Integrand size = 17, antiderivative size = 1 \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 1.93 (sec) , antiderivative size = 185, normalized size of antiderivative = 185.00 \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\frac {(-33+16 \cos (2 (a-c))+\cos (4 (a-c))) \left (\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec ^4(a-c)+16 \sec ^3(a-c) \sec (c+b x)-96 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) \sec ^3(a-c) \tan (a-c)-64 \cos \left (a+\frac {b x}{2}\right ) \csc (a) \csc (a+b x) \sec ^2(a-c) \sin \left (\frac {b x}{2}\right ) \tan (a-c)+2 \left (-\csc ^2\left (\frac {1}{2} (a+b x)\right )+\sec ^2\left (\frac {1}{2} (a+b x)\right )\right ) \tan ^2(a-c)}{16 b} \] Input:
Integrate[Csc[a + b*x]^3*Tan[c + b*x]^2,x]
Output:
((-33 + 16*Cos[2*(a - c)] + Cos[4*(a - c)])*(Log[Cos[(a + b*x)/2]] - Log[S in[(a + b*x)/2]])*Sec[a - c]^4 + 16*Sec[a - c]^3*Sec[c + b*x] - 96*ArcTanh [Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sec[a - c]^3*Tan[a - c] - 64*Cos[a + (b*x)/ 2]*Csc[a]*Csc[a + b*x]*Sec[a - c]^2*Sin[(b*x)/2]*Tan[a - c] + 2*(-Csc[(a + b*x)/2]^2 + Sec[(a + b*x)/2]^2)*Tan[a - c]^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.
\(\displaystyle \int \csc ^3(a+b x) \tan ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \csc ^3(a+b x) \tan ^2(b x+c)dx\) |
Input:
Int[Csc[a + b*x]^3*Tan[c + b*x]^2,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.59 (sec) , antiderivative size = 1267, normalized size of antiderivative = 1267.00
Input:
int(csc(b*x+a)^3*tan(b*x+c)^2,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)))+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(e xp(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)*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...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.15 (sec) , antiderivative size = 892, normalized size of antiderivative = 892.00 \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*tan(b*x+c)^2,x, algorithm="fricas")
Output:
1/4*(2*(cos(-2*a + 2*c)^2 + 8*cos(-2*a + 2*c) + 7)*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - 2*(cos(-2*a + 2*c)^3 + 9*cos(-2*a + 2*c)^2 - 9*cos(-2 *a + 2*c) - 17)*cos(b*x + a)^2 + 16*cos(-2*a + 2*c)^2 + 12*sqrt(2)*(((cos( -2*a + 2*c)^2 - 1)*cos(b*x + a)^2 - cos(-2*a + 2*c)^2 + 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)*sin(b*x + a) + cos(b*x + a)*sin(-2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - cos(-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 + 9*cos(-2*a + 2*c)^2 - 9*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)^2 - cos(-2 *a + 2*c)^2 - 8*cos(-2*a + 2*c) + 17)*sin(b*x + a)*sin(-2*a + 2*c) - (cos( -2*a + 2*c)^3 + 9*cos(-2*a + 2*c)^2 - 9*cos(-2*a + 2*c) - 17)*cos(b*x + a) )*log(1/2*cos(b*x + a) + 1/2) - ((cos(-2*a + 2*c)^3 + 9*cos(-2*a + 2*c)^2 - 9*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)^2 - cos(-2*a + 2*c)^2 - 8*cos(-2*a + 2*c) + 17 )*sin(b*x + a)*sin(-2*a + 2*c) - (cos(-2*a + 2*c)^3 + 9*cos(-2*a + 2*c)^2 - 9*cos(-2*a + 2*c) - 17)*cos(b*x + a))*log(-1/2*cos(b*x + a) + 1/2) - 16* cos(-2*a + 2*c) - 32)/((b*cos(-2*a + 2*c)^3 + 3*b*cos(-2*a + 2*c)^2 + 3...
