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

Optimal. Leaf size=33 \[ \frac {\cos (x)}{3 (a \sin (x)+a)^2}-\frac {2 \cos (x)}{3 \left (a^2 \sin (x)+a^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2750, 2648} \[ \frac {\cos (x)}{3 (a \sin (x)+a)^2}-\frac {2 \cos (x)}{3 \left (a^2 \sin (x)+a^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + a*Sin[x])^2,x]

[Out]

Cos[x]/(3*(a + a*Sin[x])^2) - (2*Cos[x])/(3*(a^2 + a^2*Sin[x]))

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]

Rule 2750

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

Rubi steps

\begin {align*} \int \frac {\sin (x)}{(a+a \sin (x))^2} \, dx &=\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {2 \int \frac {1}{a+a \sin (x)} \, dx}{3 a}\\ &=\frac {\cos (x)}{3 (a+a \sin (x))^2}-\frac {2 \cos (x)}{3 \left (a^2+a^2 \sin (x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 29, normalized size = 0.88 \[ -\frac {-4 \sin (x)+\sin (2 x)+\cos (x)+\cos (2 x)-3}{3 a^2 (\sin (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + a*Sin[x])^2,x]

[Out]

-1/3*(-3 + Cos[x] + Cos[2*x] - 4*Sin[x] + Sin[2*x])/(a^2*(1 + Sin[x])^2)

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fricas [B]  time = 0.45, size = 60, normalized size = 1.82 \[ \frac {2 \, \cos \relax (x)^{2} + {\left (2 \, \cos \relax (x) + 1\right )} \sin \relax (x) + \cos \relax (x) - 1}{3 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2} - {\left (a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.95, size = 21, normalized size = 0.64 \[ -\frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 27, normalized size = 0.82 \[ \frac {\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a+a*sin(x))^2,x)

[Out]

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

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maxima [B]  time = 0.60, size = 62, normalized size = 1.88 \[ -\frac {2 \, {\left (\frac {3 \, \sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {3 \, a^{2} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {a^{2} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.30, size = 21, normalized size = 0.64 \[ -\frac {2\,\left (3\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{3\,a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a + a*sin(x))^2,x)

[Out]

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

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sympy [B]  time = 2.13, size = 87, normalized size = 2.64 \[ - \frac {6 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {2}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*tan(x/2)/(3*a**2*tan(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2) - 2/(3*a**2*tan(x/2)**3 + 9*a
**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2)

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