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]

-(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]

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Rubi [A]  time = 0.196014, antiderivative size = 82, normalized size of antiderivative = 1., 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 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

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 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])

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*}

Mathematica [A]  time = 0.224122, 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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Maple [A]  time = 0.103, size = 70, normalized size = 0.9 \begin{align*} \sin \left ( x \right ) -{\frac{ \left ( \sqrt{5}-1 \right ) \sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}{\it Artanh} \left ( 4\,{\frac{\sin \left ( x \right ) }{\sqrt{10-2\,\sqrt{5}}}} \right ) }-{\frac{ \left ( \sqrt{5}+1 \right ) \sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}{\it Artanh} \left ( 4\,{\frac{\sin \left ( x \right ) }{\sqrt{10+2\,\sqrt{5}}}} \right ) } \end{align*}

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., 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(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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Fricas [B]  time = 2.70209, size = 423, normalized size = 5.16 \begin{align*} -\frac{1}{20} \, \sqrt{2} \sqrt{\sqrt{5} + 5} \log \left (\sqrt{2} \sqrt{\sqrt{5} + 5} + 4 \, \sin \left (x\right )\right ) + \frac{1}{20} \, \sqrt{2} \sqrt{\sqrt{5} + 5} \log \left (\sqrt{2} \sqrt{\sqrt{5} + 5} - 4 \, \sin \left (x\right )\right ) - \frac{1}{20} \, \sqrt{2} \sqrt{-\sqrt{5} + 5} \log \left (\sqrt{2} \sqrt{-\sqrt{5} + 5} + 4 \, \sin \left (x\right )\right ) + \frac{1}{20} \, \sqrt{2} \sqrt{-\sqrt{5} + 5} \log \left (\sqrt{2} \sqrt{-\sqrt{5} + 5} - 4 \, \sin \left (x\right )\right ) + \sin \left (x\right ) \end{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.29034, size = 150, normalized size = 1.83 \begin{align*} -\frac{1}{20} \, \sqrt{2 \, \sqrt{5} + 10} \log \left ({\left | \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5} + 5} + \sin \left (x\right ) \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 \left (x\right ) \right |}\right ) - \frac{1}{20} \, \sqrt{-2 \, \sqrt{5} + 10} \log \left ({\left | \sqrt{-\frac{1}{8} \, \sqrt{5} + \frac{5}{8}} + \sin \left (x\right ) \right |}\right ) + \frac{1}{20} \, \sqrt{-2 \, \sqrt{5} + 10} \log \left ({\left | -\sqrt{-\frac{1}{8} \, \sqrt{5} + \frac{5}{8}} + \sin \left (x\right ) \right |}\right ) + \sin \left (x\right ) \end{align*}

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)