Integrand size = 15, antiderivative size = 1 \[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 97.00 \[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=-\frac {\sec (a-c) \left (\csc (c+b x)+\left (\log \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec (a-c)+2 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \tan (a-c)\right )}{b} \] Input:
Integrate[Csc[c + b*x]^2*Sec[a + b*x],x]
Output:
-((Sec[a - c]*(Csc[c + b*x] + (Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]])*Sec[a - c] + 2*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Tan[a - c]))/b)
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.
| \(\displaystyle \int \sec (a+b x) \csc ^2(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sec (a+b x) \csc ^2(b x+c)dx\) |
Input:
Int[Csc[c + b*x]^2*Sec[a + b*x],x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.17 (sec) , antiderivative size = 357, normalized size of antiderivative = 357.00
| method | result | size |
| default | \(\frac {\frac {\frac {2 \left (\frac {\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )}{\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )}-\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )}{\cos \left (c \right ) \sin \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}-\sin \left (c \right ) \cos \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}+2 \tan \left (\frac {a}{2}+\frac {b x}{2}\right ) \cos \left (a \right ) \cos \left (c \right )+2 \tan \left (\frac {a}{2}+\frac {b x}{2}\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 )}+\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {\ln \left (\tan \left (\frac {a}{2}+\frac {b x}{2}\right )-1\right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}+\frac {\ln \left (\tan \left (\frac {a}{2}+\frac {b x}{2}\right )+1\right )}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}}{b}\) | \(357\) |
| risch | \(\frac {4 i {\mathrm e}^{i \left (b x +3 a +2 c \right )}}{\left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right ) \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right ) b}-\frac {4 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right ) {\mathrm e}^{2 i \left (a +c \right )}}{\left ({\mathrm e}^{4 i a}+2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}+\frac {2 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{i \left (3 a +c \right )}}{\left ({\mathrm e}^{4 i a}+2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}-\frac {2 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{i \left (a +3 c \right )}}{\left ({\mathrm e}^{4 i a}+2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}+\frac {4 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right ) {\mathrm e}^{2 i \left (a +c \right )}}{\left ({\mathrm e}^{4 i a}+2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}-\frac {2 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{i \left (3 a +c \right )}}{\left ({\mathrm e}^{4 i a}+2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}+\frac {2 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{i \left (a +3 c \right )}}{\left ({\mathrm e}^{4 i a}+2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}\) | \(381\) |
Input:
