\(\int \csc ^3(c+b x) \sec (a+b x) \, dx\) [182]

Optimal result

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

0
 

Mathematica [C] (verified)

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

Time = 0.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 165.00 \[ \int \csc ^3(c+b x) \sec (a+b x) \, dx=\frac {\csc \left (\frac {c}{2}\right ) \csc ^2(c+b x) \sec ^3(a-c) \sec \left (\frac {c}{2}\right ) (2 \sin (2 a-3 c)+(-2-4 \log (\cos (a+b x))+4 \log (\sin (c+b x))) \sin (c)-\sin (2 a-3 c-2 b x)-\sin (2 a-c+2 b x)-2 \log (\cos (a+b x)) \sin (c+2 b x)+2 \log (\sin (c+b x)) \sin (c+2 b x)+2 \log (\cos (a+b x)) \sin (3 c+2 b x)-2 \log (\sin (c+b x)) \sin (3 c+2 b x))}{16 b} \] Input:

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

Output:

(Csc[c/2]*Csc[c + b*x]^2*Sec[a - c]^3*Sec[c/2]*(2*Sin[2*a - 3*c] + (-2 - 4 
*Log[Cos[a + b*x]] + 4*Log[Sin[c + b*x]])*Sin[c] - Sin[2*a - 3*c - 2*b*x] 
- Sin[2*a - c + 2*b*x] - 2*Log[Cos[a + b*x]]*Sin[c + 2*b*x] + 2*Log[Sin[c 
+ b*x]]*Sin[c + 2*b*x] + 2*Log[Cos[a + b*x]]*Sin[3*c + 2*b*x] - 2*Log[Sin[ 
c + b*x]]*Sin[3*c + 2*b*x]))/(16*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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[Csc[c + b*x]^3*Sec[a + b*x],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 = 2.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 207.00

method	result	size
default	\(\frac {-\frac {-2 \cos \left (a \right ) \sin \left (c \right )+2 \sin \left (a \right ) \cos \left (c \right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}-\frac {\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}{2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}+\frac {\ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3}}}{b}\)	\(207\)
risch	\(\frac {8 \,{\mathrm e}^{i \left (2 b x +5 a +5 c \right )}+4 \,{\mathrm e}^{i \left (7 a +c \right )}-4 \,{\mathrm e}^{i \left (5 a +3 c \right )}}{\left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{2} b}+\frac {8 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}-\frac {8 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}\)	\(208\)

Input:

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

Output:

1/b*(-(-2*cos(a)*sin(c)+2*sin(a)*cos(c))/(cos(a)*cos(c)+sin(a)*sin(c))^3/( 
tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)-sin(a)*cos(c)+cos(a)*sin 
(c))-1/2*(cos(a)^2*cos(c)^2+sin(c)^2*cos(a)^2+cos(c)^2*sin(a)^2+sin(a)^2*s 
in(c)^2)/(cos(a)*cos(c)+sin(a)*sin(c))^3/(tan(b*x+a)*cos(a)*cos(c)+tan(b*x 
+a)*sin(a)*sin(c)-sin(a)*cos(c)+cos(a)*sin(c))^2+1/(cos(a)*cos(c)+sin(a)*s 
in(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 = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 189.00 \[ \int \csc ^3(c+b x) \sec (a+b x) \, dx=-\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (-a + c\right )^{2} - {\left (\cos \left (b x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + c\right )^{2} + \frac {1}{4}\right ) + {\left (\cos \left (b x + c\right )^{2} - 1\right )} \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right )}{2 \, {\left (b \cos \left (b x + c\right )^{2} \cos \left (-a + c\right )^{3} - b \cos \left (-a + c\right )^{3}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

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

Time = 1.38 (sec) , antiderivative size = 85585, normalized size of antiderivative = 85585.00 \[ \int \csc ^3(c+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)^3*sec(b*x+a),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 + 4*a + 4*c)^2 + 
 4*((sin(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 ...
 

Giac [C] (verification not implemented)

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

Time = 0.21 (sec) , antiderivative size = 2551, normalized size of antiderivative = 2551.00 \[ \int \csc ^3(c+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(2*(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. \[ \int \csc ^3(c+b x) \sec (a+b x) \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \csc ^3(c+b x) \sec (a+b x) \, dx=\frac {\cos \left (b x +c \right )+2 \left (\int \frac {1}{\sin \left (b x +c \right )^{3}}d x \right ) \sin \left (b x +c \right )^{2} b +2 \left (\int \frac {1}{\cos \left (b x +a \right ) \sin \left (b x +c \right )^{3}}d x \right ) \sin \left (b x +c \right )^{2} b -\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {c}{2}\right )\right ) \sin \left (b x +c \right )^{2}}{2 \sin \left (b x +c \right )^{2} b} \] Input:

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

Output:

(cos(b*x + c) + 2*int(1/sin(b*x + c)**3,x)*sin(b*x + c)**2*b + 2*int(1/(co 
s(a + b*x)*sin(b*x + c)**3),x)*sin(b*x + c)**2*b - log(tan((b*x + c)/2))*s 
in(b*x + c)**2)/(2*sin(b*x + c)**2*b)