Integrand size = 16, antiderivative size = 33 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=\frac {\text {arctanh}(\cos (c-b x)) \cos ^2(a+c)}{b}-\frac {\cos (2 a+c+b x)}{b} \] Output:
arctanh(cos(b*x-c))*cos(a+c)^2/b-cos(b*x+2*a+c)/b
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=\frac {-\cos (2 a+c+b x)+\cos ^2(a+c) \left (\log \left (\cos \left (\frac {1}{2} (c-b x)\right )\right )-\log \left (-\sin \left (\frac {1}{2} (c-b x)\right )\right )\right )}{b} \] Input:
Integrate[Cos[a + b*x]^2*Csc[c - b*x],x]
Output:
(-Cos[2*a + c + b*x] + Cos[a + c]^2*(Log[Cos[(c - b*x)/2]] - Log[-Sin[(c - 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.
| \(\displaystyle \int \cos ^2(a+b x) \csc (c-b x) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \cos ^2(a+b x) \csc (c-b x)dx\) |
Input:
Int[Cos[a + b*x]^2*Csc[c - b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.88
| method | result | size |
| risch | \(-\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right )}{2 b}-\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (2 a +2 c \right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (2 a +2 c \right )}{2 b}-\frac {\cos \left (b x +2 a +c \right )}{b}\) | \(128\) |
| default | \(-\frac {-\frac {2 \left (\left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (a \right ) \sin \left (c \right )-\cos \left (a \right ) \cos \left (c \right )\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )}+\frac {2 \left (\sin \left (a \right )^{2} \sin \left (c \right )^{2}-2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right )^{2} \cos \left (c \right )^{2}\right ) \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-2 \sin \left (a \right ) \sin \left (c \right )+2 \cos \left (a \right ) \cos \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}}{b}\) | \(290\) |
Input:
int(-cos(b*x+a)^2*csc(b*x-c),x,method=_RETURNVERBOSE)
Output:
-1/2/b*ln(-exp(I*(a+c))+exp(I*(b*x+a)))-1/2/b*ln(-exp(I*(a+c))+exp(I*(b*x+ a)))*cos(2*a+2*c)+1/2/b*ln(exp(I*(a+c))+exp(I*(b*x+a)))+1/2/b*ln(exp(I*(a+ c))+exp(I*(b*x+a)))*cos(2*a+2*c)-cos(b*x+2*a+c)/b
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (34) = 68\).
Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.45 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=\frac {\cos \left (a + c\right )^{2} \log \left (\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) + 1}{\cos \left (a + c\right ) + 1}\right ) - \cos \left (a + c\right )^{2} \log \left (-\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) - 1}{\cos \left (a + c\right ) + 1}\right ) - 2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) + 2 \, \sin \left (b x + a\right ) \sin \left (a + c\right )}{2 \, b} \] Input:
integrate(-cos(b*x+a)^2*csc(b*x-c),x, algorithm="fricas")
Output:
1/2*(cos(a + c)^2*log((cos(b*x + a)*cos(a + c) + sin(b*x + a)*sin(a + c) + 1)/(cos(a + c) + 1)) - cos(a + c)^2*log(-(cos(b*x + a)*cos(a + c) + sin(b *x + a)*sin(a + c) - 1)/(cos(a + c) + 1)) - 2*cos(b*x + a)*cos(a + c) + 2* sin(b*x + a)*sin(a + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 1460 vs. \(2 (27) = 54\).
Time = 10.58 (sec) , antiderivative size = 3216, normalized size of antiderivative = 97.45 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=\text {Too large to display} \] Input:
integrate(-cos(b*x+a)**2*csc(b*x-c),x)
Output:
-2*Piecewise((0, Eq(b, 0) & Eq(c, 0)), (-sin(b*x)/b, Eq(c, 0)), (0, Eq(b, 0)), (2*log(-tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)* *4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan (c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(-tan(c/2) + tan(b*x/2))*tan(c/2)** 3/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2 )**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(-tan(c/2) + tan(b*x/ 2))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2 *log(-tan(c/2) + tan(b*x/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan (c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)* *2 + b) - 2*log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan( c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2* b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + 1/tan(c/2))*tan( c/2)**3/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan (b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + 1 /tan(c/2))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2 )**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan (b*x/2)**2 + b) + 2*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2...
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (34) = 68\).
Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.52 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=\frac {{\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - {\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - 4 \, \cos \left (b x + 2 \, a + c\right )}{4 \, b} \] Input:
integrate(-cos(b*x+a)^2*csc(b*x-c),x, algorithm="maxima")
Output:
1/4*((cos(2*a + 2*c) + 1)*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) - (cos(2*a + 2*c) + 1)*log(cos( b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + s in(c)^2) - 4*cos(b*x + 2*a + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 1043 vs. \(2 (34) = 68\).
