3.25 \(\int \frac{\sin ^2(x)}{(a+a \sin (x))^3} \, dx\)

Optimal. Leaf size=50 \[ -\frac{7 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac{8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

[Out]

-Cos[x]/(5*(a + a*Sin[x])^3) + (8*Cos[x])/(15*a*(a + a*Sin[x])^2) - (7*Cos[x])/(15*(a^3 + a^3*Sin[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0760257, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2758, 2750, 2648} \[ -\frac{7 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac{8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a + a*Sin[x])^3,x]

[Out]

-Cos[x]/(5*(a + a*Sin[x])^3) + (8*Cos[x])/(15*a*(a + a*Sin[x])^2) - (7*Cos[x])/(15*(a^3 + a^3*Sin[x]))

Rule 2758

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

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 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))^3} \, dx &=-\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{\int \frac{-3 a+5 a \sin (x)}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=-\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{8 \cos (x)}{15 a (a+a \sin (x))^2}+\frac{7 \int \frac{1}{a+a \sin (x)} \, dx}{15 a^2}\\ &=-\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{8 \cos (x)}{15 a (a+a \sin (x))^2}-\frac{7 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.0638365, size = 47, normalized size = 0.94 \[ \frac{105 \sin (x)-12 \sin (2 x)-7 \sin (3 x)-15 \cos (x)-42 \cos (2 x)+7 \cos (3 x)+70}{60 a^3 (\sin (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(a + a*Sin[x])^3,x]

[Out]

(70 - 15*Cos[x] - 42*Cos[2*x] + 7*Cos[3*x] + 105*Sin[x] - 12*Sin[2*x] - 7*Sin[3*x])/(60*a^3*(1 + Sin[x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 37, normalized size = 0.7 \begin{align*} 8\,{\frac{1}{{a}^{3}} \left ( -1/5\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-5}+1/2\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-4}-1/3\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a+a*sin(x))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.75882, size = 140, normalized size = 2.8 \begin{align*} -\frac{4 \,{\left (\frac{5 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{10 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}}{15 \,{\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)^2/(a+a*sin(x))^3,x, algorithm="maxima")

[Out]

-4/15*(5*sin(x)/(cos(x) + 1) + 10*sin(x)^2/(cos(x) + 1)^2 + 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 = 1.40828, size = 250, normalized size = 5. \begin{align*} -\frac{7 \, \cos \left (x\right )^{3} + \cos \left (x\right )^{2} -{\left (7 \, \cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 9 \, \cos \left (x\right ) - 3}{15 \,{\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)^2/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

-1/15*(7*cos(x)^3 + cos(x)^2 - (7*cos(x)^2 + 6*cos(x) - 3)*sin(x) - 9*cos(x) - 3)/(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 = 111.999, size = 212, normalized size = 4.24 \begin{align*} \frac{4 \tan ^{5}{\left (\frac{x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 75 a^{3} \tan{\left (\frac{x}{2} \right )} + 15 a^{3}} + \frac{20 \tan ^{4}{\left (\frac{x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 75 a^{3} \tan{\left (\frac{x}{2} \right )} + 15 a^{3}} + \frac{40 \tan ^{3}{\left (\frac{x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac{x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac{x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac{x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac{x}{2} \right )} + 75 a^{3} \tan{\left (\frac{x}{2} \right )} + 15 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(a+a*sin(x))**3,x)

[Out]

4*tan(x/2)**5/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a*
*3*tan(x/2) + 15*a**3) + 20*tan(x/2)**4/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 15
0*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 40*tan(x/2)**3/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 +
150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3)

________________________________________________________________________________________

Giac [A]  time = 1.22799, size = 39, normalized size = 0.78 \begin{align*} -\frac{4 \,{\left (10 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 5 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{15 \, 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)^2/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

-4/15*(10*tan(1/2*x)^2 + 5*tan(1/2*x) + 1)/(a^3*(tan(1/2*x) + 1)^5)