3.113 \(\int \cos (x) \cot (5 x) \, dx\)
Optimal. Leaf size=110 \[ \cos (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \cos (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \cos (x)+\sqrt{5}+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \cos (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \cos (x)+\sqrt{5}+1\right )-\frac{1}{5} \tanh ^{-1}(\cos (x)) \]
[Out]
-ArcTanh[Cos[x]]/5 + Cos[x] + ((1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cos[x]])/20 + ((1 + Sqrt[5])*Log[1 + Sqrt[5]
- 4*Cos[x]])/20 - ((1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[x]])/20 - ((1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[x]])
/20
________________________________________________________________________________________
Rubi [A] time = 0.15549, antiderivative size = 110, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used =
{2075, 207, 632, 31} \[ \cos (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \cos (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \cos (x)+\sqrt{5}+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \cos (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \cos (x)+\sqrt{5}+1\right )-\frac{1}{5} \tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
[In]
Int[Cos[x]*Cot[5*x],x]
[Out]
-ArcTanh[Cos[x]]/5 + Cos[x] + ((1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cos[x]])/20 + ((1 + Sqrt[5])*Log[1 + Sqrt[5]
- 4*Cos[x]])/20 - ((1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[x]])/20 - ((1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[x]])
/20
Rule 2075
Int[(P_)^(p_)*(Qm_), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProdu
ctQ[PP, x]] /; PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 0]
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 632
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int \cos (x) \cot (5 x) \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (5-20 x^2+16 x^4\right )}{1-13 x^2+28 x^4-16 x^6} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1-\frac{1}{5 \left (-1+x^2\right )}-\frac{2 (1+x)}{5 \left (-1-2 x+4 x^2\right )}+\frac{2 (-1+x)}{5 \left (-1+2 x+4 x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\cos (x)+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cos (x)\right )+\frac{2}{5} \operatorname{Subst}\left (\int \frac{1+x}{-1-2 x+4 x^2} \, dx,x,\cos (x)\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{-1+x}{-1+2 x+4 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}(\cos (x))+\cos (x)-\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{5}+4 x} \, dx,x,\cos (x)\right )+\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{5}+4 x} \, dx,x,\cos (x)\right )+\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{5}+4 x} \, dx,x,\cos (x)\right )-\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{5}+4 x} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}(\cos (x))+\cos (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}-4 \cos (x)\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}-4 \cos (x)\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}+4 \cos (x)\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}+4 \cos (x)\right )\\ \end{align*}
Mathematica [A] time = 0.125386, size = 133, normalized size = 1.21 \[ \frac{1}{100} \left (100 \cos (x)+20 \log \left (\sin \left (\frac{x}{2}\right )\right )-20 \log \left (\cos \left (\frac{x}{2}\right )\right )+\sqrt{5} \left (\sqrt{5}-5\right ) \log \left (-4 \cos (x)-\sqrt{5}+1\right )+\sqrt{5} \left (5+\sqrt{5}\right ) \log \left (-4 \cos (x)+\sqrt{5}+1\right )-\sqrt{5} \left (\sqrt{5}-5\right ) \log \left (4 \cos (x)-\sqrt{5}+1\right )-\sqrt{5} \left (5+\sqrt{5}\right ) \log \left (4 \cos (x)+\sqrt{5}+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[x]*Cot[5*x],x]
