Integrand size = 16, antiderivative size = 33 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (c-b x)) \cos (a+c)}{b}+\frac {\csc (c-b x) \sin (a+c)}{b} \] Output:
-arctanh(cos(b*x-c))*cos(a+c)/b-csc(b*x-c)*sin(a+c)/b
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.58 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx=-\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )+\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a+c)}{b}+\frac {\csc (c-b x) \sin (a+c)}{b} \] Input:
Integrate[Csc[c - b*x]^2*Sin[a + b*x],x]
Output:
((-2*I)*ArcTan[((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])/b + (Csc [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.
\(\displaystyle \int \sin (a+b x) \csc ^2(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin (a+b x) \csc ^2(c-b x)dx\) |
Input:
Int[Csc[c - b*x]^2*Sin[a + b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.27
method | result | size |
risch | \(\frac {{\mathrm e}^{i \left (b x +3 a +2 c \right )}-{\mathrm e}^{i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (a +c \right )}-{\mathrm e}^{2 i \left (b x +a \right )}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (a +c \right )}{b}+\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (a +c \right )}{b}\) | \(108\) |
default | \(\frac {\frac {4 \left (-2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+8 \sin \left (a \right ) \cos \left (c \right )+8 \cos \left (a \right ) \sin \left (c \right )}{\left (-4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-4 \sin \left (c \right )^{2} \cos \left (a \right )^{2}\right ) \left (\sin \left (c \right ) \cos \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}+\cos \left (c \right ) \sin \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}+2 \tan \left (\frac {a}{2}+\frac {b x}{2}\right ) \cos \left (a \right ) \cos \left (c \right )-2 \tan \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (a \right ) \sin \left (c \right )-\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right )}+\frac {4 \left (-2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )\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 \sin \left (a \right ) \sin \left (c \right )+2 \cos \left (a \right ) \cos \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 (-4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-4 \sin \left (c \right )^{2} \cos \left (a \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}}}}{b}\) | \(346\) |
Input:
int(csc(b*x-c)^2*sin(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b/(exp(2*I*(a+c))-exp(2*I*(b*x+a)))*(exp(I*(b*x+3*a+2*c))-exp(I*(b*x+a)) )-ln(exp(I*(a+c))+exp(I*(b*x+a)))/b*cos(a+c)+ln(-exp(I*(a+c))+exp(I*(b*x+a )))/b*cos(a+c)
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (36) = 72\).
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.12 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx=-\frac {{\left (\cos \left (a + c\right )^{2} \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (a + c\right )\right )} \log \left (\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) + 1}{\cos \left (a + c\right ) + 1}\right ) - {\left (\cos \left (a + c\right )^{2} \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (a + c\right )\right )} \log \left (-\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) - 1}{\cos \left (a + c\right ) + 1}\right ) + 2 \, \sin \left (a + c\right )}{2 \, {\left (b \cos \left (a + c\right ) \sin \left (b x + a\right ) - b \cos \left (b x + a\right ) \sin \left (a + c\right )\right )}} \] Input:
integrate(csc(b*x-c)^2*sin(b*x+a),x, algorithm="fricas")
Output:
-1/2*((cos(a + c)^2*sin(b*x + a) - cos(b*x + a)*cos(a + c)*sin(a + c))*log ((cos(b*x + a)*cos(a + c) + sin(b*x + a)*sin(a + c) + 1)/(cos(a + c) + 1)) - (cos(a + c)^2*sin(b*x + a) - cos(b*x + a)*cos(a + c)*sin(a + c))*log(-( cos(b*x + a)*cos(a + c) + sin(b*x + a)*sin(a + c) - 1)/(cos(a + c) + 1)) + 2*sin(a + c))/(b*cos(a + c)*sin(b*x + a) - b*cos(b*x + a)*sin(a + c))
Leaf count of result is larger than twice the leaf count of optimal. 1690 vs. \(2 (29) = 58\).
Time = 62.34 (sec) , antiderivative size = 3264, normalized size of antiderivative = 98.91 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x-c)**2*sin(b*x+a),x)
Output:
Piecewise((0, Eq(b, 0) & Eq(c, 0)), (log(tan(b*x/2))/b, Eq(c, 0)), (0, Eq( b, 0)), (log(-tan(c/2) + tan(b*x/2))*tan(c/2)**4*tan(b*x/2)/(-b*tan(c/2)** 4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*ta n(b*x/2)**2 - b*tan(c/2) + b*tan(b*x/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) + b*tan(c/2)**3*tan(b*x/ 2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) + b*tan(b*x/ 2)) + log(-tan(c/2) + tan(b*x/2))*tan(c/2)**3/(-b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) + b*tan(b*x/2)) - 2*log(-tan(c/2) + tan(b*x/2))*tan(c/2)**2*tan (b*x/2)/(-b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c /2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) + b*tan(b*x/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) + b*t an(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*ta n(c/2) + b*tan(b*x/2)) - log(-tan(c/2) + tan(b*x/2))*tan(c/2)/(-b*tan(c/2) **4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)* tan(b*x/2)**2 - b*tan(c/2) + b*tan(b*x/2)) + log(-tan(c/2) + tan(b*x/2))*t an(b*x/2)/(-b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan (c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) + b*tan(b*x/2)) - log(tan (b*x/2) + 1/tan(c/2))*tan(c/2)**4*tan(b*x/2)/(-b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 ...
