3.261 \(\int \frac{\cos ^9(x)}{a-a \sin ^2(x)} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\sin ^7(x)}{7 a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^3(x)}{a}+\frac{\sin (x)}{a} \]

[Out]

Sin[x]/a - Sin[x]^3/a + (3*Sin[x]^5)/(5*a) - Sin[x]^7/(7*a)

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Rubi [A]  time = 0.0538224, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2633} \[ -\frac{\sin ^7(x)}{7 a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^3(x)}{a}+\frac{\sin (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^9/(a - a*Sin[x]^2),x]

[Out]

Sin[x]/a - Sin[x]^3/a + (3*Sin[x]^5)/(5*a) - Sin[x]^7/(7*a)

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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^9(x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \cos ^7(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (x)\right )}{a}\\ &=\frac{\sin (x)}{a}-\frac{\sin ^3(x)}{a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^7(x)}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0044287, size = 35, normalized size = 0.92 \[ \frac{\frac{35 \sin (x)}{64}+\frac{7}{64} \sin (3 x)+\frac{7}{320} \sin (5 x)+\frac{1}{448} \sin (7 x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^9/(a - a*Sin[x]^2),x]

[Out]

((35*Sin[x])/64 + (7*Sin[3*x])/64 + (7*Sin[5*x])/320 + Sin[7*x]/448)/a

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Maple [A]  time = 0.034, size = 26, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ( -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{7}}{7}}+{\frac{3\, \left ( \sin \left ( x \right ) \right ) ^{5}}{5}}- \left ( \sin \left ( x \right ) \right ) ^{3}+\sin \left ( x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^9/(a-a*sin(x)^2),x)

[Out]

1/a*(-1/7*sin(x)^7+3/5*sin(x)^5-sin(x)^3+sin(x))

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Maxima [A]  time = 0.990582, size = 38, normalized size = 1. \begin{align*} -\frac{5 \, \sin \left (x\right )^{7} - 21 \, \sin \left (x\right )^{5} + 35 \, \sin \left (x\right )^{3} - 35 \, \sin \left (x\right )}{35 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^9/(a-a*sin(x)^2),x, algorithm="maxima")

[Out]

-1/35*(5*sin(x)^7 - 21*sin(x)^5 + 35*sin(x)^3 - 35*sin(x))/a

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Fricas [A]  time = 2.45918, size = 80, normalized size = 2.11 \begin{align*} \frac{{\left (5 \, \cos \left (x\right )^{6} + 6 \, \cos \left (x\right )^{4} + 8 \, \cos \left (x\right )^{2} + 16\right )} \sin \left (x\right )}{35 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^9/(a-a*sin(x)^2),x, algorithm="fricas")

[Out]

1/35*(5*cos(x)^6 + 6*cos(x)^4 + 8*cos(x)^2 + 16)*sin(x)/a

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Sympy [B]  time = 83.0451, size = 580, normalized size = 15.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**9/(a-a*sin(x)**2),x)

[Out]

70*tan(x/2)**13/(35*a*tan(x/2)**14 + 245*a*tan(x/2)**12 + 735*a*tan(x/2)**10 + 1225*a*tan(x/2)**8 + 1225*a*tan
(x/2)**6 + 735*a*tan(x/2)**4 + 245*a*tan(x/2)**2 + 35*a) + 140*tan(x/2)**11/(35*a*tan(x/2)**14 + 245*a*tan(x/2
)**12 + 735*a*tan(x/2)**10 + 1225*a*tan(x/2)**8 + 1225*a*tan(x/2)**6 + 735*a*tan(x/2)**4 + 245*a*tan(x/2)**2 +
35*a) + 602*tan(x/2)**9/(35*a*tan(x/2)**14 + 245*a*tan(x/2)**12 + 735*a*tan(x/2)**10 + 1225*a*tan(x/2)**8 + 1
225*a*tan(x/2)**6 + 735*a*tan(x/2)**4 + 245*a*tan(x/2)**2 + 35*a) + 424*tan(x/2)**7/(35*a*tan(x/2)**14 + 245*a
*tan(x/2)**12 + 735*a*tan(x/2)**10 + 1225*a*tan(x/2)**8 + 1225*a*tan(x/2)**6 + 735*a*tan(x/2)**4 + 245*a*tan(x
/2)**2 + 35*a) + 602*tan(x/2)**5/(35*a*tan(x/2)**14 + 245*a*tan(x/2)**12 + 735*a*tan(x/2)**10 + 1225*a*tan(x/2
)**8 + 1225*a*tan(x/2)**6 + 735*a*tan(x/2)**4 + 245*a*tan(x/2)**2 + 35*a) + 140*tan(x/2)**3/(35*a*tan(x/2)**14
+ 245*a*tan(x/2)**12 + 735*a*tan(x/2)**10 + 1225*a*tan(x/2)**8 + 1225*a*tan(x/2)**6 + 735*a*tan(x/2)**4 + 245
*a*tan(x/2)**2 + 35*a) + 70*tan(x/2)/(35*a*tan(x/2)**14 + 245*a*tan(x/2)**12 + 735*a*tan(x/2)**10 + 1225*a*tan
(x/2)**8 + 1225*a*tan(x/2)**6 + 735*a*tan(x/2)**4 + 245*a*tan(x/2)**2 + 35*a)

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Giac [A]  time = 1.10332, size = 38, normalized size = 1. \begin{align*} -\frac{5 \, \sin \left (x\right )^{7} - 21 \, \sin \left (x\right )^{5} + 35 \, \sin \left (x\right )^{3} - 35 \, \sin \left (x\right )}{35 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^9/(a-a*sin(x)^2),x, algorithm="giac")

[Out]

-1/35*(5*sin(x)^7 - 21*sin(x)^5 + 35*sin(x)^3 - 35*sin(x))/a