Integrand size = 18, antiderivative size = 85 \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\frac {\text {arctanh}(\cos (c-b x)) \cos ^2(a+c)}{2 b}-\frac {\text {arctanh}(\cos (c-b x)) \cos (2 (a+c))}{b}+\frac {\cos ^2(a+c) \cot (c-b x) \csc (c-b x)}{2 b}+\frac {\csc (c-b x) \sin (2 (a+c))}{b} \] Output:
1/2*arctanh(cos(b*x-c))*cos(a+c)^2/b-arctanh(cos(b*x-c))*cos(2*a+2*c)/b+1/ 2*cos(a+c)^2*cot(b*x-c)*csc(b*x-c)/b-csc(b*x-c)*sin(2*a+2*c)/b
Time = 5.97 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.93 \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\frac {2 \cos ^2(a+c) \csc ^2\left (\frac {1}{2} (c-b x)\right )+4 (1-3 \cos (2 (a+c))) \log \left (\cos \left (\frac {1}{2} (c-b x)\right )\right )+4 (-1+3 \cos (2 (a+c))) \log \left (-\sin \left (\frac {1}{2} (c-b x)\right )\right )-2 \cos ^2(a+c) \sec ^2\left (\frac {1}{2} (c-b x)\right )+8 \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} (c-b x)\right ) \sin (2 (a+c)) \sin \left (\frac {b x}{2}\right )-8 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c-b x)\right ) \sin (2 (a+c)) \sin \left (\frac {b x}{2}\right )}{16 b} \] Input:
Integrate[Cos[a + b*x]^2*Csc[c - b*x]^3,x]
Output:
(2*Cos[a + c]^2*Csc[(c - b*x)/2]^2 + 4*(1 - 3*Cos[2*(a + c)])*Log[Cos[(c - b*x)/2]] + 4*(-1 + 3*Cos[2*(a + c)])*Log[-Sin[(c - b*x)/2]] - 2*Cos[a + c ]^2*Sec[(c - b*x)/2]^2 + 8*Csc[c/2]*Csc[(c - b*x)/2]*Sin[2*(a + c)]*Sin[(b *x)/2] - 8*Sec[c/2]*Sec[(c - b*x)/2]*Sin[2*(a + c)]*Sin[(b*x)/2])/(16*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.
\(\displaystyle \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^2(a+b x) \csc ^3(c-b x)dx\) |
Input:
Int[Cos[a + b*x]^2*Csc[c - b*x]^3,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 4.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.74
method | result | size |
risch | \(\frac {3 \,{\mathrm e}^{i \left (b x +6 a +5 c \right )}-5 \,{\mathrm e}^{3 i \left (b x +2 a +c \right )}-2 \,{\mathrm e}^{i \left (b x +4 a +3 c \right )}-2 \,{\mathrm e}^{i \left (3 b x +4 a +c \right )}-5 \,{\mathrm e}^{i \left (b x +2 a +c \right )}+3 \,{\mathrm e}^{i \left (3 b x +2 a -c \right )}}{4 \left ({\mathrm e}^{2 i \left (a +c \right )}-{\mathrm e}^{2 i \left (b x +a \right )}\right )^{2} b}-\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right )}{4 b}+\frac {3 \ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (2 a +2 c \right )}{4 b}+\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (2 a +2 c \right )}{4 b}\) | \(233\) |
default | \(\text {Expression too large to display}\) | \(1487\) |
Input:
int(-cos(b*x+a)^2*csc(b*x-c)^3,x,method=_RETURNVERBOSE)
Output:
1/4/(exp(2*I*(a+c))-exp(2*I*(b*x+a)))^2/b*(3*exp(I*(b*x+6*a+5*c))-5*exp(3* I*(b*x+2*a+c))-2*exp(I*(b*x+4*a+3*c))-2*exp(I*(3*b*x+4*a+c))-5*exp(I*(b*x+ 2*a+c))+3*exp(I*(3*b*x+2*a-c)))-1/4/b*ln(-exp(I*(a+c))+exp(I*(b*x+a)))+3/4 /b*ln(-exp(I*(a+c))+exp(I*(b*x+a)))*cos(2*a+2*c)+1/4/b*ln(exp(I*(a+c))+exp (I*(b*x+a)))-3/4/b*ln(exp(I*(a+c))+exp(I*(b*x+a)))*cos(2*a+2*c)
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (91) = 182\).
