Integrand size = 17, antiderivative size = 1 \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 1.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 74.00 \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=\frac {\csc (a) \csc (a+b x) \sin (b x) \tan ^2(a-c)+\sec ^2(a-c) (\sec (c) \sec (c+b x) \sin (b x)+2 (\log (\cos (c+b x))-\log (\sin (a+b x))) \tan (a-c))}{b} \] Input:
Integrate[Csc[a + b*x]^2*Tan[c + b*x]^2,x]
Output:
(Csc[a]*Csc[a + b*x]*Sin[b*x]*Tan[a - c]^2 + Sec[a - c]^2*(Sec[c]*Sec[c + b*x]*Sin[b*x] + 2*(Log[Cos[c + b*x]] - Log[Sin[a + b*x]])*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.
\(\displaystyle \int \csc ^2(a+b x) \tan ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \csc ^2(a+b x) \tan ^2(b x+c)dx\) |
Input:
Int[Csc[a + b*x]^2*Tan[c + b*x]^2,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.26 (sec) , antiderivative size = 384, normalized size of antiderivative = 384.00
method | result | size |
risch | \(\frac {2 i \left (5 \,{\mathrm e}^{2 i \left (b x +3 a +c \right )}-2 \,{\mathrm e}^{2 i \left (b x +2 a +2 c \right )}+{\mathrm e}^{2 i \left (b x +a +3 c \right )}+{\mathrm e}^{6 i a}-6 \,{\mathrm e}^{2 i \left (2 a +c \right )}+{\mathrm e}^{2 i \left (a +2 c \right )}\right )}{b \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}\) | \(384\) |
Input:
int(csc(b*x+a)^2*tan(b*x+c)^2,x,method=_RETURNVERBOSE)
Output:
2*I*(5*exp(2*I*(b*x+3*a+c))-2*exp(2*I*(b*x+2*a+2*c))+exp(2*I*(b*x+a+3*c))+ exp(6*I*a)-6*exp(2*I*(2*a+c))+exp(2*I*(a+2*c)))/b/(exp(2*I*a)+exp(2*I*c))^ 2/(exp(2*I*(b*x+a))-1)/(exp(2*I*(b*x+a+c))+exp(2*I*a))-8*I*ln(exp(2*I*(b*x +a))+exp(2*I*(a-c)))/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp (6*I*c))/b*exp(2*I*(2*a+c))+8*I*ln(exp(2*I*(b*x+a))+exp(2*I*(a-c)))/(exp(6 *I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))/b*exp(2*I*(a+2*c)) +8*I*ln(exp(2*I*(b*x+a))-1)/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2* c))+exp(6*I*c))/b*exp(2*I*(2*a+c))-8*I*ln(exp(2*I*(b*x+a))-1)/(exp(6*I*a)+ 3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))/b*exp(2*I*(a+2*c))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.10 (sec) , antiderivative size = 352, normalized size of antiderivative = 352.00 \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=-\frac {{\left (\cos \left (-2 \, a + 2 \, c\right ) - 3\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 2 \, \cos \left (-2 \, a + 2 \, c\right ) - 3\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 2 \, {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}{\cos \left (-2 \, a + 2 \, c\right ) + 1}\right ) - 2 \, \cos \left (-2 \, a + 2 \, c\right ) - 2}{{\left (b \cos \left (-2 \, a + 2 \, c\right )^{2} + 2 \, b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + {\left ({\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \cos \left (b x + a\right )^{2} - b \cos \left (-2 \, a + 2 \, c\right ) - b\right )} \sin \left (-2 \, a + 2 \, c\right )} \] Input:
integrate(csc(b*x+a)^2*tan(b*x+c)^2,x, algorithm="fricas")
Output:
-((cos(-2*a + 2*c) - 3)*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - (cos(- 2*a + 2*c)^2 - 2*cos(-2*a + 2*c) - 3)*cos(b*x + a)^2 + 2*((cos(-2*a + 2*c) - 1)*cos(b*x + a)^2 - cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2* a + 2*c) + 1)*log(-1/4*cos(b*x + a)^2 + 1/4) - 2*((cos(-2*a + 2*c) - 1)*co s(b*x + a)^2 - cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) + 1)*log((2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)* sin(-2*a + 2*c) - cos(-2*a + 2*c) + 1)/(cos(-2*a + 2*c) + 1)) - 2*cos(-2*a + 2*c) - 2)/((b*cos(-2*a + 2*c)^2 + 2*b*cos(-2*a + 2*c) + b)*cos(b*x + a) *sin(b*x + a) + ((b*cos(-2*a + 2*c) + b)*cos(b*x + a)^2 - b*cos(-2*a + 2*c ) - b)*sin(-2*a + 2*c))
Timed out. \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)**2*tan(b*x+c)**2,x)
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 2.73 (sec) , antiderivative size = 120050, normalized size of antiderivative = 120050.00 \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*tan(b*x+c)^2,x, algorithm="maxima")
Output:
2*(72*((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 + 72*((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 + 72*((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 + 72*((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 + 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 + 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...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.48 (sec) , antiderivative size = 3075, normalized size of antiderivative = 3075.00 \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*tan(b*x+c)^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)*l og(abs(-2*tan(b*x + a)*tan(1/2*a)^2*tan(1/2*c) + 2*tan(b*x + a)*tan(1/2*a) *tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^2 - 2*tan(b*x + a)*tan(1/2*a) + ta n(1/2*a)^2 + 2*tan(b*x + a)*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...
Timed out. \[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=\text {Hanged} \] Input:
int(tan(c + b*x)^2/sin(a + b*x)^2,x)
Output:
\text{Hanged}
\[ \int \csc ^2(a+b x) \tan ^2(c+b x) \, dx=\int \csc \left (b x +a \right )^{2} \tan \left (b x +c \right )^{2}d x \] Input:
int(csc(b*x+a)^2*tan(b*x+c)^2,x)
Output:
int(csc(a + b*x)**2*tan(b*x + c)**2,x)