Timed out. \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)**3*tan(b*x+c)**2,x)
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 72.45 (sec) , antiderivative size = 752466, normalized size of antiderivative = 752466.00 \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*tan(b*x+c)^2,x, algorithm="maxima")
Output:
-1/4*(96*(((cos(8*a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c) + 6*cos (5*a + 3*c)*cos(4*a + 4*c) - (cos(8*a) + 4*cos(6*a + 2*c) + 6*cos(4*a + 4* c) + cos(8*c))*cos(3*a + 5*c) + 4*(cos(5*a + 3*c) - cos(3*a + 5*c))*cos(2* a + 6*c) + (sin(8*a) + 4*sin(6*a + 2*c) + sin(8*c))*sin(5*a + 3*c) + 6*sin (5*a + 3*c)*sin(4*a + 4*c) - (sin(8*a) + 4*sin(6*a + 2*c) + 6*sin(4*a + 4* c) + sin(8*c))*sin(3*a + 5*c) + 4*(sin(5*a + 3*c) - sin(3*a + 5*c))*sin(2* a + 6*c))*cos(6*b*x + 10*a + 2*c)^2 + 9*((cos(8*a) + 4*cos(6*a + 2*c) + co s(8*c))*cos(5*a + 3*c) + 6*cos(5*a + 3*c)*cos(4*a + 4*c) - (cos(8*a) + 4*c os(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))*cos(3*a + 5*c) + 4*(cos(5*a + 3*c) - cos(3*a + 5*c))*cos(2*a + 6*c) + (sin(8*a) + 4*sin(6*a + 2*c) + si n(8*c))*sin(5*a + 3*c) + 6*sin(5*a + 3*c)*sin(4*a + 4*c) - (sin(8*a) + 4*s in(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))*sin(3*a + 5*c) + 4*(sin(5*a + 3*c) - sin(3*a + 5*c))*sin(2*a + 6*c))*cos(6*b*x + 8*a + 4*c)^2 + 9*((cos (8*a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c) + 6*cos(5*a + 3*c)*cos (4*a + 4*c) - (cos(8*a) + 4*cos(6*a + 2*c) + 6*cos(4*a + 4*c) + cos(8*c))* cos(3*a + 5*c) + 4*(cos(5*a + 3*c) - cos(3*a + 5*c))*cos(2*a + 6*c) + (sin (8*a) + 4*sin(6*a + 2*c) + sin(8*c))*sin(5*a + 3*c) + 6*sin(5*a + 3*c)*sin (4*a + 4*c) - (sin(8*a) + 4*sin(6*a + 2*c) + 6*sin(4*a + 4*c) + sin(8*c))* sin(3*a + 5*c) + 4*(sin(5*a + 3*c) - sin(3*a + 5*c))*sin(2*a + 6*c))*cos(6 *b*x + 6*a + 6*c)^2 + ((cos(8*a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.16 (sec) , antiderivative size = 13648, normalized size of antiderivative = 13648.00 \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*tan(b*x+c)^2,x, algorithm="giac")
Output:
1/2*(12*(tan(1/2*a)^9*tan(1/2*c)^8 - tan(1/2*a)^8*tan(1/2*c)^9 - tan(1/2*a )^9*tan(1/2*c)^7 + 2*tan(1/2*a)^8*tan(1/2*c)^8 - tan(1/2*a)^7*tan(1/2*c)^9 + 3*tan(1/2*a)^9*tan(1/2*c)^6 - tan(1/2*a)^8*tan(1/2*c)^7 + tan(1/2*a)^7* tan(1/2*c)^8 - 3*tan(1/2*a)^6*tan(1/2*c)^9 - 3*tan(1/2*a)^9*tan(1/2*c)^5 + 5*tan(1/2*a)^8*tan(1/2*c)^6 - 4*tan(1/2*a)^7*tan(1/2*c)^7 + 5*tan(1/2*a)^ 6*tan(1/2*c)^8 - 3*tan(1/2*a)^5*tan(1/2*c)^9 + 3*tan(1/2*a)^9*tan(1/2*c)^4 + 3*tan(1/2*a)^8*tan(1/2*c)^5 + 4*tan(1/2*a)^7*tan(1/2*c)^6 - 4*tan(1/2*a )^6*tan(1/2*c)^7 - 3*tan(1/2*a)^5*tan(1/2*c)^8 - 3*tan(1/2*a)^4*tan(1/2*c) ^9 - 3*tan(1/2*a)^9*tan(1/2*c)^3 + 3*tan(1/2*a)^8*tan(1/2*c)^4 - 6*tan(1/2 *a)^7*tan(1/2*c)^5 + 12*tan(1/2*a)^6*tan(1/2*c)^6 - 6*tan(1/2*a)^5*tan(1/2 *c)^7 + 3*tan(1/2*a)^4*tan(1/2*c)^8 - 3*tan(1/2*a)^3*tan(1/2*c)^9 + tan(1/ 2*a)^9*tan(1/2*c)^2 + 5*tan(1/2*a)^8*tan(1/2*c)^3 + 6*tan(1/2*a)^7*tan(1/2 *c)^4 + 6*tan(1/2*a)^6*tan(1/2*c)^5 - 6*tan(1/2*a)^5*tan(1/2*c)^6 - 6*tan( 1/2*a)^4*tan(1/2*c)^7 - 5*tan(1/2*a)^3*tan(1/2*c)^8 - tan(1/2*a)^2*tan(1/2 *c)^9 - tan(1/2*a)^9*tan(1/2*c) - tan(1/2*a)^8*tan(1/2*c)^2 - 4*tan(1/2*a) ^7*tan(1/2*c)^3 + 6*tan(1/2*a)^6*tan(1/2*c)^4 + 6*tan(1/2*a)^4*tan(1/2*c)^ 6 - 4*tan(1/2*a)^3*tan(1/2*c)^7 - tan(1/2*a)^2*tan(1/2*c)^8 - tan(1/2*a)*t an(1/2*c)^9 + 2*tan(1/2*a)^8*tan(1/2*c) + 4*tan(1/2*a)^7*tan(1/2*c)^2 + 12 *tan(1/2*a)^6*tan(1/2*c)^3 - 12*tan(1/2*a)^3*tan(1/2*c)^6 - 4*tan(1/2*a)^2 *tan(1/2*c)^7 - 2*tan(1/2*a)*tan(1/2*c)^8 - tan(1/2*a)^8 - tan(1/2*a)^7...
