Integrand size = 17, antiderivative size = 51 \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=-x \cos (2 (a-c))-\frac {\cos ^2(a-c) \cot (c+b x)}{b}-\frac {\log (\sin (c+b x)) \sin (2 (a-c))}{b} \] Output:
-x*cos(2*a-2*c)-cos(a-c)^2*cot(b*x+c)/b-ln(sin(b*x+c))*sin(2*a-2*c)/b
Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(51)=102\).
Time = 0.40 (sec) , antiderivative size = 181, normalized size of antiderivative = 3.55 \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=\frac {\csc (c) \csc (c+b x) (b x \cos (2 a-4 c-b x)-b x \cos (2 a-2 c-b x)+b x \cos (2 a+b x)-b x \cos (2 a-2 c+b x)+2 \sin (b x)+\log (\sin (c+b x)) \sin (2 a-4 c-b x)-\sin (2 a-2 c-b x)-\log (\sin (c+b x)) \sin (2 a-2 c-b x)+\log (\sin (c+b x)) \sin (2 a+b x)+\sin (2 a-2 c+b x)-\log (\sin (c+b x)) \sin (2 a-2 c+b x))}{4 b} \] Input:
Integrate[Cos[a + b*x]^2*Csc[c + b*x]^2,x]
Output:
(Csc[c]*Csc[c + b*x]*(b*x*Cos[2*a - 4*c - b*x] - b*x*Cos[2*a - 2*c - b*x] + b*x*Cos[2*a + b*x] - b*x*Cos[2*a - 2*c + b*x] + 2*Sin[b*x] + Log[Sin[c + b*x]]*Sin[2*a - 4*c - b*x] - Sin[2*a - 2*c - b*x] - Log[Sin[c + b*x]]*Sin [2*a - 2*c - b*x] + Log[Sin[c + b*x]]*Sin[2*a + b*x] + Sin[2*a - 2*c + b*x ] - Log[Sin[c + b*x]]*Sin[2*a - 2*c + b*x]))/(4*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.
\(\displaystyle \int \cos ^2(a+b x) \csc ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^2(a+b x) \csc ^2(b x+c)dx\) |
Input:
Int[Cos[a + b*x]^2*Csc[c + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.43
method | result | size |
risch | \(-x \,{\mathrm e}^{2 i \left (a -c \right )}+2 i \sin \left (2 a -2 c \right ) x +\frac {2 i \sin \left (2 a -2 c \right ) a}{b}+\frac {i {\mathrm e}^{2 i \left (2 a -c \right )}}{2 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {i {\mathrm e}^{2 i a}}{b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {i {\mathrm e}^{2 i c}}{2 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) \sin \left (2 a -2 c \right )}{b}\) | \(175\) |
default | \(\frac {\frac {\frac {\left (-2 \cos \left (a \right )^{2} \cos \left (c \right ) \sin \left (c \right )+2 \cos \left (c \right )^{2} \cos \left (a \right ) \sin \left (a \right )-2 \cos \left (a \right ) \sin \left (a \right ) \sin \left (c \right )^{2}+2 \sin \left (a \right )^{2} \cos \left (c \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (b x +a \right )^{2}+1\right )}{2}+\left (-\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+\sin \left (c \right )^{2} \cos \left (a \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \arctan \left (\tan \left (b x +a \right )\right )}{\left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right )^{2} \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )^{2}}-\frac {\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}}{\left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right ) \left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right ) \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}+\frac {\left (-2 \cos \left (c \right )^{3} \sin \left (a \right ) \cos \left (a \right )^{2}+2 \cos \left (c \right )^{2} \sin \left (c \right ) \cos \left (a \right )^{3}-4 \cos \left (c \right )^{2} \sin \left (c \right ) \sin \left (a \right )^{2} \cos \left (a \right )+4 \cos \left (c \right ) \sin \left (c \right )^{2} \sin \left (a \right ) \cos \left (a \right )^{2}-2 \cos \left (c \right ) \sin \left (c \right )^{2} \sin \left (a \right )^{3}+2 \sin \left (c \right )^{3} \sin \left (a \right )^{2} \cos \left (a \right )\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right )^{2} \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )^{2} \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}}{b}\) | \(391\) |
