Integrand size = 14, antiderivative size = 25 \[ \int \csc (c-b x) \sin (a+b x) \, dx=-x \cos (a+c)-\frac {\log (\sin (c-b x)) \sin (a+c)}{b} \] Output:
-x*cos(a+c)-ln(-sin(b*x-c))*sin(a+c)/b
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \csc (c-b x) \sin (a+b x) \, dx=-\frac {b x \cos (a+c)+\log (\sin (c-b x)) \sin (a+c)}{b} \] Input:
Integrate[Csc[c - b*x]*Sin[a + b*x],x]
Output:
-((b*x*Cos[a + c] + Log[Sin[c - b*x]]*Sin[a + c])/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.
\(\displaystyle \int \sin (a+b x) \csc (c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin (a+b x) \csc (c-b x)dx\) |
Input:
Int[Csc[c - b*x]*Sin[a + b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40
method | result | size |
risch | \(2 i \sin \left (a +c \right ) x -x \,{\mathrm e}^{i \left (a +c \right )}+\frac {2 i \sin \left (a +c \right ) a}{b}-\frac {\ln \left (-{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right ) \sin \left (a +c \right )}{b}\) | \(60\) |
default | \(-\frac {\frac {\left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )}{\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}+\frac {\frac {\left (-\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (b x +a \right )^{2}+1\right )}{2}+\left (\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\tan \left (b x +a \right )\right )}{\left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right ) \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )}}{b}\) | \(165\) |
Input:
int(-csc(b*x-c)*sin(b*x+a),x,method=_RETURNVERBOSE)
Output:
2*I*sin(a+c)*x-x*exp(I*(a+c))+2*I/b*sin(a+c)*a-ln(-exp(2*I*(a+c))+exp(2*I* (b*x+a)))/b*sin(a+c)
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \csc (c-b x) \sin (a+b x) \, dx=-\frac {b x \cos \left (a + c\right ) + \log \left (\frac {\cos \left (a + c\right ) \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \sin \left (a + c\right )}{\cos \left (a + c\right ) + 1}\right ) \sin \left (a + c\right )}{b} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a),x, algorithm="fricas")
Output:
-(b*x*cos(a + c) + log((cos(a + c)*sin(b*x + a) - cos(b*x + a)*sin(a + c)) /(cos(a + c) + 1))*sin(a + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (24) = 48\).
Time = 4.74 (sec) , antiderivative size = 337, normalized size of antiderivative = 13.48 \[ \int \csc (c-b x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a),x)
Output:
Piecewise((0, Eq(b, 0) & Eq(c, 0)), (-x, Eq(c, 0)), (0, Eq(b, 0)), (b*x*ta n(c/2)**2/(b*tan(c/2)**2 + b) - b*x/(b*tan(c/2)**2 + b) - 2*log(-tan(c/2) + tan(b*x/2))*tan(c/2)/(b*tan(c/2)**2 + b) - 2*log(tan(b*x/2) + 1/tan(c/2) )*tan(c/2)/(b*tan(c/2)**2 + b) + 2*log(tan(b*x/2)**2 + 1)*tan(c/2)/(b*tan( c/2)**2 + b), True))*cos(a) + Piecewise((zoo*x, Eq(b, 0) & Eq(c, 0)), (-lo g(sin(b*x))/b, Eq(c, 0)), (x/sin(c), Eq(b, 0)), (2*b*x*tan(c/2)/(b*tan(c/2 )**2 + b) + log(-tan(c/2) + tan(b*x/2))*tan(c/2)**2/(b*tan(c/2)**2 + b) - log(-tan(c/2) + tan(b*x/2))/(b*tan(c/2)**2 + b) + log(tan(b*x/2) + 1/tan(c /2))*tan(c/2)**2/(b*tan(c/2)**2 + b) - log(tan(b*x/2) + 1/tan(c/2))/(b*tan (c/2)**2 + b) - log(tan(b*x/2)**2 + 1)*tan(c/2)**2/(b*tan(c/2)**2 + b) + l og(tan(b*x/2)**2 + 1)/(b*tan(c/2)**2 + b), True))*sin(a)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (28) = 56\).
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.00 \[ \int \csc (c-b x) \sin (a+b x) \, dx=-\frac {2 \, b x \cos \left (a + c\right ) + \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (a + c\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (a + c\right )}{2 \, b} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a),x, algorithm="maxima")
Output:
-1/2*(2*b*x*cos(a + c) + log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + s in(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2)*sin(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)*si n(a + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (28) = 56\).
Time = 0.13 (sec) , antiderivative size = 241, normalized size of antiderivative = 9.64 \[ \int \csc (c-b x) \sin (a+b x) \, dx=-\frac {\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} {\left (b x - c\right )}}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left (\tan \left (\frac {1}{2} \, b x - \frac {1}{2} \, c\right )^{2} + 1\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x - \frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1}}{b} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a),x, algorithm="giac")
Output:
-((tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 - 4*tan(1/2*a)*tan(1/2*c) - ta n(1/2*c)^2 + 1)*(b*x - c)/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan( 1/2*c)^2 + 1) + 2*(tan(1/2*a)^2*tan(1/2*c) + tan(1/2*a)*tan(1/2*c)^2 - tan (1/2*a) - tan(1/2*c))*log(tan(1/2*b*x - 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) - 2*(tan(1/2*a)^2*tan(1/2*c) + t an(1/2*a)*tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x - 1/ 2*c)))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1))/b
Time = 0.71 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.56 \[ \int \csc (c-b x) \sin (a+b x) \, dx=-x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}}{2}\right )-x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}}{2}\right )-\frac {\ln \left (-{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}{b} \] Input:
int(sin(a + b*x)/sin(c - b*x),x)
Output:
- x*(exp(- a*1i - c*1i)/2 - exp(a*1i + c*1i)/2) - x*(exp(- a*1i - c*1i)/2 + exp(a*1i + c*1i)/2) - (log(exp(a*2i + b*x*2i) - exp(a*2i + c*2i))*((exp( - a*1i - c*1i)*1i)/2 - (exp(a*1i + c*1i)*1i)/2))/b
\[ \int \csc (c-b x) \sin (a+b x) \, dx=-\left (\int \csc \left (b x -c \right ) \sin \left (b x +a \right )d x \right ) \] Input:
int(-csc(b*x-c)*sin(b*x+a),x)
Output:
- int(csc(b*x - c)*sin(a + b*x),x)