3.251 \(\int \cos (a+b x) \cot ^2(c+b x) \, dx\)

Optimal. Leaf size=46 \[ -\frac {\cos (a-c) \csc (b x+c)}{b}+\frac {\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4577, 4578, 2637, 3770, 2606, 8} \[ -\frac {\cos (a-c) \csc (b x+c)}{b}+\frac {\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Cot[c + b*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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]

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 4578

Int[Cot[w_]^(n_.)*Sin[v_], x_Symbol] :> Int[Cos[v]*Cot[w]^(n - 1), x] + Dist[Sin[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*} \int \cos (a+b x) \cot ^2(c+b x) \, dx &=\cos (a-c) \int \cot (c+b x) \csc (c+b x) \, dx-\int \cot (c+b x) \sin (a+b x) \, dx\\ &=-\frac {\cos (a-c) \operatorname {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b}-\sin (a-c) \int \csc (c+b x) \, dx-\int \cos (a+b x) \, dx\\ &=-\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {\tanh ^{-1}(\cos (c+b x)) \sin (a-c)}{b}-\frac {\sin (a+b x)}{b}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 112, normalized size = 2.43 \[ -\frac {\cos (a-c) \csc (b x+c)}{b}+\frac {2 i \sin (a-c) \tan ^{-1}\left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{\sin (c) \cos \left (\frac {b x}{2}\right )+i \cos (c) \cos \left (\frac {b x}{2}\right )}\right )}{b}-\frac {\sin (a) \cos (b x)}{b}-\frac {\cos (a) \sin (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]*Cot[c + b*x]^2,x]

[Out]

-((Cos[a - c]*Csc[c + b*x])/b) - (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

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fricas [B]  time = 0.49, size = 316, normalized size = 6.87 \[ \frac {4 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {\sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \cos \left (b x + a\right )\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - 8 \, \cos \left (-2 \, a + 2 \, c\right ) - 8}{4 \, {\left (b \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \sin \left (b x + a\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+c)^2,x, algorithm="fricas")

[Out]

1/4*(4*(cos(-2*a + 2*c) + 1)*cos(b*x + a)^2 - 4*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) + sqrt(2)*((cos(-2*a
+ 2*c) + 1)*sin(b*x + a)*sin(-2*a + 2*c) - (cos(-2*a + 2*c)^2 - 1)*cos(b*x + a))*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) - 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))/sqrt(cos(-2*a + 2*c) + 1) - 8*cos(-2*a +
2*c) - 8)/(b*cos(b*x + a)*sin(-2*a + 2*c) + (b*cos(-2*a + 2*c) + b)*sin(b*x + a))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+c)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/b*((tan(c/2)^2*tan(a/2)-tan(c/2)*tan(a/2)^2+tan(c/2)-tan(a/2
))/(tan(c/2)^2*tan(a/2)^2+tan(c/2)^2+tan(a/2)^2+1)*ln(abs(tan(b*x/2)+tan(c/2)))+(tan(c/2)^3*tan(a/2)-tan(c/2)^
2*tan(a/2)^2+tan(c/2)^2-tan(c/2)*tan(a/2))/(-tan(c/2)^3*tan(a/2)^2-tan(c/2)^3-tan(c/2)*tan(a/2)^2-tan(c/2))*ln
(abs(tan(b*x/2)*tan(c/2)-1))+(tan(b*x/2)^3*tan(c/2)^4*tan(a/2)^2-tan(b*x/2)^3*tan(c/2)^4+4*tan(b*x/2)^3*tan(c/
2)^3*tan(a/2)-6*tan(b*x/2)^3*tan(c/2)^2*tan(a/2)^2+6*tan(b*x/2)^3*tan(c/2)^2-4*tan(b*x/2)^3*tan(c/2)*tan(a/2)+
tan(b*x/2)^3*tan(a/2)^2-tan(b*x/2)^3-6*tan(b*x/2)^2*tan(c/2)^3*tan(a/2)^2+6*tan(b*x/2)^2*tan(c/2)^3+6*tan(b*x/
2)^2*tan(c/2)*tan(a/2)^2-6*tan(b*x/2)^2*tan(c/2)+tan(b*x/2)*tan(c/2)^4*tan(a/2)^2-tan(b*x/2)*tan(c/2)^4+12*tan
(b*x/2)*tan(c/2)^3*tan(a/2)+2*tan(b*x/2)*tan(c/2)^2*tan(a/2)^2-2*tan(b*x/2)*tan(c/2)^2-12*tan(b*x/2)*tan(c/2)*
tan(a/2)+tan(b*x/2)*tan(a/2)^2-tan(b*x/2)-2*tan(c/2)^3*tan(a/2)^2+2*tan(c/2)^3-16*tan(c/2)^2*tan(a/2)+2*tan(c/
2)*tan(a/2)^2-2*tan(c/2))/(-4*tan(c/2)*tan(a/2)^2-4*tan(c/2))/(tan(b*x/2)^4*tan(c/2)+tan(b*x/2)^3*tan(c/2)^2-t
an(b*x/2)^3+tan(b*x/2)*tan(c/2)^2-tan(b*x/2)-tan(c/2)))