Time = 0.21 (sec) , antiderivative size = 1043, normalized size of antiderivative = 31.61 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=\text {Too large to display} \] Input:
integrate(-cos(b*x+a)^2*csc(b*x-c),x, algorithm="giac")
Output:
-((tan(1/2*a)^5*tan(1/2*c)^5 - 2*tan(1/2*a)^5*tan(1/2*c)^3 - 9*tan(1/2*a)^ 4*tan(1/2*c)^4 - 2*tan(1/2*a)^3*tan(1/2*c)^5 + tan(1/2*a)^5*tan(1/2*c) + 1 0*tan(1/2*a)^4*tan(1/2*c)^2 + 28*tan(1/2*a)^3*tan(1/2*c)^3 + 10*tan(1/2*a) ^2*tan(1/2*c)^4 + tan(1/2*a)*tan(1/2*c)^5 - tan(1/2*a)^4 - 10*tan(1/2*a)^3 *tan(1/2*c) - 28*tan(1/2*a)^2*tan(1/2*c)^2 - 10*tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 9*tan(1/2*a)*tan(1/2*c) + 2*tan(1/2*c)^2 - 1)*log(abs(tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c) - tan(1/2*b*x + 1/2 *a) + tan(1/2*a) + tan(1/2*c)))/(tan(1/2*a)^5*tan(1/2*c)^5 + 2*tan(1/2*a)^ 5*tan(1/2*c)^3 - tan(1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^3*tan(1/2*c)^5 + tan(1/2*a)^5*tan(1/2*c) - 2*tan(1/2*a)^4*tan(1/2*c)^2 + 4*tan(1/2*a)^3*ta n(1/2*c)^3 - 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)*tan(1/2*c)^5 - tan(1 /2*a)^4 + 2*tan(1/2*a)^3*tan(1/2*c) - 4*tan(1/2*a)^2*tan(1/2*c)^2 + 2*tan( 1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 - 2*tan(1/2*a)^2 + tan(1/2*a)*tan(1/2*c ) - 2*tan(1/2*c)^2 - 1) - (tan(1/2*a)^5*tan(1/2*c)^4 + tan(1/2*a)^4*tan(1/ 2*c)^5 - 2*tan(1/2*a)^5*tan(1/2*c)^2 - 10*tan(1/2*a)^4*tan(1/2*c)^3 - 10*t an(1/2*a)^3*tan(1/2*c)^4 - 2*tan(1/2*a)^2*tan(1/2*c)^5 + tan(1/2*a)^5 + 9* tan(1/2*a)^4*tan(1/2*c) + 28*tan(1/2*a)^3*tan(1/2*c)^2 + 28*tan(1/2*a)^2*t an(1/2*c)^3 + 9*tan(1/2*a)*tan(1/2*c)^4 + tan(1/2*c)^5 - 2*tan(1/2*a)^3 - 10*tan(1/2*a)^2*tan(1/2*c) - 10*tan(1/2*a)*tan(1/2*c)^2 - 2*tan(1/2*c)^3 + tan(1/2*a) + tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*a)*tan(1/2*a) + tan...
Time = 1.73 (sec) , antiderivative size = 223, normalized size of antiderivative = 6.76 \[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-c\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,\ln \left (-\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )}^2\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+1\right )}^2}{4\,b}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,\ln \left (\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )}^2\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+1\right )}^2}{4\,b} \] Input:
int(cos(a + b*x)^2/sin(c - b*x),x)
Output:
(exp(- a*2i - c*2i)*log(((exp(a*2i)*exp(c*2i) + 1)^2*1i)/2 + (exp(-c*1i)*e xp(b*x*1i)*(exp(a*2i)*exp(c*2i)*2i + exp(a*4i)*exp(c*4i)*1i + 1i))/2)*(exp (a*2i + c*2i) + 1)^2)/(4*b) - exp(a*2i + c*1i + b*x*1i)/(2*b) - (exp(- a*2 i - c*2i)*log((exp(-c*1i)*exp(b*x*1i)*(exp(a*2i)*exp(c*2i)*2i + exp(a*4i)* exp(c*4i)*1i + 1i))/2 - ((exp(a*2i)*exp(c*2i) + 1)^2*1i)/2)*(exp(a*2i + c* 2i) + 1)^2)/(4*b) - exp(- a*2i - c*1i - b*x*1i)/(2*b)
\[ \int \cos ^2(a+b x) \csc (c-b x) \, dx=-\left (\int \cos \left (b x +a \right )^{2} \csc \left (b x -c \right )d x \right ) \] Input:
int(-cos(b*x+a)^2*csc(b*x-c),x)
Output:
- int(cos(a + b*x)**2*csc(b*x - c),x)