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (36) = 72\).
Time = 0.05 (sec) , antiderivative size = 461, normalized size of antiderivative = 13.97 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:
integrate(csc(b*x-c)^2*sin(b*x+a),x, algorithm="maxima")
Output:
-1/2*(2*(cos(b*x + 2*a + 2*c) - cos(b*x))*cos(2*b*x + a) - 2*cos(b*x + 2*a + 2*c)*cos(a + 2*c) + 2*cos(b*x)*cos(a + 2*c) + (cos(2*b*x + a)^2*cos(a + c) - 2*cos(2*b*x + a)*cos(a + 2*c)*cos(a + c) + cos(a + 2*c)^2*cos(a + c) + cos(a + c)*sin(2*b*x + a)^2 - 2*cos(a + c)*sin(2*b*x + a)*sin(a + 2*c) + cos(a + c)*sin(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) - (cos(2*b*x + a)^2*cos(a + c) - 2*cos(2*b*x + a)*cos(a + 2*c)*cos(a + c) + cos(a + 2*c)^2*cos(a + c) + cos(a + c)*sin(2*b*x + a)^2 - 2*cos(a + c)*sin(2*b*x + a)*sin(a + 2*c) + cos(a + c)*sin(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) + 2*(sin(b*x + 2*a + 2*c) - s in(b*x))*sin(2*b*x + a) - 2*sin(b*x + 2*a + 2*c)*sin(a + 2*c) + 2*sin(b*x) *sin(a + 2*c))/(b*cos(2*b*x + a)^2 - 2*b*cos(2*b*x + a)*cos(a + 2*c) + b*c os(a + 2*c)^2 + b*sin(2*b*x + a)^2 - 2*b*sin(2*b*x + a)*sin(a + 2*c) + b*s in(a + 2*c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (36) = 72\).
Time = 0.14 (sec) , antiderivative size = 347, normalized size of antiderivative = 10.52 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:
integrate(csc(b*x-c)^2*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) - tan (1/2*c)^2 + 1)*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) + (tan(1/2*b*x - 1/2*c)*tan(1/2*a)^2*tan (1/2*c) + tan(1/2*b*x - 1/2*c)*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*b*x - 1/2 *c)*tan(1/2*a) - tan(1/2*b*x - 1/2*c)*tan(1/2*c))/(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*c)*tan(1/2*a)^2 *tan(1/2*c)^2 - tan(1/2*b*x - 1/2*c)*tan(1/2*a)^2 - 4*tan(1/2*b*x - 1/2*c) *tan(1/2*a)*tan(1/2*c) - tan(1/2*a)^2*tan(1/2*c) - tan(1/2*b*x - 1/2*c)*ta n(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*b*x - 1/2*c) + tan(1/2*a) + tan(1/2*c))/((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*c)))/b
Time = 22.00 (sec) , antiderivative size = 252, normalized size of antiderivative = 7.64 \[ \int \csc ^2(c-b x) \sin (a+b x) \, dx=-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \] Input:
int(sin(a + b*x)/sin(c - b*x)^2,x)
Output:
(log((exp(a*2i)*exp(c*2i)*(exp(a*2i)*exp(c*2i) + 1)*1i)/(exp(a*2i)*exp(c*2 i))^(1/2) - exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(c*2i)*1i + 1i))*(exp(a*2i + c*2i) + 1))/(2*b*exp(a*2i + c*2i)^(1/2)) - (log(- exp(a*1i)*exp(b*x*1i) *(exp(a*2i)*exp(c*2i)*1i + 1i) - (exp(a*2i)*exp(c*2i)*(exp(a*2i)*exp(c*2i) + 1)*1i)/(exp(a*2i)*exp(c*2i))^(1/2))*(exp(a*2i + c*2i) + 1))/(2*b*exp(a* 2i + c*2i)^(1/2)) + (exp(a*1i + b*x*1i)*(exp(a*2i + c*2i) - 1))/(b*(exp(a* 2i + c*2i) - exp(a*2i + b*x*2i)))
\[ \int \csc ^2(c-b x) \sin (a+b x) \, dx=\int \csc \left (b x -c \right )^{2} \sin \left (b x +a \right )d x \] Input:
int(csc(b*x-c)^2*sin(b*x+a),x)
Output:
int(csc(b*x - c)**2*sin(a + b*x),x)