Integrand size = 15, antiderivative size = 36 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \cos ^2(a-c)}{b}+\frac {\cos (2 a-c+b x)}{b} \] Output:
-arctanh(cos(b*x+c))*cos(a-c)^2/b+cos(b*x+2*a-c)/b
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=\frac {\cos (2 a-c+b x)+\cos ^2(a-c) \left (-\log \left (\cos \left (\frac {1}{2} (c+b x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+b x)\right )\right )\right )}{b} \] 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.
\(\displaystyle \int \cos ^2(a+b x) \csc (b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^2(a+b x) \csc (b x+c)dx\) |
Input:
Int[Cos[a + b*x]^2*Csc[c + b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 137, normalized size of antiderivative = 3.81
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}+\frac {\cos \left (b x +2 a -c \right )}{b}\) | \(137\) |
default | \(\frac {-\frac {2 \left (-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}+\frac {2 \left (-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )}}{b}\) | \(292\) |
Input:
int(cos(b*x+a)^2*csc(b*x+c),x,method=_RETURNVERBOSE)
Output:
1/2/b*ln(exp(I*(b*x+a))-exp(I*(a-c)))+1/2/b*ln(exp(I*(b*x+a))-exp(I*(a-c)) )*cos(2*a-2*c)-1/2/b*ln(exp(I*(b*x+a))+exp(I*(a-c)))-1/2/b*ln(exp(I*(b*x+a ))+exp(I*(a-c)))*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. 87 vs. \(2 (36) = 72\).
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.42 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=-\frac {\cos \left (-a + c\right )^{2} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) - \cos \left (-a + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) - 4 \, \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 2 \, {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )}{2 \, b} \] Input:
integrate(cos(b*x+a)^2*csc(b*x+c),x, algorithm="fricas")
Output:
-1/2*(cos(-a + c)^2*log(1/2*cos(b*x + c) + 1/2) - cos(-a + c)^2*log(-1/2*c os(b*x + c) + 1/2) - 4*cos(-a + c)*sin(b*x + c)*sin(-a + c) - 2*(2*cos(-a + c)^2 - 1)*cos(b*x + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 1459 vs. \(2 (27) = 54\).
Time = 10.25 (sec) , antiderivative size = 3215, normalized size of antiderivative = 89.31 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=\text {Too large to display} \] 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 + b*...
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (36) = 72\).
Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.28 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=-\frac {{\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - 4 \, \cos \left (b x + 2 \, a - c\right )}{4 \, b} \] 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(c os(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) - 4*cos(b*x + 2*a - c))/b
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 527, normalized size of antiderivative = 14.64 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^2*csc(b*x+c),x, algorithm="giac")
Output:
((tan(1/2*a)^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(2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a) - 2* tan(1/2*a)^2 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) - 2*tan(1/2*c)^2)/abs(2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c) - 2* tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1/2*c)^2 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) + 2))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)^2 - 2*(2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^ 2 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a) + tan(1/2*a)^2 - 2*tan(1/2*b*x + 1/2 *a)*tan(1/2*c) - 4*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 - 1)/((tan(1/2*a)^ 2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*(tan(1/2*b*x + 1/2*a)^2 + 1)))/b
Time = 19.36 (sec) , antiderivative size = 223, normalized size of antiderivative = 6.19 \[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}\,\ln \left (-\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )}^2\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}^2}{4\,b}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}\,\ln \left (\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )}^2\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}^2}{4\,b} \] Input:
int(cos(a + b*x)^2/sin(c + b*x),x)
Output:
exp(c*1i - a*2i - b*x*1i)/(2*b) + exp(a*2i - c*1i + b*x*1i)/(2*b) - (exp(c *2i - a*2i)*log(- ((exp(a*2i)*exp(-c*2i) + 1)^2*1i)/2 - (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*2 i - c*2i) + 1)^2)/(4*b) + (exp(c*2i - a*2i)*log(((exp(a*2i)*exp(-c*2i) + 1 )^2*1i)/2 - (exp(c*1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i)*2i + exp(a*4i)*ex p(-c*4i)*1i + 1i))/2)*(exp(a*2i - c*2i) + 1)^2)/(4*b)
\[ \int \cos ^2(a+b x) \csc (c+b x) \, dx=\int \cos \left (b x +a \right )^{2} \csc \left (b x +c \right )d x \] Input:
int(cos(b*x+a)^2*csc(b*x+c),x)
Output:
int(cos(a + b*x)**2*csc(b*x + c),x)