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]

-1/5*arctanh(cos(x))+cos(x)+1/20*ln(1-4*cos(x)-5^(1/2))*(-5^(1/2)+1)-1/20*ln(1+4*cos(x)-5^(1/2))*(-5^(1/2)+1)+
1/20*ln(1-4*cos(x)+5^(1/2))*(5^(1/2)+1)-1/20*ln(1+4*cos(x)+5^(1/2))*(5^(1/2)+1)

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Rubi [A]  time = 0.16, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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]

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*}

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Mathematica [A]  time = 0.12, 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

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fricas [A]  time = 0.62, size = 137, normalized size = 1.25 \[ \frac {1}{20} \, \sqrt {5} \log \left (-\frac {4 \, {\left (\sqrt {5} - 1\right )} \cos \relax (x) - 8 \, \cos \relax (x)^{2} + \sqrt {5} - 3}{4 \, \cos \relax (x)^{2} + 2 \, \cos \relax (x) - 1}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (-\frac {4 \, {\left (\sqrt {5} + 1\right )} \cos \relax (x) - 8 \, \cos \relax (x)^{2} - \sqrt {5} - 3}{4 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) - 1}\right ) + \cos \relax (x) - \frac {1}{20} \, \log \left (4 \, \cos \relax (x)^{2} + 2 \, \cos \relax (x) - 1\right ) + \frac {1}{20} \, \log \left (4 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) - 1\right ) - \frac {1}{10} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{10} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]

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)

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giac [A]  time = 0.16, size = 117, normalized size = 1.06 \[ \frac {1}{20} \, \sqrt {5} \log \left (\frac {{\left | -2 \, \sqrt {5} + 8 \, \cos \relax (x) + 2 \right |}}{{\left | 2 \, \sqrt {5} + 8 \, \cos \relax (x) + 2 \right |}}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (\frac {{\left | -2 \, \sqrt {5} + 8 \, \cos \relax (x) - 2 \right |}}{{\left | 2 \, \sqrt {5} + 8 \, \cos \relax (x) - 2 \right |}}\right ) + \cos \relax (x) - \frac {1}{10} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{10} \, \log \left (-\cos \relax (x) + 1\right ) - \frac {1}{20} \, \log \left ({\left | 4 \, \cos \relax (x)^{2} + 2 \, \cos \relax (x) - 1 \right |}\right ) + \frac {1}{20} \, \log \left ({\left | 4 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.44, size = 82, normalized size = 0.75 \[ -\frac {\ln \left (4 \left (\cos ^{2}\relax (x )\right )+2 \cos \relax (x )-1\right )}{20}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (8 \cos \relax (x )+2\right ) \sqrt {5}}{10}\right )}{10}+\frac {\ln \left (-1+\cos \relax (x )\right )}{10}+\frac {\ln \left (4 \left (\cos ^{2}\relax (x )\right )-2 \cos \relax (x )-1\right )}{20}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (8 \cos \relax (x )-2\right ) \sqrt {5}}{10}\right )}{10}-\frac {\ln \left (1+\cos \relax (x )\right )}{10}+\cos \relax (x ) \]

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))+1/20*ln(4*c
os(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))+cos(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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)

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mupad [B]  time = 3.03, size = 611, normalized size = 5.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(5*x)*cos(x),x)

[Out]

(atan((tan(x/2)^2*4813499234516992i)/(1220703125*((213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(
1/2)*tan(x/2)^2)/1220703125 - (4959229085483008*tan(x/2)^2)/1220703125 + 110872433262592/244140625)) - 9548732
3537408i/(244140625*((213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 -
(4959229085483008*tan(x/2)^2)/1220703125 + 110872433262592/244140625)) - (5^(1/2)*247887795585024i)/(12207031
25*((213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 - (495922908548300
8*tan(x/2)^2)/1220703125 + 110872433262592/244140625)) + (5^(1/2)*tan(x/2)^2*2217818569310208i)/(1220703125*((
213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 - (4959229085483008*tan
(x/2)^2)/1220703125 + 110872433262592/244140625)))*1i)/10 + (atan(95487323537408i/(244140625*((213485644414976
*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 + (4959229085483008*tan(x/2)^2)/122070
3125 - 110872433262592/244140625)) - (5^(1/2)*247887795585024i)/(1220703125*((213485644414976*5^(1/2))/1220703
125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 + (4959229085483008*tan(x/2)^2)/1220703125 - 1108724332
62592/244140625)) - (tan(x/2)^2*4813499234516992i)/(1220703125*((213485644414976*5^(1/2))/1220703125 - (215264
6198689792*5^(1/2)*tan(x/2)^2)/1220703125 + (4959229085483008*tan(x/2)^2)/1220703125 - 110872433262592/2441406
25)) + (5^(1/2)*tan(x/2)^2*2217818569310208i)/(1220703125*((213485644414976*5^(1/2))/1220703125 - (21526461986
89792*5^(1/2)*tan(x/2)^2)/1220703125 + (4959229085483008*tan(x/2)^2)/1220703125 - 110872433262592/244140625)))
*1i)/10 + log(tan(x/2))/5 + 2/(tan(x/2)^2 + 1) + (5^(1/2)*(atan((tan(x/2)^2*4813499234516992i)/(1220703125*((2
13485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 - (4959229085483008*tan(
x/2)^2)/1220703125 + 110872433262592/244140625)) - 95487323537408i/(244140625*((213485644414976*5^(1/2))/12207
03125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 - (4959229085483008*tan(x/2)^2)/1220703125 + 11087243
3262592/244140625)) - (5^(1/2)*247887795585024i)/(1220703125*((213485644414976*5^(1/2))/1220703125 - (21526461
98689792*5^(1/2)*tan(x/2)^2)/1220703125 - (4959229085483008*tan(x/2)^2)/1220703125 + 110872433262592/244140625
)) + (5^(1/2)*tan(x/2)^2*2217818569310208i)/(1220703125*((213485644414976*5^(1/2))/1220703125 - (2152646198689
792*5^(1/2)*tan(x/2)^2)/1220703125 - (4959229085483008*tan(x/2)^2)/1220703125 + 110872433262592/244140625))) -
atan(95487323537408i/(244140625*((213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)
/1220703125 + (4959229085483008*tan(x/2)^2)/1220703125 - 110872433262592/244140625)) - (5^(1/2)*24788779558502
4i)/(1220703125*((213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 + (49
59229085483008*tan(x/2)^2)/1220703125 - 110872433262592/244140625)) - (tan(x/2)^2*4813499234516992i)/(12207031
25*((213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 + (495922908548300
8*tan(x/2)^2)/1220703125 - 110872433262592/244140625)) + (5^(1/2)*tan(x/2)^2*2217818569310208i)/(1220703125*((
213485644414976*5^(1/2))/1220703125 - (2152646198689792*5^(1/2)*tan(x/2)^2)/1220703125 + (4959229085483008*tan
(x/2)^2)/1220703125 - 110872433262592/244140625))))*1i)/10

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \cot {\left (5 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(5*x),x)

[Out]

Integral(cos(x)*cot(5*x), x)

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