Integrand size = 17, antiderivative size = 1 \[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.60 (sec) , antiderivative size = 491, normalized size of antiderivative = 491.00 \[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=-\frac {\csc (c+b x) \sec ^3(a-c)}{b}+\frac {3 (-3+\cos (2 a-2 c)) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sec ^4(a-c)}{4 b}-\frac {3 (-3+\cos (2 a-2 c)) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sec ^4(a-c)}{4 b}-\frac {6 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{\frac {3 b}{8}+\frac {1}{8} b \cos (4 a-4 c)+\frac {1}{2} b \cos (2 a-2 c)}+\frac {\sec ^2(a-c)}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {\left (\cos \left (a-c-\frac {b x}{2}\right )-\cos \left (a-c+\frac {b x}{2}\right )\right ) \sec ^3(a-c)}{b \left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}-\frac {\sec ^2(a-c)}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {\left (-\cos \left (a-c-\frac {b x}{2}\right )+\cos \left (a-c+\frac {b x}{2}\right )\right ) \sec ^3(a-c)}{b \left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )} \] Input:
Integrate[Csc[c + b*x]^2*Sec[a + b*x]^3,x]
Output:
-((Csc[c + b*x]*Sec[a - c]^3)/b) + (3*(-3 + Cos[2*a - 2*c])*Log[Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2]]*Sec[a - c]^4)/(4*b) - (3*(-3 + Cos[2*a - 2* c])*Log[Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]]*Sec[a - c]^4)/(4*b) - ((6 *I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2] ))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/((3*b)/8 + ( b*Cos[4*a - 4*c])/8 + (b*Cos[2*a - 2*c])/2) + Sec[a - c]^2/(4*b*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])^2) + ((Cos[a - c - (b*x)/2] - Cos[a - c + (b*x)/2])*Sec[a - c]^3)/(b*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin [a/2 + (b*x)/2])) - Sec[a - c]^2/(4*b*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x )/2])^2) + ((-Cos[a - c - (b*x)/2] + Cos[a - c + (b*x)/2])*Sec[a - c]^3)/( b*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/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.
\(\displaystyle \int \sec ^3(a+b x) \csc ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sec ^3(a+b x) \csc ^2(b x+c)dx\) |
Input:
Int[Csc[c + b*x]^2*Sec[a + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.13 (sec) , antiderivative size = 858, normalized size of antiderivative = 858.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(858\) |
risch | \(\text {Expression too large to display}\) | \(922\) |
Input:
int(csc(b*x+c)^2*sec(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(2/(cos(a)*cos(c)+sin(a)*sin(c))^4*(((sin(a)^4*cos(c)^2*sin(c)^2+sin(a )^4*sin(c)^4+2*cos(a)*sin(a)^3*cos(c)^3*sin(c)+2*cos(a)*sin(a)^3*cos(c)*si n(c)^3+cos(a)^2*sin(a)^2*cos(c)^4+2*cos(a)^2*sin(a)^2*cos(c)^2*sin(c)^2+co s(a)^2*sin(a)^2*sin(c)^4+2*cos(a)^3*cos(c)^3*sin(a)*sin(c)+2*cos(a)^3*sin( a)*cos(c)*sin(c)^3+cos(a)^4*cos(c)^4+cos(a)^4*cos(c)^2*sin(c)^2)/(sin(a)*c os(c)-cos(a)*sin(c))*tan(1/2*a+1/2*b*x)-(cos(a)^2*cos(c)^2+sin(c)^2*cos(a) ^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)*(cos(a)*cos(c)+sin(a)*sin(c)))/(co s(c)*sin(a)*tan(1/2*a+1/2*b*x)^2-sin(c)*cos(a)*tan(1/2*a+1/2*b*x)^2+2*tan( 1/2*a+1/2*b*x)*cos(a)*cos(c)+2*tan(1/2*a+1/2*b*x)*sin(a)*sin(c)-sin(a)*cos (c)+cos(a)*sin(c))+3*(cos(c)^3*sin(a)^3+cos(c)*sin(c)^2*sin(a)^3-cos(c)^2* sin(c)*sin(a)^2*cos(a)-sin(c)^3*sin(a)^2*cos(a)+cos(c)^3*sin(a)*cos(a)^2+c os(c)*sin(c)^2*sin(a)*cos(a)^2-cos(c)^2*sin(c)*cos(a)^3-sin(c)^3*cos(a)^3) /(-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*c os(a)*cos(c)+2*sin(a)*sin(c))/(-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)))+1/2/(cos(a)*cos(c)+sin(a)*sin(c))^2 /(tan(1/2*a+1/2*b*x)-1)^2-1/2*(4*sin(a)*cos(c)-sin(a)*sin(c)-cos(a)*cos(c) -4*cos(a)*sin(c))/(cos(a)*cos(c)+sin(a)*sin(c))^3/(tan(1/2*a+1/2*b*x)-1)+1 /2/(cos(a)*cos(c)+sin(a)*sin(c))^4*(-6*cos(c)^2*sin(a)^2-3*sin(a)^2*sin(c) ^2+6*cos(a)*cos(c)*sin(a)*sin(c)-3*cos(a)^2*cos(c)^2-6*sin(c)^2*cos(a)^...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.15 (sec) , antiderivative size = 739, normalized size of antiderivative = 739.00 \[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^2*sec(b*x+a)^3,x, algorithm="fricas")
Output:
1/4*(18*cos(b*x + c)*cos(-a + c)^2*sin(b*x + c)*sin(-a + c) + 6*(3*cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c)^2 - 14*cos(-a + c)^3 - 3*((cos(-a + c )^4 - (2*cos(-a + c)^4 - 5*cos(-a + c)^2 + 2)*cos(b*x + c)^2 - 3*cos(-a + c)^2 + 2)*sin(b*x + c) + 2*((cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c)^3 - (cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c))*sin(-a + c))*log(2*(cos(- a + c)*sin(b*x + c) - cos(b*x + c)*sin(-a + c) + 1)/(cos(-a + c) + 1)) + 3 *((cos(-a + c)^4 - (2*cos(-a + c)^4 - 5*cos(-a + c)^2 + 2)*cos(b*x + c)^2 - 3*cos(-a + c)^2 + 2)*sin(b*x + c) + 2*((cos(-a + c)^3 - 2*cos(-a + c))*c os(b*x + c)^3 - (cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c))*sin(-a + c)) *log(-2*(cos(-a + c)*sin(b*x + c) - cos(b*x + c)*sin(-a + c) - 1)/(cos(-a + c) + 1)) - 6*(2*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)^3 + ((2*cos(- a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*sin(b*x + c)*sin(-a + c) - 2*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c))*log(1/2*cos(b*x + c) + 1/ 2) + 6*(2*(cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)^3 + ((2*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*sin(b*x + c)*sin(-a + c) - 2*(co s(-a + c)^3 - cos(-a + c))*cos(b*x + c))*log(-1/2*cos(b*x + c) + 1/2) + 12 *cos(-a + c))/((b*cos(-a + c)^6 - b*cos(-a + c)^4 - (2*b*cos(-a + c)^6 - b *cos(-a + c)^4)*cos(b*x + c)^2)*sin(b*x + c) + 2*(b*cos(b*x + c)^3*cos(-a + c)^5 - b*cos(b*x + c)*cos(-a + c)^5)*sin(-a + c))
\[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=\int \csc ^{2}{\left (b x + c \right )} \sec ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+c)**2*sec(b*x+a)**3,x)
