Integrand size = 15, antiderivative size = 1 \[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 6.66 (sec) , antiderivative size = 198, normalized size of antiderivative = 198.00 \[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\frac {-16 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \csc ^3(a-c)+\cot (a-c) \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )+9 \csc ^2(a-c) \left (\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )+\csc ^2(a-c) \left (-\cos (3 (a-c)) \csc (a-c) \left (\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )+4 \left (-\csc \left (\frac {a}{2}\right ) \csc \left (\frac {1}{2} (a+b x)\right )+\sec \left (\frac {a}{2}\right ) \sec \left (\frac {1}{2} (a+b x)\right )\right ) \sin \left (\frac {b x}{2}\right )\right )}{8 b} \] Input:
Integrate[Cot[c + b*x]*Csc[a + b*x]^3,x]
Output:
(-16*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Csc[a - c]^3 + Cot[a - c]*(Csc[ (a + b*x)/2]^2 + 9*Csc[a - c]^2*(Log[Cos[(a + b*x)/2]] - Log[Sin[(a + b*x) /2]]) - Sec[(a + b*x)/2]^2) + Csc[a - c]^2*(-(Cos[3*(a - c)]*Csc[a - c]*(L og[Cos[(a + b*x)/2]] - Log[Sin[(a + b*x)/2]])) + 4*(-(Csc[a/2]*Csc[(a + b* x)/2]) + Sec[a/2]*Sec[(a + b*x)/2])*Sin[(b*x)/2]))/(8*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.
| \(\displaystyle \int \csc ^3(a+b x) \cot (b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \csc ^3(a+b x) \cot (b x+c)dx\) |
Input:
Int[Cot[c + b*x]*Csc[a + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.20 (sec) , antiderivative size = 762, normalized size of antiderivative = 762.00
| method | result | size |
| risch | \(-\frac {i \left ({\mathrm e}^{i \left (3 b x +7 a \right )}+8 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}-{\mathrm e}^{i \left (3 b x +3 a +4 c \right )}+{\mathrm e}^{i \left (b x +5 a \right )}-8 \,{\mathrm e}^{i \left (b x +3 a +2 c \right )}-{\mathrm e}^{i \left (b x +a +4 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 )^{2}}+\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{6 i a}}{2 \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 {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{2 \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 {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{2 \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 {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{6 i c}}{2 \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}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{b \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 )}-\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{6 i a}}{2 \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 {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{2 \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 {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{2 \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 {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{6 i c}}{2 \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}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{b \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 )}\) | \(762\) |
Input:
int(cot(b*x+c)*csc(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-I/b/(exp(2*I*a)-exp(2*I*c))^2/(exp(2*I*(b*x+a))-1)^2*(exp(I*(3*b*x+7*a))+ 8*exp(I*(3*b*x+5*a+2*c))-exp(I*(3*b*x+3*a+4*c))+exp(I*(b*x+5*a))-8*exp(I*( b*x+3*a+2*c))-exp(I*(b*x+a+4*c)))+1/2*I/(exp(6*I*a)-3*exp(2*I*(2*a+c))+3*e xp(2*I*(a+2*c))-exp(6*I*c))/b*ln(exp(I*(b*x+a))+1)*exp(6*I*a)-9/2*I/(exp(6 *I*a)-3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*ln(exp(I*(b*x+a) )+1)*exp(2*I*(2*a+c))-9/2*I/(exp(6*I*a)-3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2* c))-exp(6*I*c))/b*ln(exp(I*(b*x+a))+1)*exp(2*I*(a+2*c))+1/2*I/(exp(6*I*a)- 3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*ln(exp(I*(b*x+a))+1)*e xp(6*I*c)-8*I*ln(exp(I*(b*x+a))-exp(I*(a-c)))/b/(exp(6*I*a)-3*exp(2*I*(2*a +c))+3*exp(2*I*(a+2*c))-exp(6*I*c))*exp(3*I*(a+c))-1/2*I/(exp(6*I*a)-3*exp (2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(6* I*a)+9/2*I/(exp(6*I*a)-3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b *ln(exp(I*(b*x+a))-1)*exp(2*I*(2*a+c))+9/2*I/(exp(6*I*a)-3*exp(2*I*(2*a+c) )+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(2*I*(a+2*c))-1 /2*I/(exp(6*I*a)-3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*ln(ex p(I*(b*x+a))-1)*exp(6*I*c)+8*I*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b/(exp(6*I* a)-3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))*exp(3*I*(a+c))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.12 (sec) , antiderivative size = 429, normalized size of antiderivative = 429.00 \[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\frac {2 \, {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \cos \left (b x + a\right ) - \frac {4 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - \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 ) + \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 )^{2} - 4 \, \cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \cos \left (b x + a\right )^{2} - \cos \left (-2 \, a + 2 \, c\right )^{2} + 4 \, \cos \left (-2 \, a + 2 \, c\right ) + 5\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 )^{2} - 4 \, \cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \cos \left (b x + a\right )^{2} - \cos \left (-2 \, a + 2 \, c\right )^{2} + 4 \, \cos \left (-2 \, a + 2 \, c\right ) + 5\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 8 \, \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )}{4 \, {\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(cot(b*x+c)*csc(b*x+a)^3,x, algorithm="fricas")
Output:
1/4*(2*(cos(-2*a + 2*c)^2 - 1)*cos(b*x + a) - 4*sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a)^2 - 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) + 2*sqrt(2)*((cos(-2* a + 2*c) + 1)*cos(b*x + a) - sin(b*x + a)*sin(-2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - cos(-2*a + 2*c) + 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*co s(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) - 1))/sqrt(cos(- 2*a + 2*c) + 1) - ((cos(-2*a + 2*c)^2 - 4*cos(-2*a + 2*c) - 5)*cos(b*x + a )^2 - cos(-2*a + 2*c)^2 + 4*cos(-2*a + 2*c) + 5)*log(1/2*cos(b*x + a) + 1/ 2) + ((cos(-2*a + 2*c)^2 - 4*cos(-2*a + 2*c) - 5)*cos(b*x + a)^2 - cos(-2* a + 2*c)^2 + 4*cos(-2*a + 2*c) + 5)*log(-1/2*cos(b*x + a) + 1/2) + 8*sin(b *x + a)*sin(-2*a + 2*c))/(((b*cos(-2*a + 2*c) - b)*cos(b*x + a)^2 - b*cos( -2*a + 2*c) + b)*sin(-2*a + 2*c))
\[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\int \cot {\left (b x + c \right )} \csc ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(cot(b*x+c)*csc(b*x+a)**3,x)
Output:
Integral(cot(b*x + c)*csc(a + b*x)**3, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.12 (sec) , antiderivative size = 137271, normalized size of antiderivative = 137271.00 \[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)*csc(b*x+a)^3,x, algorithm="maxima")
Output:
1/2*(((cos(6*a)^2 - 6*(2*cos(6*a) - cos(6*c))*cos(4*a + 2*c) + 27*cos(4*a + 2*c)^2 - 6*(cos(6*a) - 2*cos(6*c))*cos(2*a + 4*c) - 27*cos(2*a + 4*c)^2 - cos(6*c)^2 + sin(6*a)^2 - 6*(2*sin(6*a) - sin(6*c))*sin(4*a + 2*c) + 27* sin(4*a + 2*c)^2 - 6*(sin(6*a) - 2*sin(6*c))*sin(2*a + 4*c) - 27*sin(2*a + 4*c)^2 - sin(6*c)^2)*cos(4*b*x + 8*a)^2 + 4*(cos(6*a)^2 - 6*(2*cos(6*a) - cos(6*c))*cos(4*a + 2*c) + 27*cos(4*a + 2*c)^2 - 6*(cos(6*a) - 2*cos(6*c) )*cos(2*a + 4*c) - 27*cos(2*a + 4*c)^2 - cos(6*c)^2 + sin(6*a)^2 - 6*(2*si n(6*a) - sin(6*c))*sin(4*a + 2*c) + 27*sin(4*a + 2*c)^2 - 6*(sin(6*a) - 2* sin(6*c))*sin(2*a + 4*c) - 27*sin(2*a + 4*c)^2 - sin(6*c)^2)*cos(4*b*x + 6 *a + 2*c)^2 + (cos(6*a)^2 - 6*(2*cos(6*a) - cos(6*c))*cos(4*a + 2*c) + 27* cos(4*a + 2*c)^2 - 6*(cos(6*a) - 2*cos(6*c))*cos(2*a + 4*c) - 27*cos(2*a + 4*c)^2 - cos(6*c)^2 + sin(6*a)^2 - 6*(2*sin(6*a) - sin(6*c))*sin(4*a + 2* c) + 27*sin(4*a + 2*c)^2 - 6*(sin(6*a) - 2*sin(6*c))*sin(2*a + 4*c) - 27*s in(2*a + 4*c)^2 - sin(6*c)^2)*cos(4*b*x + 4*a + 4*c)^2 + 4*(cos(6*a)^2 - 6 *(2*cos(6*a) - cos(6*c))*cos(4*a + 2*c) + 27*cos(4*a + 2*c)^2 - 6*(cos(6*a ) - 2*cos(6*c))*cos(2*a + 4*c) - 27*cos(2*a + 4*c)^2 - cos(6*c)^2 + sin(6* a)^2 - 6*(2*sin(6*a) - sin(6*c))*sin(4*a + 2*c) + 27*sin(4*a + 2*c)^2 - 6* (sin(6*a) - 2*sin(6*c))*sin(2*a + 4*c) - 27*sin(2*a + 4*c)^2 - sin(6*c)^2) *cos(2*b*x + 6*a)^2 + 16*(cos(6*a)^2 - 6*(2*cos(6*a) - cos(6*c))*cos(4*a + 2*c) + 27*cos(4*a + 2*c)^2 - 6*(cos(6*a) - 2*cos(6*c))*cos(2*a + 4*c) ...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.34 (sec) , antiderivative size = 3565, normalized size of antiderivative = 3565.00 \[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cot(b*x+c)*csc(b*x+a)^3,x, algorithm="giac")
Output:
1/16*(2*(tan(1/2*a)^7*tan(1/2*c)^6 + 3*tan(1/2*a)^7*tan(1/2*c)^4 + 3*tan(1 /2*a)^5*tan(1/2*c)^6 - 3*tan(1/2*a)^7*tan(1/2*c)^2 + 24*tan(1/2*a)^6*tan(1 /2*c)^3 - 27*tan(1/2*a)^5*tan(1/2*c)^4 + 24*tan(1/2*a)^4*tan(1/2*c)^5 - 3* tan(1/2*a)^3*tan(1/2*c)^6 - tan(1/2*a)^7 + 27*tan(1/2*a)^5*tan(1/2*c)^2 - 32*tan(1/2*a)^4*tan(1/2*c)^3 + 27*tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)*t an(1/2*c)^6 - 3*tan(1/2*a)^5 + 24*tan(1/2*a)^4*tan(1/2*c) - 27*tan(1/2*a)^ 3*tan(1/2*c)^2 + 24*tan(1/2*a)^2*tan(1/2*c)^3 - 3*tan(1/2*a)*tan(1/2*c)^4 + 3*tan(1/2*a)^3 + 3*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)^7*tan(1/2*c)^3 - 3*tan(1/2*a)^6*tan(1/2 *c)^4 + 3*tan(1/2*a)^5*tan(1/2*c)^5 - tan(1/2*a)^4*tan(1/2*c)^6 + 3*tan(1/ 2*a)^6*tan(1/2*c)^2 - 9*tan(1/2*a)^5*tan(1/2*c)^3 + 9*tan(1/2*a)^4*tan(1/2 *c)^4 - 3*tan(1/2*a)^3*tan(1/2*c)^5 + 3*tan(1/2*a)^5*tan(1/2*c) - 9*tan(1/ 2*a)^4*tan(1/2*c)^2 + 9*tan(1/2*a)^3*tan(1/2*c)^3 - 3*tan(1/2*a)^2*tan(1/2 *c)^4 + tan(1/2*a)^4 - 3*tan(1/2*a)^3*tan(1/2*c) + 3*tan(1/2*a)^2*tan(1/2* c)^2 - tan(1/2*a)*tan(1/2*c)^3) - 2*(tan(1/2*a)^6*tan(1/2*c)^7 + 3*tan(1/2 *a)^6*tan(1/2*c)^5 + 3*tan(1/2*a)^4*tan(1/2*c)^7 + 3*tan(1/2*a)^6*tan(1/2* c)^3 + 9*tan(1/2*a)^4*tan(1/2*c)^5 + 3*tan(1/2*a)^2*tan(1/2*c)^7 + tan(1/2 *a)^6*tan(1/2*c) + 9*tan(1/2*a)^4*tan(1/2*c)^3 + 9*tan(1/2*a)^2*tan(1/2*c) ^5 + tan(1/2*c)^7 + 3*tan(1/2*a)^4*tan(1/2*c) + 9*tan(1/2*a)^2*tan(1/2*c)^ 3 + 3*tan(1/2*c)^5 + 3*tan(1/2*a)^2*tan(1/2*c) + 3*tan(1/2*c)^3 + tan(1...
Timed out. \[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\text {Hanged} \] Input:
int(cot(c + b*x)/sin(a + b*x)^3,x)
Output:
\text{Hanged}
\[ \int \cot (c+b x) \csc ^3(a+b x) \, dx=\int \cot \left (b x +c \right ) \csc \left (b x +a \right )^{3}d x \] Input:
int(cot(b*x+c)*csc(b*x+a)^3,x)
Output:
int(cot(b*x + c)*csc(a + b*x)**3,x)