int(csc(b*x+c)^2*sec(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b*(2/(cos(a)*cos(c)+sin(a)*sin(c))^2*(((cos(a)^2*cos(c)^2+2*cos(a)*cos(c )*sin(a)*sin(c)+sin(a)^2*sin(c)^2)/(sin(a)*cos(c)-cos(a)*sin(c))*tan(1/2*a +1/2*b*x)-cos(a)*cos(c)-sin(a)*sin(c))/(cos(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*t an(1/2*a+1/2*b*x)*sin(a)*sin(c)-sin(a)*cos(c)+cos(a)*sin(c))+(sin(a)*cos(c )-cos(a)*sin(c))/(-cos(c)^2*sin(a)^2-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-s in(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*cos(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/(cos(a)*cos(c)+si n(a)*sin(c))^2*ln(tan(1/2*a+1/2*b*x)-1)+1/(cos(a)*cos(c)+sin(a)*sin(c))^2* ln(tan(1/2*a+1/2*b*x)+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.11 (sec) , antiderivative size = 178, normalized size of antiderivative = 178.00 \[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=\frac {\log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + \log \left (\frac {2 \, {\left (\cos \left (-a + c\right ) \sin \left (b x + c\right ) - \cos \left (b x + c\right ) \sin \left (-a + c\right ) + 1\right )}}{\cos \left (-a + c\right ) + 1}\right ) \sin \left (b x + c\right ) - \log \left (-\frac {2 \, {\left (\cos \left (-a + c\right ) \sin \left (b x + c\right ) - \cos \left (b x + c\right ) \sin \left (-a + c\right ) - 1\right )}}{\cos \left (-a + c\right ) + 1}\right ) \sin \left (b x + c\right ) - 2 \, \cos \left (-a + c\right )}{2 \, b \cos \left (-a + c\right )^{2} \sin \left (b x + c\right )} \] Input:
integrate(csc(b*x+c)^2*sec(b*x+a),x, algorithm="fricas")
Output:
1/2*(log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a + c) - log(-1/2*cos(b *x + c) + 1/2)*sin(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))*sin(b*x + c) - log(-2*( cos(-a + c)*sin(b*x + c) - cos(b*x + c)*sin(-a + c) - 1)/(cos(-a + c) + 1) )*sin(b*x + c) - 2*cos(-a + c))/(b*cos(-a + c)^2*sin(b*x + c))
\[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=\int \csc ^{2}{\left (b x + c \right )} \sec {\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+c)**2*sec(b*x+a),x)
Output:
Integral(csc(b*x + c)**2*sec(a + b*x), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.41 (sec) , antiderivative size = 17016, normalized size of antiderivative = 17016.00 \[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^2*sec(b*x+a),x, algorithm="maxima")
Output:
-(4*(cos(4*a)^2 + 4*(cos(4*a) + cos(4*c))*cos(2*a + 2*c) + 4*cos(2*a + 2*c )^2 + 2*cos(4*a)*cos(4*c) + cos(4*c)^2 + sin(4*a)^2 + 4*(sin(4*a) + sin(4* c))*sin(2*a + 2*c) + 4*sin(2*a + 2*c)^2 + 2*sin(4*a)*sin(4*c) + sin(4*c)^2 )*cos(b*x + a + 2*c)*sin(2*b*x + 2*a + 2*c) + 4*(cos(4*a)^2 + 4*(cos(4*a) + cos(4*c))*cos(2*a + 2*c) + 4*cos(2*a + 2*c)^2 + 2*cos(4*a)*cos(4*c) + co s(4*c)^2 + sin(4*a)^2 + 4*(sin(4*a) + sin(4*c))*sin(2*a + 2*c) + 4*sin(2*a + 2*c)^2 + 2*sin(4*a)*sin(4*c) + sin(4*c)^2)*cos(b*x + a + 2*c)*sin(2*b*x + 4*c) - 4*(cos(4*a)^2 + 4*(cos(4*a) + cos(4*c))*cos(2*a + 2*c) + 4*cos(2 *a + 2*c)^2 + 2*cos(4*a)*cos(4*c) + cos(4*c)^2 + sin(4*a)^2 + 4*(sin(4*a) + sin(4*c))*sin(2*a + 2*c) + 4*sin(2*a + 2*c)^2 + 2*sin(4*a)*sin(4*c) + si n(4*c)^2)*cos(2*b*x + 2*a + 2*c)*sin(b*x + a + 2*c) - 4*(cos(4*a)^2 + 4*(c os(4*a) + cos(4*c))*cos(2*a + 2*c) + 4*cos(2*a + 2*c)^2 + 2*cos(4*a)*cos(4 *c) + cos(4*c)^2 + sin(4*a)^2 + 4*(sin(4*a) + sin(4*c))*sin(2*a + 2*c) + 4 *sin(2*a + 2*c)^2 + 2*sin(4*a)*sin(4*c) + sin(4*c)^2)*cos(2*b*x + 4*c)*sin (b*x + a + 2*c) - 4*(((sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(4*a) + c os(4*c))*sin(2*a + 2*c))*cos(2*b*x + 2*a + 2*c)^2 + ((sin(4*a) + sin(4*c)) *cos(2*a + 2*c) - (cos(4*a) + cos(4*c))*sin(2*a + 2*c))*cos(2*b*x + 4*c)^2 + ((sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(4*a) + cos(4*c))*sin(2*a + 2*c))*sin(2*b*x + 2*a + 2*c)^2 + ((sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(4*a) + cos(4*c))*sin(2*a + 2*c))*sin(2*b*x + 4*c)^2 - 2*((cos(2*a)...