Output:
Integral(csc(b*x + c)**2*sec(a + b*x)**3, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 20.64 (sec) , antiderivative size = 660194, normalized size of antiderivative = 660194.00 \[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^2*sec(b*x+a)^3,x, algorithm="maxima")
Output:
-(6*(((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))*c os(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) + c os(8*c))*sin(2*a + 6*c))*cos(6*b*x + 10*a + 2*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 + 8*a + 4*c)^2 + 9*((sin(8*a) + sin(8*c))*cos(6*a + 2*c) - 6*(s in(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*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 + 4*a + 8*c)^2 + ((sin(8*a) + sin(8*c))*c os(6*a + 2*c) - 6*(sin(8*a) + 5*sin(6*a + 2*c) + sin(8*c))*cos(4*a + 4*...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.03 (sec) , antiderivative size = 14520, normalized size of antiderivative = 14520.00 \[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^2*sec(b*x+a)^3,x, algorithm="giac")
Output:
1/2*(3*(tan(1/2*a)^9*tan(1/2*c)^9 - tan(1/2*a)^9*tan(1/2*c)^8 + tan(1/2*a) ^8*tan(1/2*c)^9 + 8*tan(1/2*a)^9*tan(1/2*c)^7 - 7*tan(1/2*a)^8*tan(1/2*c)^ 8 + 8*tan(1/2*a)^7*tan(1/2*c)^9 - 8*tan(1/2*a)^9*tan(1/2*c)^6 + 16*tan(1/2 *a)^8*tan(1/2*c)^7 - 16*tan(1/2*a)^7*tan(1/2*c)^8 + 8*tan(1/2*a)^6*tan(1/2 *c)^9 + 14*tan(1/2*a)^9*tan(1/2*c)^5 + 8*tan(1/2*a)^7*tan(1/2*c)^7 + 14*ta n(1/2*a)^5*tan(1/2*c)^9 - 14*tan(1/2*a)^9*tan(1/2*c)^4 + 22*tan(1/2*a)^8*t an(1/2*c)^5 - 24*tan(1/2*a)^7*tan(1/2*c)^6 + 24*tan(1/2*a)^6*tan(1/2*c)^7 - 22*tan(1/2*a)^5*tan(1/2*c)^8 + 14*tan(1/2*a)^4*tan(1/2*c)^9 + 8*tan(1/2* a)^9*tan(1/2*c)^3 + 22*tan(1/2*a)^8*tan(1/2*c)^4 + 8*tan(1/2*a)^7*tan(1/2* c)^5 + 8*tan(1/2*a)^6*tan(1/2*c)^6 + 8*tan(1/2*a)^5*tan(1/2*c)^7 + 22*tan( 1/2*a)^4*tan(1/2*c)^8 + 8*tan(1/2*a)^3*tan(1/2*c)^9 - 8*tan(1/2*a)^9*tan(1 /2*c)^2 - 8*tan(1/2*a)^7*tan(1/2*c)^4 + 24*tan(1/2*a)^6*tan(1/2*c)^5 - 24* tan(1/2*a)^5*tan(1/2*c)^6 + 8*tan(1/2*a)^4*tan(1/2*c)^7 + 8*tan(1/2*a)^2*t an(1/2*c)^9 + tan(1/2*a)^9*tan(1/2*c) + 16*tan(1/2*a)^8*tan(1/2*c)^2 + 24* tan(1/2*a)^7*tan(1/2*c)^3 + 24*tan(1/2*a)^6*tan(1/2*c)^4 - 4*tan(1/2*a)^5* tan(1/2*c)^5 + 24*tan(1/2*a)^4*tan(1/2*c)^6 + 24*tan(1/2*a)^3*tan(1/2*c)^7 + 16*tan(1/2*a)^2*tan(1/2*c)^8 + tan(1/2*a)*tan(1/2*c)^9 - tan(1/2*a)^9 - 7*tan(1/2*a)^8*tan(1/2*c) - 8*tan(1/2*a)^7*tan(1/2*c)^2 + 8*tan(1/2*a)^6* tan(1/2*c)^3 + 4*tan(1/2*a)^5*tan(1/2*c)^4 - 4*tan(1/2*a)^4*tan(1/2*c)^5 - 8*tan(1/2*a)^3*tan(1/2*c)^6 + 8*tan(1/2*a)^2*tan(1/2*c)^7 + 7*tan(1/2*...
Timed out. \[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^3*sin(c + b*x)^2),x)
Output:
\text{Hanged}
\[ \int \csc ^2(c+b x) \sec ^3(a+b x) \, dx=\int \csc \left (b x +c \right )^{2} \sec \left (b x +a \right )^{3}d x \] Input:
int(csc(b*x+c)^2*sec(b*x+a)^3,x)
Output:
int(csc(b*x + c)**2*sec(a + b*x)**3,x)