\(\int \csc ^2(c+b x) \sec ^2(a+b x) \, dx\) [177]

Optimal result

Integrand size = 17, antiderivative size = 1 \[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=0 \] Output:

0
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 0.86 (sec) , antiderivative size = 64, normalized size of antiderivative = 64.00 \[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=\frac {\sec ^2(a-c) (\csc (c) \csc (c+b x) \sin (b x)+\sec (a) \sec (a+b x) \sin (b x)+2 (-\log (\cos (a+b x))+\log (\sin (c+b x))) \tan (a-c))}{b} \] Input:

Integrate[Csc[c + b*x]^2*Sec[a + b*x]^2,x]
 

Output:

(Sec[a - c]^2*(Csc[c]*Csc[c + b*x]*Sin[b*x] + Sec[a]*Sec[a + b*x]*Sin[b*x] 
 + 2*(-Log[Cos[a + b*x]] + Log[Sin[c + b*x]])*Tan[a - c]))/b
 

Rubi [F]

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.

\(\displaystyle \int \sec ^2(a+b x) \csc ^2(b x+c) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sec ^2(a+b x) \csc ^2(b x+c)dx\)

Input:

Int[Csc[c + b*x]^2*Sec[a + b*x]^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.79 (sec) , antiderivative size = 176, normalized size of antiderivative = 176.00

method	result	size
default	\(\frac {\frac {\tan \left (b x +a \right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}+\frac {\left (-2 \cos \left (a \right ) \sin \left (c \right )+2 \sin \left (a \right ) \cos \left (c \right )\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3}}}{b}\)	\(176\)
risch	\(\frac {8 i \left (-{\mathrm e}^{2 i \left (b x +3 a +c \right )}+{\mathrm e}^{2 i \left (b x +2 a +2 c \right )}-2 \,{\mathrm e}^{2 i \left (2 a +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) \left ({\mathrm e}^{2 i \left (b x +a +c \right )}-{\mathrm e}^{2 i a}\right ) \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{2} b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}\)	\(362\)

Input:

int(csc(b*x+c)^2*sec(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b*(tan(b*x+a)/(cos(a)*cos(c)+sin(a)*sin(c))^2-1/(cos(a)*cos(c)+sin(a)*si 
n(c))^3*(cos(a)^2*cos(c)^2+sin(c)^2*cos(a)^2+cos(c)^2*sin(a)^2+sin(a)^2*si 
n(c)^2)/(tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)-sin(a)*cos(c)+c 
os(a)*sin(c))+(-2*cos(a)*sin(c)+2*sin(a)*cos(c))/(cos(a)*cos(c)+sin(a)*sin 
(c))^3*ln(tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)-sin(a)*cos(c)+ 
cos(a)*sin(c)))
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.10 (sec) , antiderivative size = 330, normalized size of antiderivative = 330.00 \[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=-\frac {2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right )^{2} + 2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (-a + c\right )^{2} + {\left (\cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (\cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + c\right )^{2} + \frac {1}{4}\right ) - {\left (\cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (\cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )} \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right )}{b \cos \left (b x + c\right ) \cos \left (-a + c\right )^{4} \sin \left (b x + c\right ) - {\left (b \cos \left (b x + c\right )^{2} \cos \left (-a + c\right )^{3} - b \cos \left (-a + c\right )^{3}\right )} \sin \left (-a + c\right )} \] Input:

integrate(csc(b*x+c)^2*sec(b*x+a)^2,x, algorithm="fricas")
 

Output:

-(2*cos(b*x + c)^2*cos(-a + c)^2 + 2*cos(b*x + c)*cos(-a + c)*sin(b*x + c) 
*sin(-a + c) - cos(-a + c)^2 + (cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin( 
-a + c) + (cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*log(-1/4 
*cos(b*x + c)^2 + 1/4) - (cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c 
) + (cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2 + 1)*log(4*(2*cos(b 
*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) + (2*cos(-a + c)^2 - 1)*cos(b 
*x + c)^2 - cos(-a + c)^2 + 1)/(cos(-a + c)^2 + 2*cos(-a + c) + 1)))/(b*co 
s(b*x + c)*cos(-a + c)^4*sin(b*x + c) - (b*cos(b*x + c)^2*cos(-a + c)^3 - 
b*cos(-a + c)^3)*sin(-a + c))
 

Sympy [F]

\[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=\int \csc ^{2}{\left (b x + c \right )} \sec ^{2}{\left (a + b x \right )}\, dx \] Input:

integrate(csc(b*x+c)**2*sec(b*x+a)**2,x)
 

Output:

Integral(csc(b*x + c)**2*sec(a + b*x)**2, x)
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 2.55 (sec) , antiderivative size = 114453, normalized size of antiderivative = 114453.00 \[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)^2*sec(b*x+a)^2,x, algorithm="maxima")
 

Output:

4*(36*((sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(4*a) + cos(4*c))*sin(2* 
a + 2*c))*cos(4*a + 2*c)^2 + 36*((sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (c 
os(4*a) + cos(4*c))*sin(2*a + 2*c))*cos(2*a + 4*c)^2 + 36*((sin(4*a) + sin 
(4*c))*cos(2*a + 2*c) - (cos(4*a) + cos(4*c))*sin(2*a + 2*c))*sin(4*a + 2* 
c)^2 + 36*((sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(4*a) + cos(4*c))*si 
n(2*a + 2*c))*sin(2*a + 4*c)^2 - 2*(((cos(6*a) + cos(6*c))*cos(4*a + 2*c) 
+ 3*cos(4*a + 2*c)^2 - (cos(6*a) + cos(6*c))*cos(2*a + 4*c) - 3*cos(2*a + 
4*c)^2 + (sin(6*a) + sin(6*c))*sin(4*a + 2*c) + 3*sin(4*a + 2*c)^2 - (sin( 
6*a) + sin(6*c))*sin(2*a + 4*c) - 3*sin(2*a + 4*c)^2)*cos(4*b*x + 6*a + 2* 
c)^2 + 4*((cos(6*a) + cos(6*c))*cos(4*a + 2*c) + 3*cos(4*a + 2*c)^2 - (cos 
(6*a) + cos(6*c))*cos(2*a + 4*c) - 3*cos(2*a + 4*c)^2 + (sin(6*a) + sin(6* 
c))*sin(4*a + 2*c) + 3*sin(4*a + 2*c)^2 - (sin(6*a) + sin(6*c))*sin(2*a + 
4*c) - 3*sin(2*a + 4*c)^2)*cos(4*b*x + 4*a + 4*c)^2 + ((cos(6*a) + cos(6*c 
))*cos(4*a + 2*c) + 3*cos(4*a + 2*c)^2 - (cos(6*a) + cos(6*c))*cos(2*a + 4 
*c) - 3*cos(2*a + 4*c)^2 + (sin(6*a) + sin(6*c))*sin(4*a + 2*c) + 3*sin(4* 
a + 2*c)^2 - (sin(6*a) + sin(6*c))*sin(2*a + 4*c) - 3*sin(2*a + 4*c)^2)*co 
s(4*b*x + 2*a + 6*c)^2 + ((cos(6*a) + cos(6*c))*cos(4*a + 2*c) + 3*cos(4*a 
 + 2*c)^2 - (cos(6*a) + cos(6*c))*cos(2*a + 4*c) - 3*cos(2*a + 4*c)^2 + (s 
in(6*a) + sin(6*c))*sin(4*a + 2*c) + 3*sin(4*a + 2*c)^2 - (sin(6*a) + sin( 
6*c))*sin(2*a + 4*c) - 3*sin(2*a + 4*c)^2)*cos(2*b*x + 6*a)^2 + ((cos(6...
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.24 (sec) , antiderivative size = 3227, normalized size of antiderivative = 3227.00 \[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)^2*sec(b*x+a)^2,x, algorithm="giac")
 

Output:

-1/2*(8*(tan(1/2*a)^8*tan(1/2*c)^6 - 2*tan(1/2*a)^7*tan(1/2*c)^7 + tan(1/2 
*a)^6*tan(1/2*c)^8 + 2*tan(1/2*a)^8*tan(1/2*c)^4 - 2*tan(1/2*a)^7*tan(1/2* 
c)^5 - 2*tan(1/2*a)^5*tan(1/2*c)^7 + 2*tan(1/2*a)^4*tan(1/2*c)^8 + tan(1/2 
*a)^8*tan(1/2*c)^2 + 2*tan(1/2*a)^7*tan(1/2*c)^3 - 2*tan(1/2*a)^6*tan(1/2* 
c)^4 - 2*tan(1/2*a)^5*tan(1/2*c)^5 - 2*tan(1/2*a)^4*tan(1/2*c)^6 + 2*tan(1 
/2*a)^3*tan(1/2*c)^7 + tan(1/2*a)^2*tan(1/2*c)^8 + 2*tan(1/2*a)^7*tan(1/2* 
c) + 2*tan(1/2*a)^5*tan(1/2*c)^3 - 8*tan(1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2 
*a)^3*tan(1/2*c)^5 + 2*tan(1/2*a)*tan(1/2*c)^7 + tan(1/2*a)^6 + 2*tan(1/2* 
a)^5*tan(1/2*c) - 2*tan(1/2*a)^4*tan(1/2*c)^2 - 2*tan(1/2*a)^3*tan(1/2*c)^ 
3 - 2*tan(1/2*a)^2*tan(1/2*c)^4 + 2*tan(1/2*a)*tan(1/2*c)^5 + tan(1/2*c)^6 
 + 2*tan(1/2*a)^4 - 2*tan(1/2*a)^3*tan(1/2*c) - 2*tan(1/2*a)*tan(1/2*c)^3 
+ 2*tan(1/2*c)^4 + tan(1/2*a)^2 - 2*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2)* 
log(abs(2*tan(b*x + c)*tan(1/2*a)^2*tan(1/2*c) - 2*tan(b*x + c)*tan(1/2*a) 
*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^2 + 2*tan(b*x + c)*tan(1/2*a) + ta 
n(1/2*a)^2 - 2*tan(b*x + c)*tan(1/2*c) - 4*tan(1/2*a)*tan(1/2*c) + tan(1/2 
*c)^2 - 1))/(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)^8*tan(1/2*c)^5 + 16*tan(1/2*a)^7*tan(1/2*c)^6 - 16*tan(1/2*a)^6*ta 
n(1/2*c)^7 + 3*tan(1/2*a)^5*tan(1/2*c)^8 + 3*tan(1/2*a)^8*tan(1/2*c)^3 - 3 
0*tan(1/2*a)^7*tan(1/2*c)^4 + 96*tan(1/2*a)^6*tan(1/2*c)^5 - 96*tan(1/2*a) 
^5*tan(1/2*c)^6 + 30*tan(1/2*a)^4*tan(1/2*c)^7 - 3*tan(1/2*a)^3*tan(1/2...
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=\text {Hanged} \] Input:

int(1/(cos(a + b*x)^2*sin(c + b*x)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \csc ^2(c+b x) \sec ^2(a+b x) \, dx=\int \csc \left (b x +c \right )^{2} \sec \left (b x +a \right )^{2}d x \] Input:

int(csc(b*x+c)^2*sec(b*x+a)^2,x)
 

Output:

int(csc(b*x + c)**2*sec(a + b*x)**2,x)