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maple [C]  time = 0.84, size = 145, normalized size = 3.15 \[ \frac {i {\mathrm e}^{i \left (b x +a \right )}}{2 b}-\frac {i {\mathrm e}^{-i \left (b x +a \right )}}{2 b}+\frac {i \left ({\mathrm e}^{i \left (b x +3 a \right )}+{\mathrm e}^{i \left (b x +a +2 c \right )}\right )}{b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*cot(b*x+c)^2,x)

[Out]

1/2*I*exp(I*(b*x+a))/b-1/2*I/b*exp(-I*(b*x+a))+I/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))*(exp(I*(b*x+3*a))+exp(I*(b
*x+a+2*c)))+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)

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maxima [B]  time = 0.36, size = 613, normalized size = 13.33 \[ \frac {{\left (\sin \left (3 \, b x + a + 2 \, c\right ) - \sin \left (b x + a\right )\right )} \cos \left (4 \, b x + 2 \, a + 2 \, c\right ) + 3 \, {\left (\sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, c\right )\right )} \cos \left (3 \, b x + a + 2 \, c\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (c) + \sin \relax (c)^{2}\right ) + {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (c) + \sin \relax (c)^{2}\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right ) - \cos \left (b x + a\right )\right )} \sin \left (4 \, b x + 2 \, a + 2 \, c\right ) - {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) - 1\right )} \sin \left (3 \, b x + a + 2 \, c\right ) - 3 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, c\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) \sin \left (b x + a\right ) - \sin \left (b x + a\right )}{2 \, {\left (b \cos \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) + b \cos \left (b x + a\right )^{2} + b \sin \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + b \sin \left (b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+c)^2,x, algorithm="maxima")

[Out]

1/2*((sin(3*b*x + a + 2*c) - sin(b*x + a))*cos(4*b*x + 2*a + 2*c) + 3*(sin(2*b*x + 2*a) + sin(2*b*x + 2*c))*co
s(3*b*x + a + 2*c) - (cos(3*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(3*b*x + a + 2*c)*cos(b*x + a)*sin(-a + c) + c
os(b*x + a)^2*sin(-a + c) + sin(3*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(3*b*x + a + 2*c)*sin(b*x + a)*sin(-a +
c) + sin(b*x + a)^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) + (cos(3*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(3*b*x + a + 2*c)*cos(b*x + a)*sin(-a + c) + cos(b*
x + a)^2*sin(-a + c) + sin(3*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(3*b*x + a + 2*c)*sin(b*x + a)*sin(-a + c) +
sin(b*x + a)^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) + s
in(c)^2) - (cos(3*b*x + a + 2*c) - cos(b*x + a))*sin(4*b*x + 2*a + 2*c) - (3*cos(2*b*x + 2*a) + 3*cos(2*b*x +
2*c) - 1)*sin(3*b*x + a + 2*c) - 3*cos(b*x + a)*sin(2*b*x + 2*a) - 3*cos(b*x + a)*sin(2*b*x + 2*c) + 3*cos(2*b
*x + 2*a)*sin(b*x + a) + 3*cos(2*b*x + 2*c)*sin(b*x + a) - sin(b*x + a))/(b*cos(3*b*x + a + 2*c)^2 - 2*b*cos(3
*b*x + a + 2*c)*cos(b*x + a) + b*cos(b*x + a)^2 + b*sin(3*b*x + a + 2*c)^2 - 2*b*sin(3*b*x + a + 2*c)*sin(b*x
+ a) + b*sin(b*x + a)^2)

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mupad [B]  time = 5.44, size = 289, normalized size = 6.28 \[ -\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)*cot(c + b*x)^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (a + b x \right )} \cot ^{2}{\left (b x + c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+c)**2,x)

[Out]

Integral(cos(a + b*x)*cot(b*x + c)**2, x)

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