3.7 \(\int \frac{\sin (x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a} \]

[Out]

-(x/a) - (2*Cos[x])/a + Cos[x]/(a + a*Csc[x])

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Rubi [A]  time = 0.0468814, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3819, 3787, 2638, 8} \[ -\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + a*Csc[x]),x]

[Out]

-(x/a) - (2*Cos[x])/a + Cos[x]/(a + a*Csc[x])

Rule 3819

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.077861, size = 32, normalized size = 1.28 \[ -\frac{x+\cos (x)-\frac{2 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + a*Csc[x]),x]

[Out]

-((x + Cos[x] - (2*Sin[x/2])/(Cos[x/2] + Sin[x/2]))/a)

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Maple [A]  time = 0.043, size = 40, normalized size = 1.6 \begin{align*} -2\,{\frac{1}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}-2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a+a*csc(x)),x)

[Out]

-2/a/(tan(1/2*x)^2+1)-2/a*arctan(tan(1/2*x))-2/a/(tan(1/2*x)+1)

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Maxima [B]  time = 1.45752, size = 105, normalized size = 4.2 \begin{align*} -\frac{2 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right )}}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}} - \frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*csc(x)),x, algorithm="maxima")

[Out]

-2*(sin(x)/(cos(x) + 1) + sin(x)^2/(cos(x) + 1)^2 + 2)/(a + a*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2
+ a*sin(x)^3/(cos(x) + 1)^3) - 2*arctan(sin(x)/(cos(x) + 1))/a

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Fricas [A]  time = 0.472957, size = 122, normalized size = 4.88 \begin{align*} -\frac{{\left (x + 2\right )} \cos \left (x\right ) + \cos \left (x\right )^{2} +{\left (x + \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + x + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*csc(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*csc(x)),x)

[Out]

Integral(sin(x)/(csc(x) + 1), x)/a

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Giac [A]  time = 1.33014, size = 59, normalized size = 2.36 \begin{align*} -\frac{x}{a} - \frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 2\right )}}{{\left (\tan \left (\frac{1}{2} \, x\right )^{3} + \tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 1\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*csc(x)),x, algorithm="giac")

[Out]

-x/a - 2*(tan(1/2*x)^2 + tan(1/2*x) + 2)/((tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1)*a)