Time = 0.09 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.87 \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\frac {6 \, \cos \left (a + c\right )^{2} \sin \left (b x + a\right ) \sin \left (a + c\right ) + 2 \, {\left (3 \, \cos \left (a + c\right )^{3} - 4 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right ) + {\left (3 \, \cos \left (a + c\right )^{4} - 2 \, {\left (3 \, \cos \left (a + c\right )^{3} - 2 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) - {\left (6 \, \cos \left (a + c\right )^{4} - 7 \, \cos \left (a + c\right )^{2} + 2\right )} \cos \left (b x + a\right )^{2} - 2 \, \cos \left (a + c\right )^{2}\right )} \log \left (\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) + 1}{\cos \left (a + c\right ) + 1}\right ) - {\left (3 \, \cos \left (a + c\right )^{4} - 2 \, {\left (3 \, \cos \left (a + c\right )^{3} - 2 \, \cos \left (a + c\right )\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) - {\left (6 \, \cos \left (a + c\right )^{4} - 7 \, \cos \left (a + c\right )^{2} + 2\right )} \cos \left (b x + a\right )^{2} - 2 \, \cos \left (a + c\right )^{2}\right )} \log \left (-\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) - 1}{\cos \left (a + c\right ) + 1}\right )}{4 \, {\left (2 \, b \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + {\left (2 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{2} - b \cos \left (a + c\right )^{2}\right )}} \] Input:
integrate(-cos(b*x+a)^2*csc(b*x-c)^3,x, algorithm="fricas")
Output:
1/4*(6*cos(a + c)^2*sin(b*x + a)*sin(a + c) + 2*(3*cos(a + c)^3 - 4*cos(a + c))*cos(b*x + a) + (3*cos(a + c)^4 - 2*(3*cos(a + c)^3 - 2*cos(a + c))*c os(b*x + a)*sin(b*x + a)*sin(a + c) - (6*cos(a + c)^4 - 7*cos(a + c)^2 + 2 )*cos(b*x + a)^2 - 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)) - (3*cos(a + c)^4 - 2*(3*cos(a + c) ^3 - 2*cos(a + c))*cos(b*x + a)*sin(b*x + a)*sin(a + c) - (6*cos(a + c)^4 - 7*cos(a + c)^2 + 2)*cos(b*x + a)^2 - 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)))/(2*b*cos(b*x + a)*cos(a + c)*sin(b*x + a)*sin(a + c) + (2*b*cos(a + c)^2 - b)*cos(b*x + a)^2 - b*cos(a + c)^2)
Timed out. \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\text {Timed out} \] Input:
integrate(-cos(b*x+a)**2*csc(b*x-c)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1571 vs. \(2 (91) = 182\).
Time = 0.10 (sec) , antiderivative size = 1571, normalized size of antiderivative = 18.48 \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(-cos(b*x+a)^2*csc(b*x-c)^3,x, algorithm="maxima")
Output:
1/8*(2*(3*cos(3*b*x) - 5*cos(3*b*x + 4*a + 4*c) - 2*cos(3*b*x + 2*a + 2*c) + 3*cos(b*x + 4*a + 6*c) - 2*cos(b*x + 2*a + 4*c) - 5*cos(b*x + 2*c))*cos (4*b*x + 2*a + c) + 10*(2*cos(2*b*x + 2*a + 3*c) - cos(2*a + 5*c))*cos(3*b *x + 4*a + 4*c) + 4*(2*cos(2*b*x + 2*a + 3*c) - cos(2*a + 5*c))*cos(3*b*x + 2*a + 2*c) - 4*(3*cos(3*b*x) + 3*cos(b*x + 4*a + 6*c) - 2*cos(b*x + 2*a + 4*c) - 5*cos(b*x + 2*c))*cos(2*b*x + 2*a + 3*c) + 6*cos(3*b*x)*cos(2*a + 5*c) + 6*cos(b*x + 4*a + 6*c)*cos(2*a + 5*c) - 4*cos(b*x + 2*a + 4*c)*cos (2*a + 5*c) - 10*cos(b*x + 2*c)*cos(2*a + 5*c) - ((3*cos(2*a + 2*c) - 1)*c os(4*b*x + 2*a + c)^2 + 4*(3*cos(2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c)^2 - 4*(3*cos(2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c)*cos(2*a + 5*c) + (3*cos( 2*a + 2*c) - 1)*cos(2*a + 5*c)^2 + (3*cos(2*a + 2*c) - 1)*sin(4*b*x + 2*a + c)^2 + 4*(3*cos(2*a + 2*c) - 1)*sin(2*b*x + 2*a + 3*c)^2 - 4*(3*cos(2*a + 2*c) - 1)*sin(2*b*x + 2*a + 3*c)*sin(2*a + 5*c) + (3*cos(2*a + 2*c) - 1) *sin(2*a + 5*c)^2 - 2*(2*(3*cos(2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c) - ( 3*cos(2*a + 2*c) - 1)*cos(2*a + 5*c))*cos(4*b*x + 2*a + c) - 2*(2*(3*cos(2 *a + 2*c) - 1)*sin(2*b*x + 2*a + 3*c) - (3*cos(2*a + 2*c) - 1)*sin(2*a + 5 *c))*sin(4*b*x + 2*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) + ((3*cos(2*a + 2*c) - 1)*cos( 4*b*x + 2*a + c)^2 + 4*(3*cos(2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c)^2 - 4 *(3*cos(2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c)*cos(2*a + 5*c) + (3*cos(...
Leaf count of result is larger than twice the leaf count of optimal. 6142 vs. \(2 (91) = 182\).
Time = 0.39 (sec) , antiderivative size = 6142, normalized size of antiderivative = 72.26 \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(-cos(b*x+a)^2*csc(b*x-c)^3,x, algorithm="giac")
Output:
1/8*(4*(tan(1/2*a)^5*tan(1/2*c)^5 - 10*tan(1/2*a)^5*tan(1/2*c)^3 - 25*tan( 1/2*a)^4*tan(1/2*c)^4 - 10*tan(1/2*a)^3*tan(1/2*c)^5 + tan(1/2*a)^5*tan(1/ 2*c) + 34*tan(1/2*a)^4*tan(1/2*c)^2 + 76*tan(1/2*a)^3*tan(1/2*c)^3 + 34*ta n(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)*tan(1/2*c)^5 - tan(1/2*a)^4 - 34*tan( 1/2*a)^3*tan(1/2*c) - 76*tan(1/2*a)^2*tan(1/2*c)^2 - 34*tan(1/2*a)*tan(1/2 *c)^3 - tan(1/2*c)^4 + 10*tan(1/2*a)^2 + 25*tan(1/2*a)*tan(1/2*c) + 10*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*tan(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) - 4*(tan(1/2*a)^5*tan(1/2*c)^4 + tan(1/ 2*a)^4*tan(1/2*c)^5 - 10*tan(1/2*a)^5*tan(1/2*c)^2 - 34*tan(1/2*a)^4*tan(1 /2*c)^3 - 34*tan(1/2*a)^3*tan(1/2*c)^4 - 10*tan(1/2*a)^2*tan(1/2*c)^5 + ta n(1/2*a)^5 + 25*tan(1/2*a)^4*tan(1/2*c) + 76*tan(1/2*a)^3*tan(1/2*c)^2 + 7 6*tan(1/2*a)^2*tan(1/2*c)^3 + 25*tan(1/2*a)*tan(1/2*c)^4 + tan(1/2*c)^5 - 10*tan(1/2*a)^3 - 34*tan(1/2*a)^2*tan(1/2*c) - 34*tan(1/2*a)*tan(1/2*c)^2 - 10*tan(1/2*c)^3 + tan(1/2*a) + tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*...
Timed out. \[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=\text {Hanged} \] Input:
int(cos(a + b*x)^2/sin(c - b*x)^3,x)
Output:
\text{Hanged}
\[ \int \cos ^2(a+b x) \csc ^3(c-b x) \, dx=-\left (\int \cos \left (b x +a \right )^{2} \csc \left (b x -c \right )^{3}d x \right ) \] Input:
int(-cos(b*x+a)^2*csc(b*x-c)^3,x)
Output:
- int(cos(a + b*x)**2*csc(b*x - c)**3,x)