Input:
int(cos(b*x+a)^2*csc(b*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-x*exp(2*I*(a-c))+2*I*sin(2*a-2*c)*x+2*I/b*sin(2*a-2*c)*a+1/2*I/b/(-exp(2* I*(b*x+a+c))+exp(2*I*a))*exp(2*I*(2*a-c))+I/b/(-exp(2*I*(b*x+a+c))+exp(2*I *a))*exp(2*I*a)+1/2*I/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))*exp(2*I*c)-ln(exp (2*I*(b*x+a))-exp(2*I*(a-c)))/b*sin(2*a-2*c)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.63 \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=\frac {2 \, \cos \left (-a + c\right ) \log \left (\frac {1}{2} \, \sin \left (b x + c\right )\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (b x + c\right ) \cos \left (-a + c\right )^{2} - {\left (2 \, b x \cos \left (-a + c\right )^{2} - b x\right )} \sin \left (b x + c\right )}{b \sin \left (b x + c\right )} \] Input:
integrate(cos(b*x+a)^2*csc(b*x+c)^2,x, algorithm="fricas")
Output:
(2*cos(-a + c)*log(1/2*sin(b*x + c))*sin(b*x + c)*sin(-a + c) - cos(b*x + c)*cos(-a + c)^2 - (2*b*x*cos(-a + c)^2 - b*x)*sin(b*x + c))/(b*sin(b*x + c))
Timed out. \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**2*csc(b*x+c)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (51) = 102\).
Time = 0.06 (sec) , antiderivative size = 711, normalized size of antiderivative = 13.94 \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx =\text {Too large to display} \] Input:
integrate(cos(b*x+a)^2*csc(b*x+c)^2,x, algorithm="maxima")
Output:
-1/2*(2*(b*cos(2*a + 2*c)*cos(4*c) + b*sin(2*a + 2*c)*sin(4*c))*x - (2*b*x *cos(4*c) + sin(4*a) + 2*sin(2*a + 2*c) + sin(4*c))*cos(2*b*x + 2*a + 4*c) + 2*(b*x*cos(2*b*x + 2*a + 4*c) - b*x*cos(2*a + 2*c))*cos(2*b*x + 6*c) + (sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(2*b*x + 2*a + 4*c)^2*sin(-2*a + 2*c) - 2*cos(2*b*x + 2*a + 4*c)*cos(2*a + 2*c)*sin(-2*a + 2*c) + cos(2*a + 2*c)^2*sin(-2*a + 2*c) + sin(2*b*x + 2*a + 4*c)^2*sin(-2*a + 2*c) - 2*s in(2*b*x + 2*a + 4*c)*sin(2*a + 2*c)*sin(-2*a + 2*c) + sin(2*a + 2*c)^2*si n(-2*a + 2*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) - (cos(2*b*x + 2*a + 4*c)^2*sin(-2*a + 2*c ) - 2*cos(2*b*x + 2*a + 4*c)*cos(2*a + 2*c)*sin(-2*a + 2*c) + cos(2*a + 2* c)^2*sin(-2*a + 2*c) + sin(2*b*x + 2*a + 4*c)^2*sin(-2*a + 2*c) - 2*sin(2* b*x + 2*a + 4*c)*sin(2*a + 2*c)*sin(-2*a + 2*c) + sin(2*a + 2*c)^2*sin(-2* a + 2*c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*s in(b*x)*sin(c) + sin(c)^2) - (2*b*x*sin(4*c) - cos(4*a) - 2*cos(2*a + 2*c) - cos(4*c))*sin(2*b*x + 2*a + 4*c) + 2*(b*x*sin(2*b*x + 2*a + 4*c) - b*x* sin(2*a + 2*c))*sin(2*b*x + 6*c) - (cos(4*a) + cos(4*c))*sin(2*a + 2*c))/( b*cos(2*b*x + 2*a + 4*c)^2 - 2*b*cos(2*b*x + 2*a + 4*c)*cos(2*a + 2*c) + b *cos(2*a + 2*c)^2 + b*sin(2*b*x + 2*a + 4*c)^2 - 2*b*sin(2*b*x + 2*a + 4*c )*sin(2*a + 2*c) + b*sin(2*a + 2*c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 2042 vs. \(2 (51) = 102\).
