Integrand size = 15, antiderivative size = 1 \[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.54 (sec) , antiderivative size = 82, normalized size of antiderivative = 82.00 \[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=\frac {2 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cot (a-c) \csc (a-c)-\csc (a-c) \csc (c+b x)+\cot ^2(a-c) \left (-\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )}{b} \] Input:
Integrate[Cot[c + b*x]^2*Csc[a + b*x],x]
Output:
(2*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cot[a - c]*Csc[a - c] - Csc[a - c ]*Csc[c + b*x] + Cot[a - c]^2*(-Log[Cos[(a + b*x)/2]] + Log[Sin[(a + b*x)/ 2]]))/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.
| \(\displaystyle \int \csc (a+b x) \cot ^2(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \csc (a+b x) \cot ^2(b x+c)dx\) |
Input:
Int[Cot[c + b*x]^2*Csc[a + b*x],x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.16 (sec) , antiderivative size = 550, normalized size of antiderivative = 550.00
| method | result | size |
| risch | \(-\frac {4 \,{\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 {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{4 i a}}{\left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}-\frac {2 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (a +c \right )}}{b \left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{4 i c}}{\left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}+\frac {2 \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 )}}{b \left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right )}+\frac {2 \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 )}}{b \left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{4 i a}}{\left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}+\frac {2 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (a +c \right )}}{b \left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{4 i c}}{\left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right ) b}-\frac {2 \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 )}}{b \left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right )}-\frac {2 \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 )}}{b \left ({\mathrm e}^{4 i a}-2 \,{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{4 i c}\right )}\) | \(550\) |
Input:
int(cot(b*x+c)^2*csc(b*x+a),x,method=_RETURNVERBOSE)
Output:
-4/(-exp(2*I*(b*x+a+c))+exp(2*I*a))/(exp(2*I*a)-exp(2*I*c))/b*exp(I*(b*x+3 *a+2*c))-1/(exp(4*I*a)-2*exp(2*I*(a+c))+exp(4*I*c))/b*ln(exp(I*(b*x+a))-1) *exp(4*I*a)-2*ln(exp(I*(b*x+a))-1)/b/(exp(4*I*a)-2*exp(2*I*(a+c))+exp(4*I* c))*exp(2*I*(a+c))-1/(exp(4*I*a)-2*exp(2*I*(a+c))+exp(4*I*c))/b*ln(exp(I*( b*x+a))-1)*exp(4*I*c)+2*ln(exp(I*(b*x+a))-exp(I*(a-c)))/b/(exp(4*I*a)-2*ex p(2*I*(a+c))+exp(4*I*c))*exp(I*(3*a+c))+2*ln(exp(I*(b*x+a))-exp(I*(a-c)))/ b/(exp(4*I*a)-2*exp(2*I*(a+c))+exp(4*I*c))*exp(I*(a+3*c))+1/(exp(4*I*a)-2* exp(2*I*(a+c))+exp(4*I*c))/b*ln(exp(I*(b*x+a))+1)*exp(4*I*a)+2*ln(exp(I*(b *x+a))+1)/b/(exp(4*I*a)-2*exp(2*I*(a+c))+exp(4*I*c))*exp(2*I*(a+c))+1/(exp (4*I*a)-2*exp(2*I*(a+c))+exp(4*I*c))/b*ln(exp(I*(b*x+a))+1)*exp(4*I*c)-2*l n(exp(I*(b*x+a))+exp(I*(a-c)))/b/(exp(4*I*a)-2*exp(2*I*(a+c))+exp(4*I*c))* exp(I*(3*a+c))-2*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b/(exp(4*I*a)-2*exp(2*I*( a+c))+exp(4*I*c))*exp(I*(a+3*c))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.10 (sec) , antiderivative size = 433, normalized size of antiderivative = 433.00 \[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=-\frac {\frac {\sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \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 ) + 1\right )} \sin \left (b x + a\right )\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 ) + \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{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}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \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 ) + 1\right )} \sin \left (b x + a\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \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 ) + 1\right )} \sin \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 4 \, \sin \left (-2 \, a + 2 \, c\right )}{2 \, {\left ({\left (b \cos \left (-2 \, a + 2 \, c\right ) - b\right )} \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (b \cos \left (-2 \, a + 2 \, c\right )^{2} - b\right )} \sin \left (b x + a\right )\right )}} \] Input:
integrate(cot(b*x+c)^2*csc(b*x+a),x, algorithm="fricas")
Output:
-1/2*(sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(-2*a + 2*c) + (cos(- 2*a + 2*c)^2 + 2*cos(-2*a + 2*c) + 1)*sin(b*x + a))*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) + 2*sqrt(2)* ((cos(-2*a + 2*c) + 1)*cos(b*x + a) - sin(b*x + a)*sin(-2*a + 2*c))/sqrt(c os(-2*a + 2*c) + 1) - cos(-2*a + 2*c) + 3)/(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))/s qrt(cos(-2*a + 2*c) + 1) - ((cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(-2*a + 2*c) + (cos(-2*a + 2*c)^2 + 2*cos(-2*a + 2*c) + 1)*sin(b*x + a))*log(1/2*c os(b*x + a) + 1/2) + ((cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(-2*a + 2*c) + (cos(-2*a + 2*c)^2 + 2*cos(-2*a + 2*c) + 1)*sin(b*x + a))*log(-1/2*cos(b* x + a) + 1/2) + 4*sin(-2*a + 2*c))/((b*cos(-2*a + 2*c) - b)*cos(b*x + a)*s in(-2*a + 2*c) + (b*cos(-2*a + 2*c)^2 - b)*sin(b*x + a))
\[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=\int \cot ^{2}{\left (b x + c \right )} \csc {\left (a + b x \right )}\, dx \] Input:
integrate(cot(b*x+c)**2*csc(b*x+a),x)
Output:
Integral(cot(b*x + c)**2*csc(a + b*x), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.53 (sec) , antiderivative size = 20524, normalized size of antiderivative = 20524.00 \[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)^2*csc(b*x+a),x, algorithm="maxima")
Output:
1/2*(8*(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(2*b*x + 2*a + 2*c)*cos(b*x + a + 2*c) - 8*(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(2*b*x + 4*c)*cos(b*x + a + 2*c) + 8*(cos(4*a)^2 - 4*(cos(4*a) + cos(4*c))*cos(2*a + 2*c) + 4*co s(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)*sin(2*b*x + 2*a + 2*c)*sin(b*x + a + 2*c) - 8*(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)*co s(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)*sin(2*b*x + 4*c)* sin(b*x + a + 2*c) - 8*(((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 + 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...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.22 (sec) , antiderivative size = 1206, normalized size of antiderivative = 1206.00 \[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)^2*csc(b*x+a),x, algorithm="giac")
Output:
-1/4*((tan(1/2*a)^5*tan(1/2*c)^4 - 2*tan(1/2*a)^5*tan(1/2*c)^2 + 8*tan(1/2 *a)^4*tan(1/2*c)^3 - 2*tan(1/2*a)^3*tan(1/2*c)^4 + tan(1/2*a)^5 - 8*tan(1/ 2*a)^4*tan(1/2*c) + 20*tan(1/2*a)^3*tan(1/2*c)^2 - 8*tan(1/2*a)^2*tan(1/2* c)^3 + tan(1/2*a)*tan(1/2*c)^4 - 2*tan(1/2*a)^3 + 8*tan(1/2*a)^2*tan(1/2*c ) - 2*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a))*log(abs(tan(1/2*b*x)*tan(1/2*a ) - 1))/(tan(1/2*a)^5*tan(1/2*c)^2 - 2*tan(1/2*a)^4*tan(1/2*c)^3 + tan(1/2 *a)^3*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c) - 4*tan(1/2*a)^3*tan(1/2*c) ^2 + 2*tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^3 - 2*tan(1/2*a)^2*tan(1/2*c ) + tan(1/2*a)*tan(1/2*c)^2) - (tan(1/2*a)^4*tan(1/2*c)^5 + 4*tan(1/2*a)^3 *tan(1/2*c)^4 - 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)*tan(1/2*c)^4 - tan(1/2*c)^5 + 4*tan(1/2*a)*tan(1/2*c)^2 + tan(1 /2*c))*log(abs(tan(1/2*b*x)*tan(1/2*c) - 1))/(tan(1/2*a)^4*tan(1/2*c)^3 - 2*tan(1/2*a)^3*tan(1/2*c)^4 + tan(1/2*a)^2*tan(1/2*c)^5 + 2*tan(1/2*a)^3*t an(1/2*c)^2 - 4*tan(1/2*a)^2*tan(1/2*c)^3 + 2*tan(1/2*a)*tan(1/2*c)^4 + ta n(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*c)^3) - (tan(1 /2*a)^4*tan(1/2*c)^4 - 2*tan(1/2*a)^4*tan(1/2*c)^2 + 8*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)^4 - 8*tan(1/2*a)^3*tan(1 /2*c) + 20*tan(1/2*a)^2*tan(1/2*c)^2 - 8*tan(1/2*a)*tan(1/2*c)^3 + tan(1/2 *c)^4 - 2*tan(1/2*a)^2 + 8*tan(1/2*a)*tan(1/2*c) - 2*tan(1/2*c)^2 + 1)*log (abs(tan(1/2*b*x) + tan(1/2*a)))/(tan(1/2*a)^4*tan(1/2*c)^2 - 2*tan(1/2...
