Integrand size = 17, antiderivative size = 1 \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 134.00 \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=-\frac {\cos ^3(a-c) \csc (c+b x)}{b}-\frac {\cos (b x) \sin (3 a-2 c)}{b}+\frac {6 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos ^2(a-c) \sin (a-c)}{b}-\frac {\cos (3 a-2 c) \sin (b x)}{b} \] Input:
Integrate[Cos[a + b*x]^3*Csc[c + b*x]^2,x]
Output:
-((Cos[a - c]^3*Csc[c + b*x])/b) - (Cos[b*x]*Sin[3*a - 2*c])/b + ((6*I)*Ar cTan[((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])]*Cos[a - c]^2*Sin[a - c])/b - ( Cos[3*a - 2*c]*Sin[b*x])/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.
\(\displaystyle \int \cos ^3(a+b x) \csc ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^3(a+b x) \csc ^2(b x+c)dx\) |
Input:
Int[Cos[a + b*x]^3*Csc[c + b*x]^2,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.70 (sec) , antiderivative size = 231, normalized size of antiderivative = 231.00
method | result | size |
risch | \(\frac {i \left ({\mathrm e}^{i \left (b x +5 a -2 c \right )}+3 \,{\mathrm e}^{i \left (b x +3 a \right )}+3 \,{\mathrm e}^{i \left (b x +a +2 c \right )}+{\mathrm e}^{-i \left (-b x +a -4 c \right )}\right )}{4 \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right ) b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{4 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (3 a -3 c \right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (3 a -3 c \right )}{4 b}-\frac {\sin \left (b x +3 a -2 c \right )}{b}\) | \(231\) |
default | \(\text {Expression too large to display}\) | \(690\) |
Input:
int(cos(b*x+a)^3*csc(b*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/4*I/(-exp(2*I*(b*x+a+c))+exp(2*I*a))/b*(exp(I*(b*x+5*a-2*c))+3*exp(I*(b* x+3*a))+3*exp(I*(b*x+a+2*c))+exp(-I*(-b*x+a-4*c)))+3/4*ln(exp(I*(b*x+a))+e xp(I*(a-c)))/b*sin(a-c)+3/4*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*sin(3*a-3*c) -3/4*ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*sin(a-c)-3/4*ln(exp(I*(b*x+a))-exp( I*(a-c)))/b*sin(3*a-3*c)-sin(b*x+3*a-2*c)/b
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 159.00 \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=-\frac {3 \, \cos \left (-a + c\right )^{2} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 3 \, \cos \left (-a + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 2 \, {\left (4 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 2 \, {\left (4 \, \cos \left (-a + c\right )^{3} - 3 \, \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{2} + 10 \, \cos \left (-a + c\right )^{3} - 6 \, \cos \left (-a + c\right )}{2 \, b \sin \left (b x + c\right )} \] Input:
integrate(cos(b*x+a)^3*csc(b*x+c)^2,x, algorithm="fricas")
Output:
-1/2*(3*cos(-a + c)^2*log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a + c) - 3*cos(-a + c)^2*log(-1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a + c) - 2*(4*cos(-a + c)^2 - 1)*cos(b*x + c)*sin(b*x + c)*sin(-a + c) - 2*(4*cos( -a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^2 + 10*cos(-a + c)^3 - 6*cos(-a + c))/(b*sin(b*x + c))
Timed out. \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**3*csc(b*x+c)**2,x)
Output:
Timed out
