3.27 \(\int \frac{1}{(a+a \sin (x))^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0350515, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2650, 2648} \[ -\frac{2 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac{2 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[x])^(-3),x]

[Out]

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

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

Mathematica [A]  time = 0.0578181, size = 45, normalized size = 0.9 \[ -\frac{-10 \sin \left (\frac{x}{2}\right )+\sin \left (\frac{5 x}{2}\right )+5 \cos \left (\frac{3 x}{2}\right )}{15 a^3 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[x])^(-3),x]

[Out]

-(5*Cos[(3*x)/2] - 10*Sin[x/2] + Sin[(5*x)/2])/(15*a^3*(Cos[x/2] + Sin[x/2])^5)

________________________________________________________________________________________

Maple [A]  time = 0.034, size = 57, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{{a}^{3}} \left ( -4/5\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-5}+2\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-4}+2\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-2}- \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-1}-8/3\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(x))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.62658, size = 173, normalized size = 3.46 \begin{align*} -\frac{2 \,{\left (\frac{20 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{40 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{30 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 7\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(1/(a+a*sin(x))^3,x, algorithm="maxima")

[Out]

-2/15*(20*sin(x)/(cos(x) + 1) + 40*sin(x)^2/(cos(x) + 1)^2 + 30*sin(x)^3/(cos(x) + 1)^3 + 15*sin(x)^4/(cos(x)
+ 1)^4 + 7)/(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 = 1.39788, size = 252, normalized size = 5.04 \begin{align*} -\frac{2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} -{\left (2 \, \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(1/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

-1/15*(2*cos(x)^3 - 4*cos(x)^2 - (2*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 = 3.40163, size = 348, normalized size = 6.96 \begin{align*} - \frac{30 \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{60 \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}} - \frac{80 \tan ^{2}{\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{\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{14}{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(1/(a+a*sin(x))**3,x)

[Out]

-30*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 + 150*a**3*tan(x/2)**2 + 75*
a**3*tan(x/2) + 15*a**3) - 60*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) - 80*tan(x/2)**2/(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) - 40*tan(x/2)/(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) - 14/(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 = 1.23253, size = 61, normalized size = 1.22 \begin{align*} -\frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 30 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 40 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 20 \, \tan \left (\frac{1}{2} \, x\right ) + 7\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(1/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

-2/15*(15*tan(1/2*x)^4 + 30*tan(1/2*x)^3 + 40*tan(1/2*x)^2 + 20*tan(1/2*x) + 7)/(a^3*(tan(1/2*x) + 1)^5)