Time = 39.43 (sec) , antiderivative size = 14480, normalized size of antiderivative = 14480.00 \[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=\text {Too large to display} \] Input:
int(cot(c + b*x)^2/sin(a + b*x),x)
Output:
((2*(cot(c)^2 + 1))/(cos(a) - cot(c)*sin(a)) + (2*tan((b*x)/2)*cot(c)*(cot (c)^2 + 1))/(cos(a) - cot(c)*sin(a)))/(b*(2*tan((b*x)/2)*cot(c) - tan((b*x )/2)^2 + 1)) + (atan((((sin(a) + cos(a)*cot(c))*(cot(c)^2 + 1)^(1/2)*((32* (cos(a)^2*sin(a)^4 - cot(c)^2*sin(a)^6 + 2*cos(a)^3*cot(c)*sin(a)^3 - 6*co s(a)*cot(c)^3*sin(a)^5 - cos(a)*cot(c)^5*sin(a)^5 + cos(a)^5*cot(c)^5*sin( a) + 4*cos(a)^2*cot(c)^2*sin(a)^4 + cos(a)^4*cot(c)^2*sin(a)^2 + 9*cos(a)^ 3*cot(c)^3*sin(a)^3 - 9*cos(a)^2*cot(c)^4*sin(a)^4 + 6*cos(a)^4*cot(c)^4*s in(a)^2 - 4*cos(a)^3*cot(c)^5*sin(a)^3 - 2*cos(a)^2*cot(c)^6*sin(a)^4 - co s(a)^3*cot(c)^7*sin(a)^3))/(cos(a)^3 - cot(c)^3*sin(a)^3 - 3*cos(a)^2*cot( c)*sin(a) + 3*cos(a)*cot(c)^2*sin(a)^2) + (32*tan((b*x)/2)*(2*cos(a)*sin(a )^5 - 2*cot(c)*sin(a)^6 + 2*cos(a)^3*sin(a)^3 - 4*cot(c)^3*sin(a)^6 - cot( c)^5*sin(a)^6 + 4*cos(a)^2*cot(c)*sin(a)^4 + 4*cos(a)^4*cot(c)*sin(a)^2 - 2*cos(a)*cot(c)^2*sin(a)^5 + 2*cos(a)^5*cot(c)^2*sin(a) - 11*cos(a)*cot(c) ^4*sin(a)^5 + 5*cos(a)^5*cot(c)^4*sin(a) - 2*cos(a)*cot(c)^6*sin(a)^5 + 4* cos(a)^5*cot(c)^6*sin(a) + 7*cos(a)^3*cot(c)^2*sin(a)^3 + cos(a)^2*cot(c)^ 3*sin(a)^4 + 10*cos(a)^4*cot(c)^3*sin(a)^2 + 4*cos(a)^3*cot(c)^4*sin(a)^3 - 14*cos(a)^2*cot(c)^5*sin(a)^4 + 7*cos(a)^4*cot(c)^5*sin(a)^2 - 11*cos(a) ^3*cot(c)^6*sin(a)^3 - cos(a)^2*cot(c)^7*sin(a)^4 - 4*cos(a)^4*cot(c)^7*si n(a)^2))/(cos(a)^3 - cot(c)^3*sin(a)^3 - 3*cos(a)^2*cot(c)*sin(a) + 3*cos( a)*cot(c)^2*sin(a)^2) - ((sin(a) + cos(a)*cot(c))*(cot(c)^2 + 1)^(1/2)*...
\[ \int \cot ^2(c+b x) \csc (a+b x) \, dx=\int \cot \left (b x +c \right )^{2} \csc \left (b x +a \right )d x \] Input:
int(cot(b*x+c)^2*csc(b*x+a),x)
Output:
int(cot(b*x + c)**2*csc(a + b*x),x)