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.08 (sec) , antiderivative size = 1042, normalized size of antiderivative = 1042.00 \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^3*csc(b*x+c)^2,x, algorithm="maxima")
Output:
1/8*(4*(sin(3*b*x + 3*a + 4*c) - sin(b*x + 3*a + 2*c))*cos(4*b*x + 6*a + 2 *c) + 2*(3*sin(2*b*x + 6*a) + 3*sin(2*b*x + 4*a + 2*c) + 3*sin(2*b*x + 2*a + 4*c) + 3*sin(2*b*x + 6*c) - 2*sin(4*c))*cos(3*b*x + 3*a + 4*c) - 3*((si n(-a + c) + sin(-3*a + 3*c))*cos(3*b*x + 3*a + 4*c)^2 - 2*(sin(-a + c) + s in(-3*a + 3*c))*cos(3*b*x + 3*a + 4*c)*cos(b*x + 3*a + 2*c) + (sin(-a + c) + sin(-3*a + 3*c))*cos(b*x + 3*a + 2*c)^2 + (sin(-a + c) + sin(-3*a + 3*c ))*sin(3*b*x + 3*a + 4*c)^2 - 2*(sin(-a + c) + sin(-3*a + 3*c))*sin(3*b*x + 3*a + 4*c)*sin(b*x + 3*a + 2*c) + (sin(-a + c) + sin(-3*a + 3*c))*sin(b* x + 3*a + 2*c)^2)*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*((sin(-a + c) + sin(-3*a + 3*c))*co s(3*b*x + 3*a + 4*c)^2 - 2*(sin(-a + c) + sin(-3*a + 3*c))*cos(3*b*x + 3*a + 4*c)*cos(b*x + 3*a + 2*c) + (sin(-a + c) + sin(-3*a + 3*c))*cos(b*x + 3 *a + 2*c)^2 + (sin(-a + c) + sin(-3*a + 3*c))*sin(3*b*x + 3*a + 4*c)^2 - 2 *(sin(-a + c) + sin(-3*a + 3*c))*sin(3*b*x + 3*a + 4*c)*sin(b*x + 3*a + 2* c) + (sin(-a + c) + sin(-3*a + 3*c))*sin(b*x + 3*a + 2*c)^2)*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) - 4*(cos(3*b*x + 3*a + 4*c) - cos(b*x + 3*a + 2*c))*sin(4*b*x + 6*a + 2*c) - 2*(3*cos(2*b*x + 6*a) + 3*cos(2*b*x + 4*a + 2*c) + 3*cos(2*b*x + 2* a + 4*c) + 3*cos(2*b*x + 6*c) - 2*cos(4*c))*sin(3*b*x + 3*a + 4*c) - 6*cos (b*x + 3*a + 2*c)*sin(2*b*x + 6*a) - 6*cos(b*x + 3*a + 2*c)*sin(2*b*x +...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.59 (sec) , antiderivative size = 5372, normalized size of antiderivative = 5372.00 \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^3*csc(b*x+c)^2,x, algorithm="giac")
Output:
-1/2*(12*(tan(1/2*a)^7*tan(1/2*c)^6 - tan(1/2*a)^6*tan(1/2*c)^7 - 2*tan(1/ 2*a)^7*tan(1/2*c)^4 + 12*tan(1/2*a)^6*tan(1/2*c)^5 - 12*tan(1/2*a)^5*tan(1 /2*c)^6 + 2*tan(1/2*a)^4*tan(1/2*c)^7 + tan(1/2*a)^7*tan(1/2*c)^2 - 13*tan (1/2*a)^6*tan(1/2*c)^3 + 49*tan(1/2*a)^5*tan(1/2*c)^4 - 49*tan(1/2*a)^4*ta n(1/2*c)^5 + 13*tan(1/2*a)^3*tan(1/2*c)^6 - tan(1/2*a)^2*tan(1/2*c)^7 + 2* tan(1/2*a)^6*tan(1/2*c) - 22*tan(1/2*a)^5*tan(1/2*c)^2 + 76*tan(1/2*a)^4*t an(1/2*c)^3 - 76*tan(1/2*a)^3*tan(1/2*c)^4 + 22*tan(1/2*a)^2*tan(1/2*c)^5 - 2*tan(1/2*a)*tan(1/2*c)^6 + tan(1/2*a)^5 - 13*tan(1/2*a)^4*tan(1/2*c) + 49*tan(1/2*a)^3*tan(1/2*c)^2 - 49*tan(1/2*a)^2*tan(1/2*c)^3 + 13*tan(1/2*a )*tan(1/2*c)^4 - tan(1/2*c)^5 - 2*tan(1/2*a)^3 + 12*tan(1/2*a)^2*tan(1/2*c ) - 12*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(1/2*c) + tan(1/2*b*x + 1/2*a) - tan(1/2*a) + tan(1/2*c)))/(tan(1/2*a)^7*tan(1/2*c)^7 + 3*tan(1/2*a)^7*t an(1/2*c)^5 + tan(1/2*a)^6*tan(1/2*c)^6 + 3*tan(1/2*a)^5*tan(1/2*c)^7 + 3* tan(1/2*a)^7*tan(1/2*c)^3 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 9*tan(1/2*a)^5*t an(1/2*c)^5 + 3*tan(1/2*a)^4*tan(1/2*c)^6 + 3*tan(1/2*a)^3*tan(1/2*c)^7 + tan(1/2*a)^7*tan(1/2*c) + 3*tan(1/2*a)^6*tan(1/2*c)^2 + 9*tan(1/2*a)^5*tan (1/2*c)^3 + 9*tan(1/2*a)^4*tan(1/2*c)^4 + 9*tan(1/2*a)^3*tan(1/2*c)^5 + 3* tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2*a)*tan(1/2*c)^7 + tan(1/2*a)^6 + 3*tan (1/2*a)^5*tan(1/2*c) + 9*tan(1/2*a)^4*tan(1/2*c)^2 + 9*tan(1/2*a)^3*tan...
