Integrand size = 13, antiderivative size = 1 \[ \int \cot (c+b x) \csc (a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 122.00 \[ \int \cot (c+b x) \csc (a+b x) \, dx=-\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \csc (a-c)}{b}+\frac {\cot (a-c) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}-\frac {\cot (a-c) \log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \] Input:
Integrate[Cot[c + b*x]*Csc[a + b*x],x]
Output:
((-2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x )/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Csc[a - c])/b + (Cot [a - c]*Log[Cos[a/2 + (b*x)/2]])/b - (Cot[a - c]*Log[Sin[a/2 + (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.
\(\displaystyle \int \csc (a+b x) \cot (b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \csc (a+b x) \cot (b x+c)dx\) |
Input:
Int[Cot[c + b*x]*Csc[a + b*x],x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.13 (sec) , antiderivative size = 252, normalized size of antiderivative = 252.00
method | result | size |
risch | \(-\frac {2 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{i \left (a +c \right )}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}-\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i a}}{\left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right ) b}-\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i c}}{\left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right ) b}+\frac {2 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{i \left (a +c \right )}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i a}}{\left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right ) b}+\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i c}}{\left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right ) b}\) | \(252\) |
Input:
int(cot(b*x+c)*csc(b*x+a),x,method=_RETURNVERBOSE)
Output:
-2*I*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b/(exp(2*I*a)-exp(2*I*c))*exp(I*(a+c) )-I/(exp(2*I*a)-exp(2*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(2*I*a)-I/(exp(2*I*a )-exp(2*I*c))/b*ln(exp(I*(b*x+a))-1)*exp(2*I*c)+2*I*ln(exp(I*(b*x+a))-exp( I*(a-c)))/b/(exp(2*I*a)-exp(2*I*c))*exp(I*(a+c))+I/(exp(2*I*a)-exp(2*I*c)) /b*ln(exp(I*(b*x+a))+1)*exp(2*I*a)+I/(exp(2*I*a)-exp(2*I*c))/b*ln(exp(I*(b *x+a))+1)*exp(2*I*c)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 236, normalized size of antiderivative = 236.00 \[ \int \cot (c+b x) \csc (a+b x) \, dx=\frac {\sqrt {2} \sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1} \log \left (\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{2 \, b \sin \left (-2 \, a + 2 \, c\right )} \] Input:
integrate(cot(b*x+c)*csc(b*x+a),x, algorithm="fricas")
Output:
1/2*(sqrt(2)*sqrt(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*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)*log(1/2*cos(b*x + a) + 1/2) + (cos(-2*a + 2*c) + 1)*log(-1/2*cos (b*x + a) + 1/2))/(b*sin(-2*a + 2*c))
\[ \int \cot (c+b x) \csc (a+b x) \, dx=\int \cot {\left (b x + c \right )} \csc {\left (a + b x \right )}\, dx \] Input:
integrate(cot(b*x+c)*csc(b*x+a),x)
Output:
Integral(cot(b*x + c)*csc(a + b*x), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.07 (sec) , antiderivative size = 527, normalized size of antiderivative = 527.00 \[ \int \cot (c+b x) \csc (a+b x) \, dx =\text {Too large to display} \] Input:
integrate(cot(b*x+c)*csc(b*x+a),x, algorithm="maxima")
Output:
-((cos(2*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) + s in(a), cos(b*x) - cos(a)) - (cos(2*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2* c)^2)*arctan2(sin(b*x) - sin(a), cos(b*x) + cos(a)) - 2*((cos(2*a) - cos(2 *c))*cos(a + c) + (sin(2*a) - sin(2*c))*sin(a + c))*arctan2(sin(b*x) + sin (c), cos(b*x) - cos(c)) + 2*((cos(2*a) - cos(2*c))*cos(a + c) + (sin(2*a) - sin(2*c))*sin(a + c))*arctan2(sin(b*x) - sin(c), cos(b*x) + cos(c)) + (c os(2*c)*sin(2*a) - cos(2*a)*sin(2*c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - (cos(2*c)*sin(2*a ) - cos(2*a)*sin(2*c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin (b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) - ((sin(2*a) - sin(2*c))*cos(a + c ) - (cos(2*a) - cos(2*c))*sin(a + c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + sin(c)^2) + ((sin(2*a) - sin( 2*c))*cos(a + c) - (cos(2*a) - cos(2*c))*sin(a + c))*log(cos(b*x)^2 - 2*co s(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2))/(2* b*cos(2*a)*cos(2*c) - b*cos(2*c)^2 + 2*b*sin(2*a)*sin(2*c) - b*sin(2*c)^2 - (cos(2*a)^2 + sin(2*a)^2)*b)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.15 (sec) , antiderivative size = 378, normalized size of antiderivative = 378.00 \[ \int \cot (c+b x) \csc (a+b x) \, dx =\text {Too large to display} \] Input:
integrate(cot(b*x+c)*csc(b*x+a),x, algorithm="giac")
Output:
1/2*((tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^3 + 4*tan(1/2*a)^2*tan(1/2*c) - 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)^3*tan(1/2*c) - tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 - tan(1/2*a)*tan(1/2*c)) - (tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^2*tan(1/ 2*c) + tan(1/2*c)^3 + tan(1/2*c))*log(abs(tan(1/2*b*x)*tan(1/2*c) - 1))/(t an(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2) - (tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a )*tan(1/2*c) - tan(1/2*c)^2 + 1)*log(abs(tan(1/2*b*x) + tan(1/2*a)))/(tan( 1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c)) + (tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*log(abs(tan (1/2*b*x) + tan(1/2*c)))/(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^ 2 + tan(1/2*a) - tan(1/2*c)))/b
Time = 57.48 (sec) , antiderivative size = 46101, normalized size of antiderivative = 46101.00 \[ \int \cot (c+b x) \csc (a+b x) \, dx=\text {Too large to display} \] Input:
int(cot(c + b*x)/sin(a + b*x),x)
Output:
(atan(((-(16*sin(a)^4*sin(c)^6 - 8*sin(a + c)^4*sin(c)^2 - 8*sin(a + c)^4* cos(c)^2 - 4*sin(a)^2*sin(c)^4*cos(a - c)^2 - 8*sin(a + c)*cos(c)^2*(-(2*s in(a)^2*sin(c)^2 - sin(a + c)^2 + 3*cos(a + c)*sin(a)*sin(c) + sin(a)*sin( c)*cos(a - c))^3)^(1/2) + 8*sin(a + c)*sin(c)^2*(-(2*sin(a)^2*sin(c)^2 - s in(a + c)^2 + 3*cos(a + c)*sin(a)*sin(c) + sin(a)*sin(c)*cos(a - c))^3)^(1 /2) - 9*cos(a + c)^2*sin(a + c)^2*cos(c)^2 - 36*cos(a + c)^2*sin(a)^2*sin( c)^4 + 8*sin(a + c)^2*sin(a)^2*sin(c)^4 - sin(a + c)^2*cos(c)^2*cos(a - c) ^2 - 8*cos(c)^2*sin(a)^4*sin(c)^4 + 4*cos(c)*sin(c)*cos(a - c)*(-(2*sin(a) ^2*sin(c)^2 - sin(a + c)^2 + 3*cos(a + c)*sin(a)*sin(c) + sin(a)*sin(c)*co s(a - c))^3)^(1/2) + 27*cos(a + c)^3*cos(c)^2*sin(a)*sin(c) - 24*sin(a + c )^3*cos(c)*sin(a)*sin(c)^2 + 48*sin(a + c)*cos(c)*sin(a)^3*sin(c)^4 + 2*co s(c)^2*sin(a)^2*sin(c)^2*cos(a - c)^2 - 24*cos(a + c)*sin(a)^2*sin(c)^4*co s(a - c) + 12*sin(a + c)^2*sin(a)*sin(c)^3*cos(a - c) - 8*cos(c)*sin(a)*si n(c)^2*(-(2*sin(a)^2*sin(c)^2 - sin(a + c)^2 + 3*cos(a + c)*sin(a)*sin(c) + sin(a)*sin(c)*cos(a - c))^3)^(1/2) + cos(c)^2*sin(a)*sin(c)*cos(a - c)^3 - 12*cos(a + c)*cos(c)^2*sin(a)^3*sin(c)^3 - 12*cos(a + c)*sin(a + c)^3*c os(c)*sin(c) + 12*cos(a + c)*cos(c)*sin(c)*(-(2*sin(a)^2*sin(c)^2 - sin(a + c)^2 + 3*cos(a + c)*sin(a)*sin(c) + sin(a)*sin(c)*cos(a - c))^3)^(1/2) - 4*cos(c)^2*sin(a)^3*sin(c)^3*cos(a - c) + 18*cos(a + c)^2*cos(c)^2*sin(a) ^2*sin(c)^2 - 4*sin(a + c)^3*cos(c)*sin(c)*cos(a - c) + 20*sin(a + c)^2...
\[ \int \cot (c+b x) \csc (a+b x) \, dx=\int \cot \left (b x +c \right ) \csc \left (b x +a \right )d x \] Input:
int(cot(b*x+c)*csc(b*x+a),x)
Output:
int(cot(b*x + c)*csc(a + b*x),x)