 - 6*sin(a)^6 + 6*cos(a)^4*sin(a)^2 - 6*cot(c)^2*sin(a)^6 - 2*cot(c)^4*s in(a)^6 + 9*cos(a)*cot(c)*sin(a)^5 + 4*cos(a)^5*cot(c)*sin(a) + 20*cos(a)^ 3*cot(c)*sin(a)^3 + 24*cos(a)*cot(c)^3*sin(a)^5 + 6*cos(a)^5*cot(c)^3*sin( a) + 8*cos(a)*cot(c)^5*sin(a)^5 + 6*cos(a)^2*cot(c)^2*sin(a)^4 + 13*cos(a) ^4*cot(c)^2*sin(a)^2 + 33*cos(a)^3*cot(c)^3*sin(a)^3 + 8*cos(a)^2*cot(c)^4 *sin(a)^4 + 12*cos(a)^4*cot(c)^4*sin(a)^2 + 8*cos(a)^3*cot(c)^5*sin(a)^3)) /(sin(a)^2*(cos(a)^2 + sin(a)^2)*(cos(a)^3 - cot(c)^3*sin(a)^3 - 3*cos(a)^ 2*cot(c)*sin(a) + 3*cos(a)*cot(c)^2*sin(a)^2)) + (2*tan((b*x)/2)^2*(3*cos( a)^2*sin(a)^4 - cos(a)^6*cot(c)^2 - 6*sin(a)^6 + 7*cos(a)^4*sin(a)^2 - 9*c ot(c)^2*sin(a)^6 - 2*cot(c)^4*sin(a)^6 + 12*cos(a)*cot(c)*sin(a)^5 + 2*cos (a)^5*cot(c)*sin(a) + 17*cos(a)^3*cot(c)*sin(a)^3 + 14*cos(a)*cot(c)^3*sin (a)^5 + 3*cos(a)^5*cot(c)^3*sin(a) + 4*cos(a)*cot(c)^5*sin(a)^5 + 18*cos(a )^2*cot(c)^2*sin(a)^4 + 23*cos(a)^4*cot(c)^2*sin(a)^2 + 24*cos(a)^3*cot(c) ^3*sin(a)^3 + 13*cos(a)^2*cot(c)^4*sin(a)^4 + 18*cos(a)^4*cot(c)^4*sin(...
\[ \int \cot ^2(c+b x) \csc ^3(a+b x) \, dx=\int \cot \left (b x +c \right )^{2} \csc \left (b x +a \right )^{3}d x \] Input:
int(cot(b*x+c)^2*csc(b*x+a)^3,x)
Output:
int(cot(b*x + c)**2*csc(a + b*x)**3,x)