\(\int \csc (c+b x) \sec ^3(a+b x) \, dx\) [172]

Optimal result

Integrand size = 15, antiderivative size = 1 \[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=0 \] Output:

0
 

Mathematica [C] (verified)

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

Time = 0.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 151.00 \[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=-\frac {(-2 \cos (3 a-2 c)+\cos (3 a-2 c+2 b x)-\cos (a-2 (c+b x))+2 \cos (a+2 b x) \log (\cos (a+b x))+2 \cos (3 a+2 b x) \log (\cos (a+b x))+\cos (a) (-2+4 \log (\cos (a+b x))-4 \log (\sin (c+b x)))-2 \cos (a+2 b x) \log (\sin (c+b x))-2 \cos (3 a+2 b x) \log (\sin (c+b x))) \sec (a) \sec ^3(a-c) \sec ^2(a+b x)}{8 b} \] Input:

Integrate[Csc[c + b*x]*Sec[a + b*x]^3,x]
 

Output:

-1/8*((-2*Cos[3*a - 2*c] + Cos[3*a - 2*c + 2*b*x] - Cos[a - 2*(c + b*x)] + 
 2*Cos[a + 2*b*x]*Log[Cos[a + b*x]] + 2*Cos[3*a + 2*b*x]*Log[Cos[a + b*x]] 
 + Cos[a]*(-2 + 4*Log[Cos[a + b*x]] - 4*Log[Sin[c + b*x]]) - 2*Cos[a + 2*b 
*x]*Log[Sin[c + b*x]] - 2*Cos[3*a + 2*b*x]*Log[Sin[c + b*x]])*Sec[a]*Sec[a 
 - c]^3*Sec[a + b*x]^2)/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.

\(\displaystyle \int \sec ^3(a+b x) \csc (b x+c) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sec ^3(a+b x) \csc (b x+c)dx\)

Input:

Int[Csc[c + b*x]*Sec[a + b*x]^3,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 = 1.45 (sec) , antiderivative size = 158, normalized size of antiderivative = 158.00

method	result	size
default	\(\frac {\frac {\frac {\tan \left (b x +a \right )^{2} \cos \left (a \right ) \cos \left (c \right )}{2}+\frac {\tan \left (b x +a \right )^{2} \sin \left (a \right ) \sin \left (c \right )}{2}-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}+\frac {\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3}}}{b}\)	\(158\)
risch	\(-\frac {4 \left (-2 \,{\mathrm e}^{i \left (2 b x +5 a +c \right )}-{\mathrm e}^{i \left (3 a +c \right )}+{\mathrm e}^{i \left (a +3 c \right )}\right )}{\left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2} \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{2} b}+\frac {8 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}-\frac {8 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}\)	\(197\)

Input:

int(csc(b*x+c)*sec(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/b*(1/(cos(a)*cos(c)+sin(a)*sin(c))^2*(1/2*tan(b*x+a)^2*cos(a)*cos(c)+1/2 
*tan(b*x+a)^2*sin(a)*sin(c)-tan(b*x+a)*cos(a)*sin(c)+tan(b*x+a)*sin(a)*cos 
(c))+(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))^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 = 392, normalized size of antiderivative = 392.00 \[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=\frac {2 \, \cos \left (-a + c\right )^{4} - 2 \, {\left (2 \, \cos \left (-a + c\right )^{3} - \cos \left (-a + c\right )\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 4 \, {\left (\cos \left (-a + c\right )^{4} - \cos \left (-a + c\right )^{2}\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + c\right )^{2} + \frac {1}{4}\right ) - {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )} \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right )}{2 \, {\left (2 \, b \cos \left (b x + c\right ) \cos \left (-a + c\right )^{4} \sin \left (b x + c\right ) \sin \left (-a + c\right ) - b \cos \left (-a + c\right )^{5} + b \cos \left (-a + c\right )^{3} + {\left (2 \, b \cos \left (-a + c\right )^{5} - b \cos \left (-a + c\right )^{3}\right )} \cos \left (b x + c\right )^{2}\right )}} \] Input:

integrate(csc(b*x+c)*sec(b*x+a)^3,x, algorithm="fricas")
 

Output:

