\(\int \cos (a+b x) \cot (c+b x) \, dx\) [250]

Optimal result

Integrand size = 13, antiderivative size = 29 \[ \int \cos (a+b x) \cot (c+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}+\frac {\cos (a+b x)}{b} \]

[Out]

-arctanh(cos(b*x+c))*cos(a-c)/b+cos(b*x+a)/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4673, 2718, 3855} \[ \int \cos (a+b x) \cot (c+b x) \, dx=\frac {\cos (a+b x)}{b}-\frac {\cos (a-c) \text {arctanh}(\cos (b x+c))}{b} \]

[In]

Int[Cos[a + b*x]*Cot[c + b*x],x]

[Out]

-((ArcTanh[Cos[c + b*x]]*Cos[a - c])/b) + Cos[a + b*x]/b

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4673

Int[Cos[v_]*Cot[w_]^(n_.), x_Symbol] :> -Int[Sin[v]*Cot[w]^(n - 1), x] + Dist[Cos[v - w], Int[Csc[w]*Cot[w]^(n
- 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]

Rubi steps \begin{align*} \text {integral}& = \cos (a-c) \int \csc (c+b x) \, dx-\int \sin (a+b x) \, dx \\ & = -\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}+\frac {\cos (a+b x)}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.24 \[ \int \cos (a+b x) \cot (c+b x) \, dx=-\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a-c)}{b}+\frac {\cos (a) \cos (b x)}{b}-\frac {\sin (a) \sin (b x)}{b} \]

[In]

Integrate[Cos[a + b*x]*Cot[c + b*x],x]

[Out]

((-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 + (Cos[a]*Cos[b*x])/b - (Sin[a]*Sin[b*x])/b

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21

method	result	size
risch	\(\frac {{\mathrm e}^{i \left (x b +a \right )}}{2 b}+\frac {{\mathrm e}^{-i \left (x b +a \right )}}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\)	\(93\)

[In]

int(cos(b*x+a)*cot(b*x+c),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(I*(b*x+a))/b+1/2/b*exp(-I*(b*x+a))+ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*cos(a-c)-ln(exp(I*(b*x+a))+exp(I*
(a-c)))/b*cos(a-c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (29) = 58\).

Time = 0.26 (sec) , antiderivative size = 190, normalized size of antiderivative = 6.55 \[ \int \cos (a+b x) \cot (c+b x) \, dx=\frac {\sqrt {2} \sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right ) + 4 \, \cos \left (b x + a\right )}{4 \, b} \]

[In]

integrate(cos(b*x+a)*cot(b*x+c),x, algorithm="fricas")

[Out]

1/4*(sqrt(2)*sqrt(cos(-2*a + 2*c) + 1)*log(-(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*si
n(-2*a + 2*c) - 2*sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a) - sin(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)) + 4*cos(b*x + a))/b

Sympy [F]

\[ \int \cos (a+b x) \cot (c+b x) \, dx=\int \cos {\left (a + b x \right )} \cot {\left (b x + c \right )}\, dx \]

[In]

integrate(cos(b*x+a)*cot(b*x+c),x)

[Out]

Integral(cos(a + b*x)*cot(b*x + c), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (29) = 58\).

Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.62 \[ \int \cos (a+b x) \cot (c+b x) \, dx=-\frac {\cos \left (-a + c\right ) \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - \cos \left (-a + c\right ) \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - 2 \, \cos \left (b x + a\right )}{2 \, b} \]

[In]

integrate(cos(b*x+a)*cot(b*x+c),x, algorithm="maxima")

[Out]

-1/2*(cos(-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) -
cos(-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) - 2*co
s(b*x + a))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (29) = 58\).

Time = 0.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 8.07 \[ \int \cos (a+b x) \cot (c+b x) \, dx=-\frac {\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x\right ) + \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (2 \, \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} - 1\right )}}{{\left (\tan \left (\frac {1}{2} \, b x\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} + 1\right )}}}{b} \]

[In]

integrate(cos(b*x+a)*cot(b*x+c),x, algorithm="giac")

[Out]

-((tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)^2*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*c)^3 + tan(1/2*c)
)*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)) - (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) + 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) + 2*(2*tan(1/2*b*x)*
tan(1/2*a) + tan(1/2*a)^2 - 1)/((tan(1/2*b*x)^2 + 1)*(tan(1/2*a)^2 + 1)))/b

Mupad [B] (verification not implemented)

Time = 26.57 (sec) , antiderivative size = 231, normalized size of antiderivative = 7.97 \[ \int \cos (a+b x) \cot (c+b x) \, dx=\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]

[In]

int(cos(a + b*x)*cot(c + b*x),x)

[Out]

exp(- a*1i - b*x*1i)/(2*b) + exp(a*1i + b*x*1i)/(2*b) - (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)) + (log((exp(a*2i)*exp(-c*2i)*(exp(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)*1i + 1i))*(exp(a*2i - c*2i) + 1))/(2*b*exp(a*2i - c
*2i)^(1/2))