Integrand size = 17, antiderivative size = 1 \[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.40 (sec) , antiderivative size = 404, normalized size of antiderivative = 404.00 \[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\frac {2 i \arctan (\tan (c+b x)) \cot (a-c) \csc ^2(a-c)}{b}+\frac {2 \cot (a-c) \csc ^2(a-c) \log (\sin (a+b x))}{b}-\frac {\cot (a-c) \csc ^2(a-c) \log \left (\sin ^2(c+b x)\right )}{b}+x \left (-2 \cot (a-c) \csc (a) \csc (a-c) \csc (c)-\frac {\cos (a) \cos (c) \cot (a)}{(\cos (c) \sin (a)-\cos (a) \sin (c))^3}-\frac {\cos (c) \csc (a)}{(\cos (c) \sin (a)-\cos (a) \sin (c))^3}+\frac {\cos (c) \sin (a)}{(\cos (c) \sin (a)-\cos (a) \sin (c))^3}-\frac {2 \cos (a) \sin (c)}{(\cos (c) \sin (a)-\cos (a) \sin (c))^3}+\frac {2 i \cos (a) \cos (c)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}-\frac {\cos (a) \cos (c) \cot (c)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}-\frac {\cos (a) \csc (c)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}-\frac {2 \cos (c) \sin (a)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}-\frac {i \cos (c) \cot (c) \sin (a)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}+\frac {i \csc (c) \sin (a)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}+\frac {\cos (a) \sin (c)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}+\frac {i \sin (a) \sin (c)}{(-\cos (c) \sin (a)+\cos (a) \sin (c))^3}\right )+\frac {\csc (a) \csc ^2(a-c) \csc (a+b x) \sin (b x)}{b}+\frac {\csc ^2(a-c) \csc (c) \csc (c+b x) \sin (b x)}{b} \] Input:
Integrate[Csc[a + b*x]^2*Csc[c + b*x]^2,x]
Output:
((2*I)*ArcTan[Tan[c + b*x]]*Cot[a - c]*Csc[a - c]^2)/b + (2*Cot[a - c]*Csc [a - c]^2*Log[Sin[a + b*x]])/b - (Cot[a - c]*Csc[a - c]^2*Log[Sin[c + b*x] ^2])/b + x*(-2*Cot[a - c]*Csc[a]*Csc[a - c]*Csc[c] - (Cos[a]*Cos[c]*Cot[a] )/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^3 - (Cos[c]*Csc[a])/(Cos[c]*Sin[a] - Cos [a]*Sin[c])^3 + (Cos[c]*Sin[a])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^3 - (2*Cos [a]*Sin[c])/(Cos[c]*Sin[a] - Cos[a]*Sin[c])^3 + ((2*I)*Cos[a]*Cos[c])/(-(C os[c]*Sin[a]) + Cos[a]*Sin[c])^3 - (Cos[a]*Cos[c]*Cot[c])/(-(Cos[c]*Sin[a] ) + Cos[a]*Sin[c])^3 - (Cos[a]*Csc[c])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^ 3 - (2*Cos[c]*Sin[a])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^3 - (I*Cos[c]*Cot [c]*Sin[a])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^3 + (I*Csc[c]*Sin[a])/(-(Co s[c]*Sin[a]) + Cos[a]*Sin[c])^3 + (Cos[a]*Sin[c])/(-(Cos[c]*Sin[a]) + Cos[ a]*Sin[c])^3 + (I*Sin[a]*Sin[c])/(-(Cos[c]*Sin[a]) + Cos[a]*Sin[c])^3) + ( Csc[a]*Csc[a - c]^2*Csc[a + b*x]*Sin[b*x])/b + (Csc[a - c]^2*Csc[c]*Csc[c + b*x]*Sin[b*x])/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.
