3.209 \(\int \frac{\sin (x)}{-\cot (x)+\csc (x)} \, dx\)

Optimal. Leaf size=4 \[ x+\sin (x) \]

[Out]

x + Sin[x]

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Rubi [A]  time = 0.0697868, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4392, 2682, 8} \[ x+\sin (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(-Cot[x] + Csc[x]),x]

[Out]

x + Sin[x]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{-\cot (x)+\csc (x)} \, dx &=\int \frac{\sin ^2(x)}{1-\cos (x)} \, dx\\ &=\sin (x)+\int 1 \, dx\\ &=x+\sin (x)\\ \end{align*}

Mathematica [B]  time = 0.006656, size = 14, normalized size = 3.5 \[ 2 \left (\frac{x}{2}+\frac{\sin (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(-Cot[x] + Csc[x]),x]

[Out]

2*(x/2 + Sin[x]/2)

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Maple [B]  time = 0.079, size = 19, normalized size = 4.8 \begin{align*} 2\,{\frac{\tan \left ( x/2 \right ) }{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1}}+x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(-cot(x)+csc(x)),x)

[Out]

2*tan(1/2*x)/(tan(1/2*x)^2+1)+x

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Maxima [B]  time = 1.53232, size = 51, normalized size = 12.75 \begin{align*} \frac{2 \, \sin \left (x\right )}{{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(-cot(x)+csc(x)),x, algorithm="maxima")

[Out]

2*sin(x)/((sin(x)^2/(cos(x) + 1)^2 + 1)*(cos(x) + 1)) + 2*arctan(sin(x)/(cos(x) + 1))

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Fricas [A]  time = 0.469711, size = 16, normalized size = 4. \begin{align*} x + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(-cot(x)+csc(x)),x, algorithm="fricas")

[Out]

x + sin(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(-cot(x)+csc(x)),x)

[Out]

-Integral(sin(x)/(cot(x) - csc(x)), x)

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Giac [B]  time = 1.12148, size = 24, normalized size = 6. \begin{align*} x + \frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(-cot(x)+csc(x)),x, algorithm="giac")

[Out]

x + 2*tan(1/2*x)/(tan(1/2*x)^2 + 1)