3.250 \(\int \cos (a+b x) \cot (c+b x) \, dx\)

Optimal. Leaf size=29 \[ \frac{\cos (a+b x)}{b}-\frac{\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b} \]

[Out]

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

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Rubi [A]  time = 0.0209471, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4577, 2638, 3770} \[ \frac{\cos (a+b x)}{b}-\frac{\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b} \]

Antiderivative was successfully verified.

[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 4577

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]

Rule 2638

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

Rule 3770

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

Rubi steps

\begin{align*} \int \cos (a+b x) \cot (c+b x) \, dx &=\cos (a-c) \int \csc (c+b x) \, dx-\int \sin (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{b}+\frac{\cos (a+b x)}{b}\\ \end{align*}

Mathematica [C]  time = 0.0508546, size = 94, normalized size = 3.24 \[ -\frac{2 i \cos (a-c) \tan ^{-1}\left (\frac{(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac{b x}{2}\right )-\sin (c) \sin \left (\frac{b x}{2}\right )\right )}{\sin (c) \cos \left (\frac{b x}{2}\right )+i \cos (c) \cos \left (\frac{b x}{2}\right )}\right )}{b}-\frac{\sin (a) \sin (b x)}{b}+\frac{\cos (a) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[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

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Maple [C]  time = 0.079, size = 93, normalized size = 3.2 \begin{align*}{\frac{{{\rm e}^{i \left ( bx+a \right ) }}}{2\,b}}+{\frac{{{\rm e}^{-i \left ( bx+a \right ) }}}{2\,b}}-{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+{{\rm e}^{i \left ( a-c \right ) }} \right ) \cos \left ( a-c \right ) }{b}}+{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-{{\rm e}^{i \left ( a-c \right ) }} \right ) \cos \left ( a-c \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.18219, size = 142, normalized size = 4.9 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 0.518793, size = 510, normalized size = 17.59 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (a + b x \right )} \cot{\left (b x + c \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.16001, size = 316, normalized size = 10.9 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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