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.23 (sec) , antiderivative size = 2468, normalized size of antiderivative = 2468.00 \[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)^2*sec(b*x+a),x, algorithm="giac")
Output:
1/2*(2*(tan(1/2*a)^5*tan(1/2*c)^5 - tan(1/2*a)^5*tan(1/2*c)^4 + tan(1/2*a) ^4*tan(1/2*c)^5 + 2*tan(1/2*a)^5*tan(1/2*c)^3 + 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)^5*tan(1/2*c)^2 + 2*tan(1/2*a) ^4*tan(1/2*c)^3 - 2*tan(1/2*a)^3*tan(1/2*c)^4 + 2*tan(1/2*a)^2*tan(1/2*c)^ 5 + tan(1/2*a)^5*tan(1/2*c) + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 4*tan(1/2*a)^3 *tan(1/2*c)^3 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)*tan(1/2*c)^5 - ta n(1/2*a)^5 + tan(1/2*a)^4*tan(1/2*c) - 4*tan(1/2*a)^3*tan(1/2*c)^2 + 4*tan (1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)*tan(1/2*c)^4 + tan(1/2*c)^5 + tan(1/2* a)^4 + 2*tan(1/2*a)^3*tan(1/2*c) + 4*tan(1/2*a)^2*tan(1/2*c)^2 + 2*tan(1/2 *a)*tan(1/2*c)^3 + tan(1/2*c)^4 - 2*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 + 2*tan(1/2*c)^3 + 2*tan(1/2*a)^2 + tan(1/2 *a)*tan(1/2*c) + 2*tan(1/2*c)^2 - tan(1/2*a) + tan(1/2*c) + 1)*log(abs(-ta n(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*c)*tan(1/2*a) - tan(1/2*b*x + 1/2*c)*tan(1/2*c) - tan(1/2*a)*tan(1/2*c) - tan(1/2*b*x + 1/2*c) - tan(1/2*a) + tan(1/2*c) - 1))/(tan(1/2*a)^5*tan(1/2*c)^5 - tan(1 /2*a)^5*tan(1/2*c)^4 + tan(1/2*a)^4*tan(1/2*c)^5 - 2*tan(1/2*a)^5*tan(1/2* c)^3 + 9*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)^5*tan(1/2*c)^2 - 10*tan(1/2*a)^4*tan(1/2*c)^3 + 10*tan(1/2*a)^3*tan( 1/2*c)^4 - 2*tan(1/2*a)^2*tan(1/2*c)^5 + tan(1/2*a)^5*tan(1/2*c) - 10*tan( 1/2*a)^4*tan(1/2*c)^2 + 28*tan(1/2*a)^3*tan(1/2*c)^3 - 10*tan(1/2*a)^2*...
Timed out. \[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)*sin(c + b*x)^2),x)
Output:
\text{Hanged}
\[ \int \csc ^2(c+b x) \sec (a+b x) \, dx=\frac {2 \cos \left (b x +c \right )-\left (\int \frac {\tan \left (\frac {b x}{2}+\frac {c}{2}\right )^{2}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}d x \right ) \sin \left (b x +c \right ) b -\left (\int \frac {1}{\tan \left (\frac {b x}{2}+\frac {c}{2}\right )^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-\tan \left (\frac {b x}{2}+\frac {c}{2}\right )^{2}}d x \right ) \sin \left (b x +c \right ) b -\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +c \right )+\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +c \right )+\sin \left (b x +c \right ) b x}{2 \sin \left (b x +c \right ) b} \] Input:
int(csc(b*x+c)^2*sec(b*x+a),x)
Output:
(2*cos(b*x + c) - int(tan((b*x + c)/2)**2/(tan((a + b*x)/2)**2 - 1),x)*sin (b*x + c)*b - int(1/(tan((b*x + c)/2)**2*tan((a + b*x)/2)**2 - tan((b*x + c)/2)**2),x)*sin(b*x + c)*b - log(tan((a + b*x)/2) - 1)*sin(b*x + c) + log (tan((a + b*x)/2) + 1)*sin(b*x + c) + sin(b*x + c)*b*x)/(2*sin(b*x + c)*b)