3.5 \(\int \frac{\sin ^2(x)}{a+a \sin (x)} \, dx\)

Optimal. Leaf size=27 \[ -\frac{x}{a}-\frac{\cos (x)}{a}-\frac{\cos (x)}{a (\sin (x)+1)} \]

[Out]

-(x/a) - Cos[x]/a - Cos[x]/(a*(1 + Sin[x]))

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Rubi [A]  time = 0.0664069, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2746, 12, 2735, 2648} \[ -\frac{x}{a}-\frac{\cos (x)}{a}-\frac{\cos (x)}{a (\sin (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a + a*Sin[x]),x]

[Out]

-(x/a) - Cos[x]/a - Cos[x]/(a*(1 + Sin[x]))

Rule 2746

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b^2
*Cos[e + f*x])/(d*f), x] + Dist[1/d, Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0606443, size = 48, normalized size = 1.78 \[ -\frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (\cos \left (\frac{x}{2}\right ) (x+\cos (x))+\sin \left (\frac{x}{2}\right ) (x+\cos (x)-2)\right )}{a (\sin (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(a + a*Sin[x]),x]

[Out]

-(((Cos[x/2] + Sin[x/2])*(Cos[x/2]*(x + Cos[x]) + (-2 + x + Cos[x])*Sin[x/2]))/(a*(1 + Sin[x])))

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Maple [A]  time = 0.024, size = 40, normalized size = 1.5 \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)^2/(a+a*sin(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 = 2.5776, size = 105, normalized size = 3.89 \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)^2/(a+a*sin(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 = 1.72242, size = 122, normalized size = 4.52 \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)^2/(a+a*sin(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 [B]  time = 1.45736, size = 250, normalized size = 9.26 \begin{align*} - \frac{x \tan ^{3}{\left (\frac{x}{2} \right )}}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} - \frac{x \tan ^{2}{\left (\frac{x}{2} \right )}}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} - \frac{x \tan{\left (\frac{x}{2} \right )}}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} - \frac{x}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} + \frac{3 \tan ^{3}{\left (\frac{x}{2} \right )}}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} + \frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} + \frac{\tan{\left (\frac{x}{2} \right )}}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} - \frac{1}{a \tan ^{3}{\left (\frac{x}{2} \right )} + a \tan ^{2}{\left (\frac{x}{2} \right )} + a \tan{\left (\frac{x}{2} \right )} + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(a+a*sin(x)),x)

[Out]

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

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Giac [A]  time = 1.89314, size = 59, normalized size = 2.19 \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)^2/(a+a*sin(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)