Time = 50.20 (sec) , antiderivative size = 74535, normalized size of antiderivative = 74535.00 \[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\text {Too large to display} \] Input:
int(tan(c + b*x)^2/sin(a + b*x)^3,x)
Output:
((tan((b*x)/2)*(9*sin(a)^5*tan(c) - 21*cos(a)*sin(a)^4 - 18*cos(a)^3*sin(a )^2 + 2*cos(a)^5*tan(c)^2 + 12*sin(a)^5*tan(c)^3 + 2*sin(a)^5*tan(c)^5 + 8 *cos(a)^4*sin(a)*tan(c) + 24*cos(a)^2*sin(a)^3*tan(c) - 36*cos(a)*sin(a)^4 *tan(c)^2 + 12*cos(a)^4*sin(a)*tan(c)^3 - 16*cos(a)*sin(a)^4*tan(c)^4 - 37 *cos(a)^3*sin(a)^2*tan(c)^2 + 27*cos(a)^2*sin(a)^3*tan(c)^3 - 18*cos(a)^3* sin(a)^2*tan(c)^4 + 2*cos(a)^2*sin(a)^3*tan(c)^5))/(sin(a)*(cos(a)^2 + sin (a)^2)*(cos(a)^3 + sin(a)^3*tan(c)^3 + 3*cos(a)^2*sin(a)*tan(c) + 3*cos(a) *sin(a)^2*tan(c)^2)) - (6*sin(a)^4 + 5*cos(a)^2*sin(a)^2 - cos(a)^4*tan(c) ^2 + 8*sin(a)^4*tan(c)^2 + 2*sin(a)^4*tan(c)^4 - 5*cos(a)*sin(a)^3*tan(c) - 2*cos(a)^3*sin(a)*tan(c) - 4*cos(a)*sin(a)^3*tan(c)^3 - 5*cos(a)^3*sin(a )*tan(c)^3 + 10*cos(a)^2*sin(a)^2*tan(c)^2 + 2*cos(a)^2*sin(a)^2*tan(c)^4) /((cos(a)^2 + sin(a)^2)*(cos(a)^3 + sin(a)^3*tan(c)^3 + 3*cos(a)^2*sin(a)* tan(c) + 3*cos(a)*sin(a)^2*tan(c)^2)) + (tan((b*x)/2)^5*(3*sin(a)^5*tan(c) - 3*cos(a)*sin(a)^4 - 2*cos(a)^3*sin(a)^2 + 2*cos(a)^5*tan(c)^2 + 4*sin(a )^5*tan(c)^3 + 2*sin(a)^5*tan(c)^5 + 6*cos(a)^2*sin(a)^3*tan(c) - 6*cos(a) *sin(a)^4*tan(c)^2 + 6*cos(a)^4*sin(a)*tan(c)^3 - 7*cos(a)^3*sin(a)^2*tan( c)^2 + 9*cos(a)^2*sin(a)^3*tan(c)^3 + 2*cos(a)^2*sin(a)^3*tan(c)^5))/(sin( a)*(cos(a)^2 + sin(a)^2)*(cos(a)^3 + sin(a)^3*tan(c)^3 + 3*cos(a)^2*sin(a) *tan(c) + 3*cos(a)*sin(a)^2*tan(c)^2)) - (tan((b*x)/2)^4*(6*sin(a)^6 - 3*c os(a)^2*sin(a)^4 - 6*cos(a)^4*sin(a)^2 + 2*cos(a)^6*tan(c)^2 + 6*sin(a)...
\[ \int \csc ^3(a+b x) \tan ^2(c+b x) \, dx=\int \csc \left (b x +a \right )^{3} \tan \left (b x +c \right )^{2}d x \] Input:
int(csc(b*x+a)^3*tan(b*x+c)^2,x)
Output:
int(csc(a + b*x)**3*tan(b*x + c)**2,x)