Time = 0.20 (sec) , antiderivative size = 2042, normalized size of antiderivative = 40.04 \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^2*csc(b*x+c)^2,x, algorithm="giac")
Output:
-((tan(1/2*a)^4*tan(1/2*c)^4 - 6*tan(1/2*a)^4*tan(1/2*c)^2 + 16*tan(1/2*a) ^3*tan(1/2*c)^3 - 6*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 - 16*tan(1/2* a)^3*tan(1/2*c) + 36*tan(1/2*a)^2*tan(1/2*c)^2 - 16*tan(1/2*a)*tan(1/2*c)^ 3 + tan(1/2*c)^4 - 6*tan(1/2*a)^2 + 16*tan(1/2*a)*tan(1/2*c) - 6*tan(1/2*c )^2 + 1)*(b*x + a)/(tan(1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^ 2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c) ^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) - 2*(tan(1/2*a)^4 *tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)^4*tan(1/2*c) + 6*ta n(1/2*a)^3*tan(1/2*c)^2 - 6*tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)*tan(1/2 *c)^4 - tan(1/2*a)^3 + 6*tan(1/2*a)^2*tan(1/2*c) - 6*tan(1/2*a)*tan(1/2*c) ^2 + tan(1/2*c)^3 + tan(1/2*a) - tan(1/2*c))*log(tan(b*x + a)^2 + 1)/(tan( 1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1 /2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*ta n(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) + 4*(tan(1/2*a)^6*tan(1/2*c)^5 - tan(1/2* a)^5*tan(1/2*c)^6 - 2*tan(1/2*a)^6*tan(1/2*c)^3 + 11*tan(1/2*a)^5*tan(1/2* c)^4 - 11*tan(1/2*a)^4*tan(1/2*c)^5 + 2*tan(1/2*a)^3*tan(1/2*c)^6 + tan(1/ 2*a)^6*tan(1/2*c) - 11*tan(1/2*a)^5*tan(1/2*c)^2 + 38*tan(1/2*a)^4*tan(1/2 *c)^3 - 38*tan(1/2*a)^3*tan(1/2*c)^4 + 11*tan(1/2*a)^2*tan(1/2*c)^5 - tan( 1/2*a)*tan(1/2*c)^6 + tan(1/2*a)^5 - 11*tan(1/2*a)^4*tan(1/2*c) + 38*tan(1 /2*a)^3*tan(1/2*c)^2 - 38*tan(1/2*a)^2*tan(1/2*c)^3 + 11*tan(1/2*a)*tan...
Time = 18.82 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.92 \[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=-x\,\left (\cos \left (2\,a-2\,c\right )-\sin \left (2\,a-2\,c\right )\,1{}\mathrm {i}\right )+\frac {\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}-c\,4{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{2\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}+c\,4{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\right )\,\left (2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-2\,b\,{\mathrm {e}}^{a\,6{}\mathrm {i}-c\,6{}\mathrm {i}}\right )\,1{}\mathrm {i}}{4\,b^2} \] Input:
int(cos(a + b*x)^2/sin(c + b*x)^2,x)
Output:
((2*exp(a*2i - c*2i) + exp(a*4i - c*4i) + 1)*1i)/(2*b*(exp(a*2i - c*2i) - exp(a*2i + b*x*2i))) - x*(cos(2*a - 2*c) - sin(2*a - 2*c)*1i) - (exp(c*4i - a*4i)*log(exp(a*2i)*exp(b*x*2i) - exp(a*2i)*exp(-c*2i))*(2*b*exp(a*2i - c*2i) - 2*b*exp(a*6i - c*6i))*1i)/(4*b^2)
\[ \int \cos ^2(a+b x) \csc ^2(c+b x) \, dx=\int \cos \left (b x +a \right )^{2} \csc \left (b x +c \right )^{2}d x \] Input:
int(cos(b*x+a)^2*csc(b*x+c)^2,x)
Output:
int(cos(a + b*x)**2*csc(b*x + c)**2,x)