3.82 \(\int \cot (4 x) \sin (x) \, dx\)

Optimal. Leaf size=28 \[ \sin (x)-\frac{1}{4} \tanh ^{-1}(\sin (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}} \]

[Out]

-ArcTanh[Sin[x]]/4 - ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + Sin[x]

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Rubi [A]  time = 0.0511705, antiderivative size = 28, 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} \[ \sin (x)-\frac{1}{4} \tanh ^{-1}(\sin (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[4*x]*Sin[x],x]

[Out]

-ArcTanh[Sin[x]]/4 - ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + 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 (4 x) \sin (x) \, dx &=\operatorname{Subst}\left (\int \frac{1-8 x^2+8 x^4}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1-\frac{3-4 x^2}{4-12 x^2+8 x^4}\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)-\operatorname{Subst}\left (\int \frac{3-4 x^2}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=\sin (x)+2 \operatorname{Subst}\left (\int \frac{1}{-8+8 x^2} \, dx,x,\sin (x)\right )+2 \operatorname{Subst}\left (\int \frac{1}{-4+8 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac{1}{4} \tanh ^{-1}(\sin (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}}+\sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0320893, size = 28, normalized size = 1. \[ \sin (x)-\frac{1}{4} \tanh ^{-1}(\sin (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[4*x]*Sin[x],x]

[Out]

-ArcTanh[Sin[x]]/4 - ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + Sin[x]

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Maple [A]  time = 0.047, size = 30, normalized size = 1.1 \begin{align*} \sin \left ( x \right ) -{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{4}}-{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{8}}+{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(4*x)*sin(x),x)

[Out]

sin(x)-1/4*arctanh(sin(x)*2^(1/2))*2^(1/2)-1/8*ln(1+sin(x))+1/8*ln(sin(x)-1)

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Maxima [B]  time = 1.57866, size = 234, normalized size = 8.36 \begin{align*} -\frac{1}{16} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) + \frac{1}{16} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{16} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) + \frac{1}{16} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(4*x)*sin(x),x, algorithm="maxima")

[Out]

-1/16*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 2*sqrt(2)*sin(x) + 2) + 1/16*sqrt(2)*log(2*cos(
x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) - 2*sqrt(2)*sin(x) + 2) - 1/16*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sq
rt(2)*cos(x) + 2*sqrt(2)*sin(x) + 2) + 1/16*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) - 2*sqrt(2)
*sin(x) + 2) - 1/8*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + 1/8*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) + sin
(x)

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Fricas [B]  time = 2.50194, size = 170, normalized size = 6.07 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (-\frac{2 \, \cos \left (x\right )^{2} + 2 \, \sqrt{2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \frac{1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(4*x)*sin(x),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log(-(2*cos(x)^2 + 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1)) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x)
+ 1) + sin(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (x \right )} \cot{\left (4 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(4*x)*sin(x),x)

[Out]

Integral(sin(x)*cot(4*x), x)

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Giac [B]  time = 1.16234, size = 68, normalized size = 2.43 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac{1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(4*x)*sin(x),x, algorithm="giac")

[Out]

1/8*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x))) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x)
+ 1) + sin(x)