| \(\displaystyle \int \csc ^2(a+b x) \csc ^2(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \csc ^2(a+b x) \csc ^2(b x+c)dx\) |
Input:
Int[Csc[a + b*x]^2*Csc[c + b*x]^2,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.07 (sec) , antiderivative size = 261, normalized size of antiderivative = 261.00
| method | result | size |
| default | \(\frac {-\frac {1}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{2} \tan \left (b x +a \right )}+\frac {\left (2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (b x +a \right )\right )}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{3}}+\frac {\left (-2 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )-2 \sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{3} \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}-\frac {\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{2} \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}}{b}\) | \(261\) |
| risch | \(\frac {8 i \left ({\mathrm e}^{2 i \left (b x +3 a +c \right )}+{\mathrm e}^{2 i \left (b x +2 a +2 c \right )}-2 \,{\mathrm e}^{2 i \left (2 a +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) \left ({\mathrm e}^{2 i \left (b x +a +c \right )}-{\mathrm e}^{2 i a}\right ) \left (-{\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{2} b}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{\left ({\mathrm e}^{6 i a}-3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}-{\mathrm e}^{6 i c}\right ) b}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{\left ({\mathrm e}^{6 i a}-3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}-{\mathrm e}^{6 i c}\right ) b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{\left ({\mathrm e}^{6 i a}-3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}-{\mathrm e}^{6 i c}\right ) b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{\left ({\mathrm e}^{6 i a}-3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}-{\mathrm e}^{6 i c}\right ) b}\) | \(370\) |
Input:
int(csc(b*x+a)^2*csc(b*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/(sin(a)*cos(c)-cos(a)*sin(c))^2/tan(b*x+a)+1/(sin(a)*cos(c)-cos(a) *sin(c))^3*(2*cos(a)*cos(c)+2*sin(a)*sin(c))*ln(tan(b*x+a))+(-2*cos(a)^2*c os(c)^2-4*cos(a)*cos(c)*sin(a)*sin(c)-2*sin(a)^2*sin(c)^2)/(sin(a)*cos(c)- cos(a)*sin(c))^3/(cos(a)*cos(c)+sin(a)*sin(c))*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))-(cos(a)^2*cos(c)^2+ sin(c)^2*cos(a)^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)/(sin(a)*cos(c)-cos( a)*sin(c))^2/(cos(a)*cos(c)+sin(a)*sin(c))/(tan(b*x+a)*cos(a)*cos(c)+tan(b *x+a)*sin(a)*sin(c)-sin(a)*cos(c)+cos(a)*sin(c)))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 358, normalized size of antiderivative = 358.00 \[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + 2 \, {\left (\cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + {\left (\cos \left (b x + c\right )^{2} \cos \left (-a + c\right )^{2} + \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (-a + c\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + c\right )^{2} + \frac {1}{4}\right ) - {\left (\cos \left (b x + c\right )^{2} \cos \left (-a + c\right )^{2} + \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (-a + c\right )^{2}\right )} \log \left (-\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right ) + 1}{{\left (b \cos \left (-a + c\right )^{4} - 2 \, b \cos \left (-a + c\right )^{2} + b\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) + {\left (b \cos \left (-a + c\right )^{3} - {\left (b \cos \left (-a + c\right )^{3} - b \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{2} - b \cos \left (-a + c\right )\right )} \sin \left (-a + c\right )} \] Input:
integrate(csc(b*x+a)^2*csc(b*x+c)^2,x, algorithm="fricas")
Output:
(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 + (cos(b*x + c)^2*cos(-a + c)^2 + cos(b* x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) - cos(-a + c)^2)*log(-1/4*cos( b*x + c)^2 + 1/4) - (cos(b*x + c)^2*cos(-a + c)^2 + cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) - cos(-a + c)^2)*log(-(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)/(cos(-a + c)^2 + 2*cos(-a + c) + 1)) + 1)/((b*cos(-a + c)^4 - 2 *b*cos(-a + c)^2 + b)*cos(b*x + c)*sin(b*x + c) + (b*cos(-a + c)^3 - (b*co s(-a + c)^3 - b*cos(-a + c))*cos(b*x + c)^2 - b*cos(-a + c))*sin(-a + c))
\[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\int \csc ^{2}{\left (a + b x \right )} \csc ^{2}{\left (b x + c \right )}\, dx \] Input:
integrate(csc(b*x+a)**2*csc(b*x+c)**2,x)
Output:
Integral(csc(a + b*x)**2*csc(b*x + c)**2, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.88 (sec) , antiderivative size = 146269, normalized size of antiderivative = 146269.00 \[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*csc(b*x+c)^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) - ( cos(4*a) + cos(4*c))*sin(2*a + 2*c))*cos(2*a + 4*c)^2 + 36*((sin(4*a) + si n(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))*s in(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 + (co s(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)*c os(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 + ( 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(2*b*x + 6*a)^2 + ((cos(...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 3043, normalized size of antiderivative = 3043.00 \[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^2*csc(b*x+c)^2,x, algorithm="giac")
Output:
-1/4*((tan(1/2*a)^8*tan(1/2*c)^8 + 8*tan(1/2*a)^7*tan(1/2*c)^7 - 2*tan(1/2 *a)^8*tan(1/2*c)^4 + 8*tan(1/2*a)^7*tan(1/2*c)^5 + 16*tan(1/2*a)^6*tan(1/2 *c)^6 + 8*tan(1/2*a)^5*tan(1/2*c)^7 - 2*tan(1/2*a)^4*tan(1/2*c)^8 - 8*tan( 1/2*a)^7*tan(1/2*c)^3 + 32*tan(1/2*a)^6*tan(1/2*c)^4 + 8*tan(1/2*a)^5*tan( 1/2*c)^5 + 32*tan(1/2*a)^4*tan(1/2*c)^6 - 8*tan(1/2*a)^3*tan(1/2*c)^7 + ta n(1/2*a)^8 - 8*tan(1/2*a)^7*tan(1/2*c) + 16*tan(1/2*a)^6*tan(1/2*c)^2 - 8* tan(1/2*a)^5*tan(1/2*c)^3 + 68*tan(1/2*a)^4*tan(1/2*c)^4 - 8*tan(1/2*a)^3* tan(1/2*c)^5 + 16*tan(1/2*a)^2*tan(1/2*c)^6 - 8*tan(1/2*a)*tan(1/2*c)^7 + tan(1/2*c)^8 - 8*tan(1/2*a)^5*tan(1/2*c) + 32*tan(1/2*a)^4*tan(1/2*c)^2 + 8*tan(1/2*a)^3*tan(1/2*c)^3 + 32*tan(1/2*a)^2*tan(1/2*c)^4 - 8*tan(1/2*a)* tan(1/2*c)^5 - 2*tan(1/2*a)^4 + 8*tan(1/2*a)^3*tan(1/2*c) + 16*tan(1/2*a)^ 2*tan(1/2*c)^2 + 8*tan(1/2*a)*tan(1/2*c)^3 - 2*tan(1/2*c)^4 + 8*tan(1/2*a) *tan(1/2*c) + 1)*log(abs(tan(b*x + a)*tan(1/2*a)^2*tan(1/2*c)^2 - tan(b*x + a)*tan(1/2*a)^2 + 4*tan(b*x + a)*tan(1/2*a)*tan(1/2*c) - 2*tan(1/2*a)^2* tan(1/2*c) - tan(b*x + a)*tan(1/2*c)^2 + 2*tan(1/2*a)*tan(1/2*c)^2 + tan(b *x + a) - 2*tan(1/2*a) + 2*tan(1/2*c)))/(tan(1/2*a)^8*tan(1/2*c)^5 - 3*tan (1/2*a)^7*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^7 - tan(1/2*a)^5*tan(1/ 2*c)^8 - tan(1/2*a)^8*tan(1/2*c)^3 + 10*tan(1/2*a)^7*tan(1/2*c)^4 - 25*tan (1/2*a)^6*tan(1/2*c)^5 + 25*tan(1/2*a)^5*tan(1/2*c)^6 - 10*tan(1/2*a)^4*ta n(1/2*c)^7 + tan(1/2*a)^3*tan(1/2*c)^8 - 3*tan(1/2*a)^7*tan(1/2*c)^2 + ...
Timed out. \[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(sin(a + b*x)^2*sin(c + b*x)^2),x)
Output:
\text{Hanged}
\[ \int \csc ^2(a+b x) \csc ^2(c+b x) \, dx=\int \csc \left (b x +c \right )^{2} \csc \left (b x +a \right )^{2}d x \] Input:
int(csc(b*x+a)^2*csc(b*x+c)^2,x)
Output:
int(csc(b*x + c)**2*csc(a + b*x)**2,x)