3.83 \(\int \cot (5 x) \sin (x) \, dx\)

Optimal. Leaf size=82 \[ \sin (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right ) \]

[Out]

sin(x)-1/10*arctanh(1/5*sin(x)*(50+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-1/10*arctanh(2*sin(x)*2^(1/2)/(5+5^
(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ \sin (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[5*x]*Sin[x],x]

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Sin[x]])/5 - (Sqrt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5
+ Sqrt[5]))/5]*Sin[x]])/5 + Sin[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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rubi steps

\begin {align*} \int \cot (5 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1-12 x^2+16 x^4}{5-20 x^2+16 x^4} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {4 \left (1-2 x^2\right )}{5-20 x^2+16 x^4}\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)-4 \operatorname {Subst}\left (\int \frac {1-2 x^2}{5-20 x^2+16 x^4} \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac {1}{5} \left (4 \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10+2 \sqrt {5}+16 x^2} \, dx,x,\sin (x)\right )+\frac {1}{5} \left (4 \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10-2 \sqrt {5}+16 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right )+\sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 76, normalized size = 0.93 \[ \frac {1}{10} \left (10 \sin (x)-\sqrt {10-2 \sqrt {5}} \tanh ^{-1}\left (\sqrt {2+\frac {2}{\sqrt {5}}} \sin (x)\right )-\sqrt {2 \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[5*x]*Sin[x],x]

[Out]

(-(Sqrt[10 - 2*Sqrt[5]]*ArcTanh[Sqrt[2 + 2/Sqrt[5]]*Sin[x]]) - Sqrt[2*(5 + Sqrt[5])]*ArcTanh[2*Sqrt[2/(5 + Sqr
t[5])]*Sin[x]] + 10*Sin[x])/10

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fricas [B]  time = 0.56, size = 127, normalized size = 1.55 \[ -\frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} + 4 \, \sin \relax (x)\right ) + \frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} - 4 \, \sin \relax (x)\right ) - \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} + 4 \, \sin \relax (x)\right ) + \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} - 4 \, \sin \relax (x)\right ) + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(5*x)*sin(x),x, algorithm="fricas")

[Out]

-1/20*sqrt(2)*sqrt(sqrt(5) + 5)*log(sqrt(2)*sqrt(sqrt(5) + 5) + 4*sin(x)) + 1/20*sqrt(2)*sqrt(sqrt(5) + 5)*log
(sqrt(2)*sqrt(sqrt(5) + 5) - 4*sin(x)) - 1/20*sqrt(2)*sqrt(-sqrt(5) + 5)*log(sqrt(2)*sqrt(-sqrt(5) + 5) + 4*si
n(x)) + 1/20*sqrt(2)*sqrt(-sqrt(5) + 5)*log(sqrt(2)*sqrt(-sqrt(5) + 5) - 4*sin(x)) + sin(x)

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giac [B]  time = 0.33, size = 111, normalized size = 1.35 \[ -\frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} + \sin \relax (x) \right |}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | -\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} + \sin \relax (x) \right |}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | \sqrt {-\frac {1}{8} \, \sqrt {5} + \frac {5}{8}} + \sin \relax (x) \right |}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {-\frac {1}{8} \, \sqrt {5} + \frac {5}{8}} + \sin \relax (x) \right |}\right ) + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(5*x)*sin(x),x, algorithm="giac")

[Out]

-1/20*sqrt(2*sqrt(5) + 10)*log(abs(1/2*sqrt(1/2)*sqrt(sqrt(5) + 5) + sin(x))) + 1/20*sqrt(2*sqrt(5) + 10)*log(
abs(-1/2*sqrt(1/2)*sqrt(sqrt(5) + 5) + sin(x))) - 1/20*sqrt(-2*sqrt(5) + 10)*log(abs(sqrt(-1/8*sqrt(5) + 5/8)
+ sin(x))) + 1/20*sqrt(-2*sqrt(5) + 10)*log(abs(-sqrt(-1/8*sqrt(5) + 5/8) + sin(x))) + sin(x)

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maple [A]  time = 0.21, size = 70, normalized size = 0.85 \[ \sin \relax (x )-\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \sin \relax (x )}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \sin \relax (x )}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)-1/5*(5^(1/2)-1)*5^(1/2)/(10-2*5^(1/2))^(1/2)*arctanh(4*sin(x)/(10-2*5^(1/2))^(1/2))-1/5*(5^(1/2)+1)*5^(
1/2)/(10+2*5^(1/2))^(1/2)*arctanh(4*sin(x)/(10+2*5^(1/2))^(1/2))

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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(cot(5*x)*sin(x),x, algorithm="maxima")

[Out]

-integrate(1/2*((cos(3*x) + cos(2*x) + cos(x))*cos(4*x) + (2*cos(2*x) + 2*cos(x) + 1)*cos(3*x) + cos(3*x)^2 +
(2*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + (sin(3*x) + sin(2*x) + sin(x))*sin(4*x) + 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 + cos(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) + co
s(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 + sin(2*x)^2 + 2*sin(2*x)*sin(x) + sin(x)^2 + 2*cos(x) + 1), x) - integrate(-1/2*((cos(3*x) - cos
(2*x) + cos(x))*cos(4*x) + (2*cos(2*x) - 2*cos(x) + 1)*cos(3*x) - cos(3*x)^2 + (2*cos(x) - 1)*cos(2*x) - cos(2
*x)^2 - cos(x)^2 + (sin(3*x) - sin(2*x) + sin(x))*sin(4*x) + 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 + cos(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 - sin(2*x)^2 + 2*s
in(2*x)*sin(x) - sin(x)^2 + 2*cos(x) - 1), x) + sin(x)

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mupad [B]  time = 2.61, size = 119, normalized size = 1.45 \[ \sin \relax (x)-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\frac {25\,\sqrt {2}\,\sin \relax (x)\,\sqrt {\sqrt {5}+5}}{2}+\frac {11\,\sqrt {2}\,\sqrt {5}\,\sin \relax (x)\,\sqrt {\sqrt {5}+5}}{2}}{20\,\sqrt {5}+45}\right )\,\sqrt {\sqrt {5}+5}}{10}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\frac {25\,\sqrt {2}\,\sin \relax (x)\,\sqrt {5-\sqrt {5}}}{2}-\frac {11\,\sqrt {2}\,\sqrt {5}\,\sin \relax (x)\,\sqrt {5-\sqrt {5}}}{2}}{20\,\sqrt {5}-45}\right )\,\sqrt {5-\sqrt {5}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x) - (2^(1/2)*atanh(((25*2^(1/2)*sin(x)*(5^(1/2) + 5)^(1/2))/2 + (11*2^(1/2)*5^(1/2)*sin(x)*(5^(1/2) + 5)^
(1/2))/2)/(20*5^(1/2) + 45))*(5^(1/2) + 5)^(1/2))/10 + (2^(1/2)*atanh(((25*2^(1/2)*sin(x)*(5 - 5^(1/2))^(1/2))
/2 - (11*2^(1/2)*5^(1/2)*sin(x)*(5 - 5^(1/2))^(1/2))/2)/(20*5^(1/2) - 45))*(5 - 5^(1/2))^(1/2))/10

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)*cot(5*x), x)

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