[Out]
(100*Cos[x] - 20*Log[Cos[x/2]] + Sqrt[5]*(-5 + Sqrt[5])*Log[1 - Sqrt[5] - 4*Cos[x]] + Sqrt[5]*(5 + Sqrt[5])*Lo
g[1 + Sqrt[5] - 4*Cos[x]] - Sqrt[5]*(-5 + Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[x]] - Sqrt[5]*(5 + Sqrt[5])*Log[1 +
Sqrt[5] + 4*Cos[x]] + 20*Log[Sin[x/2]])/100
________________________________________________________________________________________
Maple [A] time = 0.153, size = 82, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\cos \left ( x \right ) -1 \right ) }{20}}-{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 8\,\cos \left ( x \right ) -2 \right ) \sqrt{5}}{10}} \right ) }-{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{10}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{10}}-{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -1 \right ) }{20}}-{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 8\,\cos \left ( x \right ) +2 \right ) \sqrt{5}}{10}} \right ) }+\cos \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(x)*cot(5*x),x)
[Out]
1/20*ln(4*cos(x)^2-2*cos(x)-1)-1/10*5^(1/2)*arctanh(1/10*(8*cos(x)-2)*5^(1/2))-1/10*ln(1+cos(x))+1/10*ln(-1+co
s(x))-1/20*ln(4*cos(x)^2+2*cos(x)-1)-1/10*5^(1/2)*arctanh(1/10*(8*cos(x)+2)*5^(1/2))+cos(x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)*cot(5*x),x, algorithm="maxima")
[Out]
cos(x) + 1/10*integrate(-(cos(2*x)*sin(4*x) - cos(4*x)*sin(2*x) + cos(3/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x
) + cos(1/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x) - cos(2*x)*sin(3/2*arctan2(sin(2*x), cos(2*x))) - cos(2*x)*s
in(1/2*arctan2(sin(2*x), cos(2*x))) - sin(2*x))/(2*(cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 - 2*(cos(
4*x) + cos(2*x) - cos(1/2*arctan2(sin(2*x), cos(2*x))) + 1)*cos(3/2*arctan2(sin(2*x), cos(2*x))) + cos(3/2*arc
tan2(sin(2*x), cos(2*x)))^2 - 2*(cos(4*x) + cos(2*x) + 1)*cos(1/2*arctan2(sin(2*x), cos(2*x))) + cos(1/2*arcta
n2(sin(2*x), cos(2*x)))^2 + sin(4*x)^2 + 2*sin(4*x)*sin(2*x) + sin(2*x)^2 - 2*(sin(4*x) + sin(2*x) - sin(1/2*a
rctan2(sin(2*x), cos(2*x))))*sin(3/2*arctan2(sin(2*x), cos(2*x))) + sin(3/2*arctan2(sin(2*x), cos(2*x)))^2 - 2
*(sin(4*x) + sin(2*x))*sin(1/2*arctan2(sin(2*x), cos(2*x))) + sin(1/2*arctan2(sin(2*x), cos(2*x)))^2 + 2*cos(2
*x) + 1), x) + 1/10*integrate((cos(2*x)*sin(4*x) - cos(4*x)*sin(2*x) - cos(3/2*arctan2(sin(2*x), cos(2*x)))*si
n(2*x) - cos(1/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x) + cos(2*x)*sin(3/2*arctan2(sin(2*x), cos(2*x))) + cos(2
*x)*sin(1/2*arctan2(sin(2*x), cos(2*x))) - sin(2*x))/(2*(cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 + 2*
(cos(4*x) + cos(2*x) + cos(1/2*arctan2(sin(2*x), cos(2*x))) + 1)*cos(3/2*arctan2(sin(2*x), cos(2*x))) + cos(3/
2*arctan2(sin(2*x), cos(2*x)))^2 + 2*(cos(4*x) + cos(2*x) + 1)*cos(1/2*arctan2(sin(2*x), cos(2*x))) + cos(1/2*
arctan2(sin(2*x), cos(2*x)))^2 + sin(4*x)^2 + 2*sin(4*x)*sin(2*x) + sin(2*x)^2 + 2*(sin(4*x) + sin(2*x) + sin(
1/2*arctan2(sin(2*x), cos(2*x))))*sin(3/2*arctan2(sin(2*x), cos(2*x))) + sin(3/2*arctan2(sin(2*x), cos(2*x)))^
2 + 2*(sin(4*x) + sin(2*x))*sin(1/2*arctan2(sin(2*x), cos(2*x))) + sin(1/2*arctan2(sin(2*x), cos(2*x)))^2 + 2*