Time = 25.92 (sec) , antiderivative size = 393, normalized size of antiderivative = 393.00 \[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=-\frac {{\mathrm {e}}^{-a\,3{}\mathrm {i}+c\,2{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,2{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,2{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{a\,4{}\mathrm {i}-c\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}-c\,6{}\mathrm {i}}+1\right )}{4\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}-\frac {3\,\ln \left (\frac {3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (\sin \left (2\,a-2\,c\right )\,1{}\mathrm {i}+\sin \left (2\,a-2\,c\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}{2}-\frac {\sin \left (2\,a-2\,c\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,3{}\mathrm {i}}{2\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\sin \left (2\,a-2\,c\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{4\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {3\,\ln \left (\frac {3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (\sin \left (2\,a-2\,c\right )\,1{}\mathrm {i}+\sin \left (2\,a-2\,c\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}{2}+\frac {\sin \left (2\,a-2\,c\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,3{}\mathrm {i}}{2\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\sin \left (2\,a-2\,c\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{4\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \] Input:
int(cos(a + b*x)^3/sin(c + b*x)^2,x)
Output:
(exp(a*3i - c*2i + b*x*1i)*1i)/(2*b) - (exp(c*2i - a*3i - b*x*1i)*1i)/(2*b ) - (exp(c*2i - a*1i + b*x*1i)*(3*exp(a*2i - c*2i) + 3*exp(a*4i - c*4i) + exp(a*6i - c*6i) + 1))/(4*b*(exp(a*2i - c*2i)*1i - exp(a*2i + b*x*2i)*1i)) - (3*log((3*exp(a*1i)*exp(b*x*1i)*(sin(2*a - 2*c)*1i + sin(2*a - 2*c)*exp (a*2i)*exp(-c*2i)*1i))/2 - (sin(2*a - 2*c)*exp(a*2i)*exp(-c*2i)*(exp(a*2i) *exp(-c*2i) + 1)*3i)/(2*(exp(a*2i)*exp(-c*2i))^(1/2)))*sin(2*a - 2*c)*(exp (a*2i - c*2i) + 1))/(4*b*exp(a*2i - c*2i)^(1/2)) + (3*log((3*exp(a*1i)*exp (b*x*1i)*(sin(2*a - 2*c)*1i + sin(2*a - 2*c)*exp(a*2i)*exp(-c*2i)*1i))/2 + (sin(2*a - 2*c)*exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) + 1)*3i)/(2*(e xp(a*2i)*exp(-c*2i))^(1/2)))*sin(2*a - 2*c)*(exp(a*2i - c*2i) + 1))/(4*b*e xp(a*2i - c*2i)^(1/2))
\[ \int \cos ^3(a+b x) \csc ^2(c+b x) \, dx=\int \cos \left (b x +a \right )^{3} \csc \left (b x +c \right )^{2}d x \] Input:
int(cos(b*x+a)^3*csc(b*x+c)^2,x)
Output:
int(cos(a + b*x)**3*csc(b*x + c)**2,x)