1/2*(2*cos(-a + c)^4 - 2*(2*cos(-a + c)^3 - cos(-a + c))*cos(b*x + c)*sin( 
b*x + c)*sin(-a + c) - 4*(cos(-a + c)^4 - cos(-a + c)^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) + (2* 
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) - (2*cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) + (2*co 
s(-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)))/(2*b*cos(b*x 
 + c)*cos(-a + c)^4*sin(b*x + c)*sin(-a + c) - b*cos(-a + c)^5 + b*cos(-a 
+ c)^3 + (2*b*cos(-a + c)^5 - b*cos(-a + c)^3)*cos(b*x + c)^2)
 

Sympy [F]

\[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=\int \csc {\left (b x + c \right )} \sec ^{3}{\left (a + b x \right )}\, dx \] Input:

integrate(csc(b*x+c)*sec(b*x+a)**3,x)
 

Output:

Integral(csc(b*x + c)*sec(a + b*x)**3, x)
 

Maxima [C] (verification not implemented)

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

Time = 1.26 (sec) , antiderivative size = 85407, normalized size of antiderivative = 85407.00 \[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)*sec(b*x+a)^3,x, algorithm="maxima")
 

Output:

4*(9*((cos(4*a) + cos(4*c))*cos(3*a + c) + 2*cos(3*a + c)*cos(2*a + 2*c) + 
 (sin(4*a) + sin(4*c))*sin(3*a + c) + 2*sin(3*a + c)*sin(2*a + 2*c))*cos(4 
*a + 2*c)^2 + 9*((cos(4*a) + cos(4*c))*cos(3*a + c) + 2*cos(3*a + c)*cos(2 
*a + 2*c) - (cos(4*a) + 2*cos(2*a + 2*c) + cos(4*c))*cos(a + 3*c) + (sin(4 
*a) + sin(4*c))*sin(3*a + c) + 2*sin(3*a + c)*sin(2*a + 2*c) - (sin(4*a) + 
 2*sin(2*a + 2*c) + sin(4*c))*sin(a + 3*c))*cos(2*a + 4*c)^2 + 2*(cos(6*a) 
^2 + 2*cos(6*a)*cos(6*c) + cos(6*c)^2 + sin(6*a)^2 + 2*sin(6*a)*sin(6*c) + 
 sin(6*c)^2)*cos(3*a + c)*cos(2*a + 2*c) + 9*((cos(4*a) + cos(4*c))*cos(3* 
a + c) + 2*cos(3*a + c)*cos(2*a + 2*c) + (sin(4*a) + sin(4*c))*sin(3*a + c 
) + 2*sin(3*a + c)*sin(2*a + 2*c))*sin(4*a + 2*c)^2 + 9*((cos(4*a) + cos(4 
*c))*cos(3*a + c) + 2*cos(3*a + c)*cos(2*a + 2*c) - (cos(4*a) + 2*cos(2*a 
+ 2*c) + cos(4*c))*cos(a + 3*c) + (sin(4*a) + sin(4*c))*sin(3*a + c) + 2*s 
in(3*a + c)*sin(2*a + 2*c) - (sin(4*a) + 2*sin(2*a + 2*c) + sin(4*c))*sin( 
a + 3*c))*sin(2*a + 4*c)^2 + 2*(cos(6*a)^2 + 2*cos(6*a)*cos(6*c) + cos(6*c 
)^2 + sin(6*a)^2 + 2*sin(6*a)*sin(6*c) + sin(6*c)^2)*sin(3*a + c)*sin(2*a 
+ 2*c) - 2*(((sin(6*a) + 3*sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - (co 
s(6*a) + 3*cos(4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)*si 
n(3*a + 3*c) + 3*cos(3*a + 3*c)*sin(2*a + 4*c))*cos(4*b*x + 8*a)^2 + 4*((s 
in(6*a) + 3*sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - (cos(6*a) + 3*cos( 
4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)*sin(3*a + 3*c)...
 

Giac [C] (verification not implemented)

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

Time = 0.37 (sec) , antiderivative size = 6981, normalized size of antiderivative = 6981.00 \[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)*sec(b*x+a)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \] Input:

int(1/(cos(a + b*x)^3*sin(c + b*x)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \csc (c+b x) \sec ^3(a+b x) \, dx=\int \csc \left (b x +c \right ) \sec \left (b x +a \right )^{3}d x \] Input:

int(csc(b*x+c)*sec(b*x+a)^3,x)
 

Output:

int(csc(b*x + c)*sec(a + b*x)**3,x)