cos(2*x) + 1), x) - 1/10*integrate((cos(x)*sin(4*x) + cos(x)*sin(3*x) + cos(x)*sin(2*x) - cos(4*x)*sin(x) - co
s(3*x)*sin(x) - cos(2*x)*sin(x) - sin(x))/(2*(cos(3*x) + cos(2*x) + cos(x) + 1)*cos(4*x) + cos(4*x)^2 + 2*(cos
(2*x) + cos(x) + 1)*cos(3*x) + cos(3*x)^2 + 2*(cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + 2*(sin(3*x) + si
n(2*x) + sin(x))*sin(4*x) + sin(4*x)^2 + 2*(sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + sin(2*x)^2 + 2*sin(2*x)
*sin(x) + sin(x)^2 + 2*cos(x) + 1), x) - 1/10*integrate(-(cos(x)*sin(4*x) - cos(x)*sin(3*x) + cos(x)*sin(2*x)
- cos(4*x)*sin(x) + cos(3*x)*sin(x) - cos(2*x)*sin(x) - sin(x))/(2*(cos(3*x) - cos(2*x) + cos(x) - 1)*cos(4*x)
- cos(4*x)^2 + 2*(cos(2*x) - cos(x) + 1)*cos(3*x) - cos(3*x)^2 + 2*(cos(x) - 1)*cos(2*x) - cos(2*x)^2 - cos(x
)^2 + 2*(sin(3*x) - sin(2*x) + sin(x))*sin(4*x) - sin(4*x)^2 + 2*(sin(2*x) - sin(x))*sin(3*x) - sin(3*x)^2 - s
in(2*x)^2 + 2*sin(2*x)*sin(x) - sin(x)^2 + 2*cos(x) - 1), x) + 3/10*integrate(-(cos(4/3*arctan2(sin(3*x), cos(
3*x)))*sin(3*x) + cos(2/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) + cos(1/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x
) - cos(3*x)*sin(4/3*arctan2(sin(3*x), cos(3*x))) - cos(3*x)*sin(2/3*arctan2(sin(3*x), cos(3*x))) - cos(3*x)*s
in(1/3*arctan2(sin(3*x), cos(3*x))) + sin(3*x))/(cos(3*x)^2 + 2*(cos(3*x) + cos(2/3*arctan2(sin(3*x), cos(3*x)
)) + cos(1/3*arctan2(sin(3*x), cos(3*x))) + 1)*cos(4/3*arctan2(sin(3*x), cos(3*x))) + cos(4/3*arctan2(sin(3*x)
, cos(3*x)))^2 + 2*(cos(3*x) + cos(1/3*arctan2(sin(3*x), cos(3*x))) + 1)*cos(2/3*arctan2(sin(3*x), cos(3*x)))
+ cos(2/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*(cos(3*x) + 1)*cos(1/3*arctan2(sin(3*x), cos(3*x))) + cos(1/3*arc
tan2(sin(3*x), cos(3*x)))^2 + sin(3*x)^2 + 2*(sin(3*x) + sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arctan
2(sin(3*x), cos(3*x))))*sin(4/3*arctan2(sin(3*x), cos(3*x))) + sin(4/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*(sin
(3*x) + sin(1/3*arctan2(sin(3*x), cos(3*x))))*sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(2/3*arctan2(sin(3*x),
cos(3*x)))^2 + 2*sin(3*x)*sin(1/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*c
os(3*x) + 1), x) + 3/10*integrate(-(cos(4/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) + cos(2/3*arctan2(sin(3*x),
cos(3*x)))*sin(3*x) - cos(1/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) - cos(3*x)*sin(4/3*arctan2(sin(3*x), cos(3
*x))) - cos(3*x)*sin(2/3*arctan2(sin(3*x), cos(3*x))) + cos(3*x)*sin(1/3*arctan2(sin(3*x), cos(3*x))) + sin(3*
x))/(cos(3*x)^2 - 2*(cos(3*x) - cos(2/3*arctan2(sin(3*x), cos(3*x))) + cos(1/3*arctan2(sin(3*x), cos(3*x))) -
1)*cos(4/3*arctan2(sin(3*x), cos(3*x))) + cos(4/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*(cos(3*x) + cos(1/3*arcta
n2(sin(3*x), cos(3*x))) - 1)*cos(2/3*arctan2(sin(3*x), cos(3*x))) + cos(2/3*arctan2(sin(3*x), cos(3*x)))^2 + 2
*(cos(3*x) - 1)*cos(1/3*arctan2(sin(3*x), cos(3*x))) + cos(1/3*arctan2(sin(3*x), cos(3*x)))^2 + sin(3*x)^2 - 2
*(sin(3*x) - sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arctan2(sin(3*x), cos(3*x))))*sin(4/3*arctan2(sin(
3*x), cos(3*x))) + sin(4/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*(sin(3*x) + sin(1/3*arctan2(sin(3*x), cos(3*x)))
)*sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(2/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*sin(3*x)*sin(1/3*arctan2(s
in(3*x), cos(3*x))) + sin(1/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*cos(3*x) + 1), x) + 1/5*integrate((sin(4*x) +
