3.80 \(\int \cot (2 x) \sin (x) \, dx\)

Optimal. Leaf size=10 \[ \sin (x)-\frac{1}{2} \tanh ^{-1}(\sin (x)) \]

[Out]

-ArcTanh[Sin[x]]/2 + Sin[x]

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Rubi [A]  time = 0.0218386, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {388, 206} \[ \sin (x)-\frac{1}{2} \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cot[2*x]*Sin[x],x]

[Out]

-ArcTanh[Sin[x]]/2 + Sin[x]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0090133, size = 10, normalized size = 1. \[ \sin (x)-\frac{1}{2} \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[2*x]*Sin[x],x]

[Out]

-ArcTanh[Sin[x]]/2 + Sin[x]

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Maple [A]  time = 0.026, size = 12, normalized size = 1.2 \begin{align*} \sin \left ( x \right ) -{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(2*x)*sin(x),x)

[Out]

sin(x)-1/2*ln(sec(x)+tan(x))

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Maxima [B]  time = 1.49586, size = 50, normalized size = 5. \begin{align*} -\frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac{1}{4} \, \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(2*x)*sin(x),x, algorithm="maxima")

[Out]

-1/4*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + 1/4*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) + sin(x)

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Fricas [B]  time = 2.41989, size = 73, normalized size = 7.3 \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{4} \, \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(2*x)*sin(x),x, algorithm="fricas")

[Out]

-1/4*log(sin(x) + 1) + 1/4*log(-sin(x) + 1) + sin(x)

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Sympy [B]  time = 1.20567, size = 19, normalized size = 1.9 \begin{align*} \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4} + \sin{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(2*x)*sin(x),x)

[Out]

log(sin(x) - 1)/4 - log(sin(x) + 1)/4 + sin(x)

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Giac [B]  time = 1.12964, size = 26, normalized size = 2.6 \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{4} \, \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(2*x)*sin(x),x, algorithm="giac")

[Out]

-1/4*log(sin(x) + 1) + 1/4*log(-sin(x) + 1) + sin(x)