3.26 \(\int \frac{\sin (x)}{(a+a \sin (x))^3} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\cos (x)}{5 \left (a^3 \sin (x)+a^3\right )}-\frac{\cos (x)}{5 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

[Out]

Cos[x]/(5*(a + a*Sin[x])^3) - Cos[x]/(5*a*(a + a*Sin[x])^2) - Cos[x]/(5*(a^3 + a^3*Sin[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0460519, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2750, 2650, 2648} \[ -\frac{\cos (x)}{5 \left (a^3 \sin (x)+a^3\right )}-\frac{\cos (x)}{5 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + a*Sin[x])^3,x]

[Out]

Cos[x]/(5*(a + a*Sin[x])^3) - Cos[x]/(5*a*(a + a*Sin[x])^2) - Cos[x]/(5*(a^3 + a^3*Sin[x]))

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)]

Rule 2650

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

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 (x)}{(a+a \sin (x))^3} \, dx &=\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{3 \int \frac{1}{(a+a \sin (x))^2} \, dx}{5 a}\\ &=\frac{\cos (x)}{5 (a+a \sin (x))^3}-\frac{\cos (x)}{5 a (a+a \sin (x))^2}+\frac{\int \frac{1}{a+a \sin (x)} \, dx}{5 a^2}\\ &=\frac{\cos (x)}{5 (a+a \sin (x))^3}-\frac{\cos (x)}{5 a (a+a \sin (x))^2}-\frac{\cos (x)}{5 \left (a^3+a^3 \sin (x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.0403954, size = 41, normalized size = 0.82 \[ \frac{\sin ^2\left (\frac{x}{2}\right ) (8 \sin (x)+\sin (2 x)+4 \cos (x)-\cos (2 x)+7)}{5 a^3 (\sin (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + a*Sin[x])^3,x]

[Out]

(Sin[x/2]^2*(7 + 4*Cos[x] - Cos[2*x] + 8*Sin[x] + Sin[2*x]))/(5*a^3*(1 + Sin[x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.035, size = 45, normalized size = 0.9 \begin{align*} 4\,{\frac{1}{{a}^{3}} \left ( \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-3}+2/5\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-5}- \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-4}-1/2\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a+a*sin(x))^3,x)

[Out]

4/a^3*(1/(tan(1/2*x)+1)^3+2/5/(tan(1/2*x)+1)^5-1/(tan(1/2*x)+1)^4-1/2/(tan(1/2*x)+1)^2)

________________________________________________________________________________________

Maxima [B]  time = 1.62675, size = 157, normalized size = 3.14 \begin{align*} -\frac{2 \,{\left (\frac{5 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{5 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 1\right )}}{5 \,{\left (a^{3} + \frac{5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))^3,x, algorithm="maxima")

[Out]

-2/5*(5*sin(x)/(cos(x) + 1) + 5*sin(x)^2/(cos(x) + 1)^2 + 5*sin(x)^3/(cos(x) + 1)^3 + 1)/(a^3 + 5*a^3*sin(x)/(
cos(x) + 1) + 10*a^3*sin(x)^2/(cos(x) + 1)^2 + 10*a^3*sin(x)^3/(cos(x) + 1)^3 + 5*a^3*sin(x)^4/(cos(x) + 1)^4
+ a^3*sin(x)^5/(cos(x) + 1)^5)

________________________________________________________________________________________

Fricas [B]  time = 1.38556, size = 246, normalized size = 4.92 \begin{align*} -\frac{\cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) + 1}{5 \,{\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} +{\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

-1/5*(cos(x)^3 - 2*cos(x)^2 - (cos(x)^2 + 3*cos(x) + 1)*sin(x) - 2*cos(x) + 1)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2
- 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))

________________________________________________________________________________________

Sympy [B]  time = 5.19911, size = 418, normalized size = 8.36 \begin{align*} \frac{12 \tan ^{5}{\left (\frac{x}{2} \right )}}{55 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan{\left (\frac{x}{2} \right )} + 55 a^{3}} + \frac{60 \tan ^{4}{\left (\frac{x}{2} \right )}}{55 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan{\left (\frac{x}{2} \right )} + 55 a^{3}} + \frac{10 \tan ^{3}{\left (\frac{x}{2} \right )}}{55 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan{\left (\frac{x}{2} \right )} + 55 a^{3}} + \frac{10 \tan ^{2}{\left (\frac{x}{2} \right )}}{55 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan{\left (\frac{x}{2} \right )} + 55 a^{3}} - \frac{50 \tan{\left (\frac{x}{2} \right )}}{55 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan{\left (\frac{x}{2} \right )} + 55 a^{3}} - \frac{10}{55 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 550 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 275 a^{3} \tan{\left (\frac{x}{2} \right )} + 55 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))**3,x)

[Out]

12*tan(x/2)**5/(55*a**3*tan(x/2)**5 + 275*a**3*tan(x/2)**4 + 550*a**3*tan(x/2)**3 + 550*a**3*tan(x/2)**2 + 275
*a**3*tan(x/2) + 55*a**3) + 60*tan(x/2)**4/(55*a**3*tan(x/2)**5 + 275*a**3*tan(x/2)**4 + 550*a**3*tan(x/2)**3
+ 550*a**3*tan(x/2)**2 + 275*a**3*tan(x/2) + 55*a**3) + 10*tan(x/2)**3/(55*a**3*tan(x/2)**5 + 275*a**3*tan(x/2
)**4 + 550*a**3*tan(x/2)**3 + 550*a**3*tan(x/2)**2 + 275*a**3*tan(x/2) + 55*a**3) + 10*tan(x/2)**2/(55*a**3*ta
n(x/2)**5 + 275*a**3*tan(x/2)**4 + 550*a**3*tan(x/2)**3 + 550*a**3*tan(x/2)**2 + 275*a**3*tan(x/2) + 55*a**3)
- 50*tan(x/2)/(55*a**3*tan(x/2)**5 + 275*a**3*tan(x/2)**4 + 550*a**3*tan(x/2)**3 + 550*a**3*tan(x/2)**2 + 275*
a**3*tan(x/2) + 55*a**3) - 10/(55*a**3*tan(x/2)**5 + 275*a**3*tan(x/2)**4 + 550*a**3*tan(x/2)**3 + 550*a**3*ta
n(x/2)**2 + 275*a**3*tan(x/2) + 55*a**3)

________________________________________________________________________________________

Giac [A]  time = 1.32119, size = 50, normalized size = 1. \begin{align*} -\frac{2 \,{\left (5 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 5 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 5 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{5 \, a^{3}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

-2/5*(5*tan(1/2*x)^3 + 5*tan(1/2*x)^2 + 5*tan(1/2*x) + 1)/(a^3*(tan(1/2*x) + 1)^5)