sin(3*x) + sin(2*x) + sin(x))/(2*(cos(3*x) + cos(2*x) + cos(x) + 1)*cos(4*x) + cos(4*x)^2 + 2*(cos(2*x) + cos
(x) + 1)*cos(3*x) + cos(3*x)^2 + 2*(cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + 2*(sin(3*x) + sin(2*x) + si
n(x))*sin(4*x) + sin(4*x)^2 + 2*(sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin(x) + s
in(x)^2 + 2*cos(x) + 1), x) - 1/5*integrate(-(sin(4*x) - sin(3*x) + sin(2*x) - sin(x))/(2*(cos(3*x) - cos(2*x)
+ cos(x) - 1)*cos(4*x) - cos(4*x)^2 + 2*(cos(2*x) - cos(x) + 1)*cos(3*x) - cos(3*x)^2 + 2*(cos(x) - 1)*cos(2*
x) - cos(2*x)^2 - cos(x)^2 + 2*(sin(3*x) - sin(2*x) + sin(x))*sin(4*x) - sin(4*x)^2 + 2*(sin(2*x) - sin(x))*si
n(3*x) - sin(3*x)^2 - sin(2*x)^2 + 2*sin(2*x)*sin(x) - sin(x)^2 + 2*cos(x) - 1), x) - 1/10*log(cos(x)^2 + sin(
x)^2 + 2*cos(x) + 1) + 1/10*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)
________________________________________________________________________________________
Fricas [A] time = 2.60458, size = 466, normalized size = 4.24 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (-\frac{4 \,{\left (\sqrt{5} - 1\right )} \cos \left (x\right ) - 8 \, \cos \left (x\right )^{2} + \sqrt{5} - 3}{4 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) - 1}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (-\frac{4 \,{\left (\sqrt{5} + 1\right )} \cos \left (x\right ) - 8 \, \cos \left (x\right )^{2} - \sqrt{5} - 3}{4 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 1}\right ) + \cos \left (x\right ) - \frac{1}{20} \, \log \left (4 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) - 1\right ) + \frac{1}{20} \, \log \left (4 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 1\right ) - \frac{1}{10} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{10} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)*cot(5*x),x, algorithm="fricas")
[Out]
1/20*sqrt(5)*log(-(4*(sqrt(5) - 1)*cos(x) - 8*cos(x)^2 + sqrt(5) - 3)/(4*cos(x)^2 + 2*cos(x) - 1)) + 1/20*sqrt
(5)*log(-(4*(sqrt(5) + 1)*cos(x) - 8*cos(x)^2 - sqrt(5) - 3)/(4*cos(x)^2 - 2*cos(x) - 1)) + cos(x) - 1/20*log(
4*cos(x)^2 + 2*cos(x) - 1) + 1/20*log(4*cos(x)^2 - 2*cos(x) - 1) - 1/10*log(1/2*cos(x) + 1/2) + 1/10*log(-1/2*
cos(x) + 1/2)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)*cot(5*x),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.19695, size = 158, normalized size = 1.44 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{{\left | -2 \, \sqrt{5} + 8 \, \cos \left (x\right ) + 2 \right |}}{{\left | 2 \, \sqrt{5} + 8 \, \cos \left (x\right ) + 2 \right |}}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (\frac{{\left | -2 \, \sqrt{5} + 8 \, \cos \left (x\right ) - 2 \right |}}{{\left | 2 \, \sqrt{5} + 8 \, \cos \left (x\right ) - 2 \right |}}\right ) + \cos \left (x\right ) - \frac{1}{10} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{10} \, \log \left (-\cos \left (x\right ) + 1\right ) - \frac{1}{20} \, \log \left ({\left | 4 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) - 1 \right |}\right ) + \frac{1}{20} \, \log \left ({\left | 4 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)*cot(5*x),x, algorithm="giac")
[Out]
1/20*sqrt(5)*log(abs(-2*sqrt(5) + 8*cos(x) + 2)/abs(2*sqrt(5) + 8*cos(x) + 2)) + 1/20*sqrt(5)*log(abs(-2*sqrt(
5) + 8*cos(x) - 2)/abs(2*sqrt(5) + 8*cos(x) - 2)) + cos(x) - 1/10*log(cos(x) + 1) + 1/10*log(-cos(x) + 1) - 1/
20*log(abs(4*cos(x)^2 + 2*cos(x) - 1)) + 1/20*log(abs(4*cos(x)^2 - 2*cos(x) - 1))