Integrand size = 17, antiderivative size = 1 \[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.61 (sec) , antiderivative size = 381, normalized size of antiderivative = 381.00 \[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=-\frac {\csc ^2\left (\frac {c}{2}+\frac {b x}{2}\right ) \sec ^2(a-c)}{8 b}+\frac {\left (\cos \left (a-c-\frac {b x}{2}\right )-\cos \left (a-c+\frac {b x}{2}\right )\right ) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {b x}{2}\right ) \sec ^3(a-c)}{2 b}+\frac {3 (-3+\cos (2 a-2 c)) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )\right ) \sec ^4(a-c)}{4 b}-\frac {3 (-3+\cos (2 a-2 c)) \log \left (\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right ) \sec ^4(a-c)}{4 b}+\frac {\left (-\cos \left (a-c-\frac {b x}{2}\right )+\cos \left (a-c+\frac {b x}{2}\right )\right ) \sec ^3(a-c) \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {b x}{2}\right )}{2 b}+\frac {\sec ^2(a-c) \sec ^2\left (\frac {c}{2}+\frac {b x}{2}\right )}{8 b}+\frac {\sec ^3(a-c) \sec (a+b x)}{b}-\frac {6 i \arctan \left (\frac {(i \cos (a)+\sin (a)) \left (\cos \left (\frac {b x}{2}\right ) \sin (a)+\cos (a) \sin \left (\frac {b x}{2}\right )\right )}{\cos (a) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (a)}\right ) \sin (a-c)}{\frac {3 b}{8}+\frac {1}{8} b \cos (4 a-4 c)+\frac {1}{2} b \cos (2 a-2 c)} \] Input:
Integrate[Csc[c + b*x]^3*Sec[a + b*x]^2,x]
Output:
-1/8*(Csc[c/2 + (b*x)/2]^2*Sec[a - c]^2)/b + ((Cos[a - c - (b*x)/2] - Cos[ a - c + (b*x)/2])*Csc[c/2]*Csc[c/2 + (b*x)/2]*Sec[a - c]^3)/(2*b) + (3*(-3 + Cos[2*a - 2*c])*Log[Cos[c/2 + (b*x)/2]]*Sec[a - c]^4)/(4*b) - (3*(-3 + Cos[2*a - 2*c])*Log[Sin[c/2 + (b*x)/2]]*Sec[a - c]^4)/(4*b) + ((-Cos[a - c - (b*x)/2] + Cos[a - c + (b*x)/2])*Sec[a - c]^3*Sec[c/2]*Sec[c/2 + (b*x)/ 2])/(2*b) + (Sec[a - c]^2*Sec[c/2 + (b*x)/2]^2)/(8*b) + (Sec[a - c]^3*Sec[ a + b*x])/b - ((6*I)*ArcTan[((I*Cos[a] + Sin[a])*(Cos[(b*x)/2]*Sin[a] + Co s[a]*Sin[(b*x)/2]))/(Cos[a]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[a])]*Sin[a - c])/((3*b)/8 + (b*Cos[4*a - 4*c])/8 + (b*Cos[2*a - 2*c])/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.
