3.279 \(\int \frac{\cos ^8(x)}{(a-a \sin ^2(x))^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{3 x}{8 a^2}+\frac{\sin (x) \cos ^3(x)}{4 a^2}+\frac{3 \sin (x) \cos (x)}{8 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0501656, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3175, 2635, 8} \[ \frac{3 x}{8 a^2}+\frac{\sin (x) \cos ^3(x)}{4 a^2}+\frac{3 \sin (x) \cos (x)}{8 a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^8/(a - a*Sin[x]^2)^2,x]

[Out]

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

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \cos ^4(x) \, dx}{a^2}\\ &=\frac{\cos ^3(x) \sin (x)}{4 a^2}+\frac{3 \int \cos ^2(x) \, dx}{4 a^2}\\ &=\frac{3 \cos (x) \sin (x)}{8 a^2}+\frac{\cos ^3(x) \sin (x)}{4 a^2}+\frac{3 \int 1 \, dx}{8 a^2}\\ &=\frac{3 x}{8 a^2}+\frac{3 \cos (x) \sin (x)}{8 a^2}+\frac{\cos ^3(x) \sin (x)}{4 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0027006, size = 26, normalized size = 0.79 \[ \frac{\frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^8/(a - a*Sin[x]^2)^2,x]

[Out]

((3*x)/8 + Sin[2*x]/4 + Sin[4*x]/32)/a^2

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Maple [A]  time = 0.039, size = 40, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( x \right ) }{4\,{a}^{2} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{3\,\tan \left ( x \right ) }{8\,{a}^{2} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{3\,\arctan \left ( \tan \left ( x \right ) \right ) }{8\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^8/(a-a*sin(x)^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.51192, size = 58, normalized size = 1.76 \begin{align*} \frac{3 \, \tan \left (x\right )^{3} + 5 \, \tan \left (x\right )}{8 \,{\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} + \frac{3 \, x}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^8/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94281, size = 65, normalized size = 1.97 \begin{align*} \frac{{\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 3 \, x}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^8/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

1/8*((2*cos(x)^3 + 3*cos(x))*sin(x) + 3*x)/a^2

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Sympy [B]  time = 123.222, size = 549, normalized size = 16.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**8/(a-a*sin(x)**2)**2,x)

[Out]

3*x*tan(x/2)**8/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/2)**2 + 8*a**2
) + 12*x*tan(x/2)**6/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/2)**2 + 8
*a**2) + 18*x*tan(x/2)**4/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/2)**
2 + 8*a**2) + 12*x*tan(x/2)**2/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x
/2)**2 + 8*a**2) + 3*x/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/2)**2 +
8*a**2) - 10*tan(x/2)**7/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/2)**
2 + 8*a**2) + 6*tan(x/2)**5/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/2)
**2 + 8*a**2) - 6*tan(x/2)**3/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/
2)**2 + 8*a**2) + 10*tan(x/2)/(8*a**2*tan(x/2)**8 + 32*a**2*tan(x/2)**6 + 48*a**2*tan(x/2)**4 + 32*a**2*tan(x/
2)**2 + 8*a**2)

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Giac [A]  time = 1.10106, size = 42, normalized size = 1.27 \begin{align*} \frac{3 \, x}{8 \, a^{2}} + \frac{3 \, \tan \left (x\right )^{3} + 5 \, \tan \left (x\right )}{8 \,{\left (\tan \left (x\right )^{2} + 1\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^8/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

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