Integrand size = 13, antiderivative size = 29 \[ \int \cot (c+b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \sin (a-c)}{b}+\frac {\sin (a+b x)}{b} \] Output:
-arctanh(cos(b*x+c))*sin(a-c)/b+sin(b*x+a)/b
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21 \[ \int \cot (c+b x) \sin (a+b x) \, dx=\frac {\cos (b x) \sin (a)}{b}-\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{b}+\frac {\cos (a) \sin (b x)}{b} \] Input:
Integrate[Cot[c + b*x]*Sin[a + b*x],x]
Output:
(Cos[b*x]*Sin[a])/b - ((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])] *Sin[a - c])/b + (Cos[a]*Sin[b*x])/b
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5089, 3042, 3117, 4257}
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.
\(\displaystyle \int \sin (a+b x) \cot (b x+c) \, dx\) |
\(\Big \downarrow \) 5089 |
\(\displaystyle \sin (a-c) \int \csc (c+b x)dx+\int \cos (a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sin (a-c) \int \csc (c+b x)dx+\int \sin \left (a+b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \sin (a-c) \int \csc (c+b x)dx+\frac {\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\sin (a+b x)}{b}-\frac {\sin (a-c) \text {arctanh}(\cos (b x+c))}{b}\) |
Input:
Int[Cot[c + b*x]*Sin[a + b*x],x]
Output:
-((ArcTanh[Cos[c + b*x]]*Sin[a - c])/b) + Sin[a + b*x]/b
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cot[w_]^(n_.)*Sin[v_], x_Symbol] :> Int[Cos[v]*Cot[w]^(n - 1), x] + Sim p[Sin[v - w] Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.28
method | result | size |
risch | \(-\frac {i {\mathrm e}^{i \left (b x +a \right )}}{2 b}+\frac {i {\mathrm e}^{-i \left (b x +a \right )}}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\) | \(95\) |
Input:
int(cot(b*x+c)*sin(b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/2*I/b*exp(I*(b*x+a))+1/2*I/b*exp(-I*(b*x+a))+ln(exp(I*(b*x+a))-exp(I*(a -c)))/b*sin(a-c)-ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*sin(a-c)
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 6.79 \[ \int \cot (c+b x) \sin (a+b x) \, dx=\frac {\frac {\sqrt {2} \log \left (\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right ) \sin \left (-2 \, a + 2 \, c\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} + 4 \, \sin \left (b x + a\right )}{4 \, b} \] Input:
integrate(cot(b*x+c)*sin(b*x+a),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*log((2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b* x + a)*sin(-2*a + 2*c) + 2*sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a) - s in(b*x + a)*sin(-2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - cos(-2*a + 2*c) + 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2 *a + 2*c) - cos(-2*a + 2*c) - 1))*sin(-2*a + 2*c)/sqrt(cos(-2*a + 2*c) + 1 ) + 4*sin(b*x + a))/b
\[ \int \cot (c+b x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \cot {\left (b x + c \right )}\, dx \] Input:
integrate(cot(b*x+c)*sin(b*x+a),x)
Output:
Integral(sin(a + b*x)*cot(b*x + c), x)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (29) = 58\).
Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.62 \[ \int \cot (c+b x) \sin (a+b x) \, dx=\frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (-a + c\right ) - \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (-a + c\right ) + 2 \, \sin \left (b x + a\right )}{2 \, b} \] Input:
integrate(cot(b*x+c)*sin(b*x+a),x, algorithm="maxima")
Output:
1/2*(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)*sin(-a + c) - log(cos(b*x)^2 - 2*cos(b*x)*cos(c) + c os(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2)*sin(-a + c) + 2*sin(b *x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (29) = 58\).
Time = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 7.79 \[ \int \cot (c+b x) \sin (a+b x) \, dx=-\frac {2 \, {\left (\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x\right ) + \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {\tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - \tan \left (\frac {1}{2} \, b x\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )}{{\left (\tan \left (\frac {1}{2} \, b x\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} + 1\right )}}\right )}}{b} \] Input:
integrate(cot(b*x+c)*sin(b*x+a),x, algorithm="giac")
Output:
-2*((tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)*tan( 1/2*c) - tan(1/2*c)^2)*log(abs(tan(1/2*b*x)*tan(1/2*c) - 1))/(tan(1/2*a)^2 *tan(1/2*c)^3 + tan(1/2*a)^2*tan(1/2*c) + tan(1/2*c)^3 + tan(1/2*c)) - (ta n(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(abs(tan(1/2*b*x) + 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)*tan(1/2*a)^2 - tan(1/2*b*x) - 2*t an(1/2*a))/((tan(1/2*b*x)^2 + 1)*(tan(1/2*a)^2 + 1)))/b
Time = 20.98 (sec) , antiderivative size = 233, normalized size of antiderivative = 8.03 \[ \int \cot (c+b x) \sin (a+b x) \, dx=\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \] Input:
int(cot(c + b*x)*sin(a + b*x),x)
Output:
(exp(- a*1i - b*x*1i)*1i)/(2*b) - (exp(a*1i + b*x*1i)*1i)/(2*b) - (log(- e xp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i) - 1) - (exp(a*2i)*exp(-c*2i)*(e xp(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)) + (log((exp(a*2i)*exp(-c*2i)*(ex p(a*2i)*exp(-c*2i) - 1)*1i)/(-exp(a*2i)*exp(-c*2i))^(1/2) - exp(a*1i)*exp( b*x*1i)*(exp(a*2i)*exp(-c*2i) - 1))*(exp(a*2i - c*2i) - 1))/(2*b*(-exp(a*2 i - c*2i))^(1/2))
\[ \int \cot (c+b x) \sin (a+b x) \, dx=\int \cot \left (b x +c \right ) \sin \left (b x +a \right )d x \] Input:
int(cot(b*x+c)*sin(b*x+a),x)
Output:
int(cot(b*x + c)*sin(a + b*x),x)