\(\displaystyle \int \sec ^2(a+b x) \csc ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sec ^2(a+b x) \csc ^3(b x+c)dx\) |
Input:
Int[Csc[c + b*x]^3*Sec[a + b*x]^2,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.50 (sec) , antiderivative size = 934, normalized size of antiderivative = 934.00
method | result | size |
risch | \(\text {Expression too large to display}\) | \(934\) |
default | \(\text {Expression too large to display}\) | \(1072\) |
Input:
int(csc(b*x+c)^3*sec(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
4/(exp(2*I*(b*x+a))+1)/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^2/(exp(2*I*a)+exp( 2*I*c))^3/b*(-3*exp(5*I*(b*x+2*a+c))+9*exp(I*(5*b*x+8*a+7*c))+5*exp(I*(3*b *x+10*a+3*c))-14*exp(I*(3*b*x+8*a+5*c))+5*exp(I*(3*b*x+6*a+7*c))+9*exp(I*( b*x+8*a+3*c))-3*exp(I*(b*x+6*a+5*c)))-24*I*ln(exp(I*(b*x+a))+I)/(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*exp (I*(5*a+3*c))+24*I*ln(exp(I*(b*x+a))+I)/(exp(8*I*a)+4*exp(2*I*(3*a+c))+6*e xp(4*I*(a+c))+4*exp(2*I*(a+3*c))+exp(8*I*c))/b*exp(I*(3*a+5*c))+24*I*ln(ex p(I*(b*x+a))-I)/(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*exp(I*(5*a+3*c))-24*I*ln(exp(I*(b*x+a))-I)/(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*e xp(I*(3*a+5*c))-6*ln(exp(I*(b*x+a))-exp(I*(a-c)))/(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*exp(2*I*(3*a+c))+ 36*ln(exp(I*(b*x+a))-exp(I*(a-c)))/(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*exp(4*I*(a+c))-6*ln(exp(I*(b*x+a ))-exp(I*(a-c)))/(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*exp(2*I*(a+3*c))+6*ln(exp(I*(b*x+a))+exp(I*(a-c))) /(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*exp(2*I*(3*a+c))-36*ln(exp(I*(b*x+a))+exp(I*(a-c)))/(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*exp(4*I *(a+c))+6*ln(exp(I*(b*x+a))+exp(I*(a-c)))/(exp(8*I*a)+4*exp(2*I*(3*a+c)...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.14 (sec) , antiderivative size = 617, normalized size of antiderivative = 617.00 \[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sec(b*x+a)^2,x, algorithm="fricas")
Output:
-1/4*(6*cos(b*x + c)*cos(-a + c)^2*sin(b*x + c)*sin(-a + c) + 6*(cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c)^2 - 8*cos(-a + c)^3 - 6*(((cos(-a + c)^ 2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*sin(b*x + c) - (cos(b*x + c)^3* cos(-a + c) - cos(b*x + c)*cos(-a + c))*sin(-a + c))*log(2*(cos(-a + c)*si n(b*x + c) - cos(b*x + c)*sin(-a + c) + 1)/(cos(-a + c) + 1)) + 6*(((cos(- a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*sin(b*x + c) - (cos(b*x + c)^3*cos(-a + c) - cos(b*x + c)*cos(-a + 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)^3 - 2*cos(-a + c))*cos(b*x + c)^3 + ((cos(-a + c)^2 - 2)*cos (b*x + c)^2 - cos(-a + c)^2 + 2)*sin(b*x + c)*sin(-a + c) - (cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c))*log(1/2*cos(b*x + c) + 1/2) + 3*((cos(-a + c)^3 - 2*cos(-a + c))*cos(b*x + c)^3 + ((cos(-a + c)^2 - 2)*cos(b*x + c)^ 2 - cos(-a + c)^2 + 2)*sin(b*x + c)*sin(-a + c) - (cos(-a + c)^3 - 2*cos(- a + c))*cos(b*x + c))*log(-1/2*cos(b*x + c) + 1/2) + 12*cos(-a + c))/(b*co s(b*x + c)^3*cos(-a + c)^5 - b*cos(b*x + c)*cos(-a + c)^5 + (b*cos(b*x + c )^2*cos(-a + c)^4 - b*cos(-a + c)^4)*sin(b*x + c)*sin(-a + c))
\[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=\int \csc ^{3}{\left (b x + c \right )} \sec ^{2}{\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+c)**3*sec(b*x+a)**2,x)
Output:
Integral(csc(b*x + c)**3*sec(a + b*x)**2, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 22.86 (sec) , antiderivative size = 643576, normalized size of antiderivative = 643576.00 \[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sec(b*x+a)^2,x, algorithm="maxima")
Output:
(24*(((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 + 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 + 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 + 4*a + 8*c)^2 + ((cos(8*a) + 4*cos(6*a + 2*c) + cos(8*c))*cos(5*a + 3*c...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.77 (sec) , antiderivative size = 13498, normalized size of antiderivative = 13498.00 \[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^3*sec(b*x+a)^2,x, algorithm="giac")
Output:
1/8*(48*(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...
Timed out. \[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)^2*sin(c + b*x)^3),x)
Output:
\text{Hanged}
\[ \int \csc ^3(c+b x) \sec ^2(a+b x) \, dx=\int \csc \left (b x +c \right )^{3} \sec \left (b x +a \right )^{2}d x \] Input:
int(csc(b*x+c)^3*sec(b*x+a)^2,x)
Output:
int(csc(b*x + c)**3*sec(a + b*x)**2,x)