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

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Rubi [A]  time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.45, size = 27, normalized size = 0.71 \[ \frac {{\left (5 \, \cos \relax (x)^{6} + 6 \, \cos \relax (x)^{4} + 8 \, \cos \relax (x)^{2} + 16\right )} \sin \relax (x)}{35 \, a} \]

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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giac [A]  time = 0.14, size = 28, normalized size = 0.74 \[ -\frac {5 \, \sin \relax (x)^{7} - 21 \, \sin \relax (x)^{5} + 35 \, \sin \relax (x)^{3} - 35 \, \sin \relax (x)}{35 \, a} \]

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

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maple [A]  time = 0.17, size = 26, normalized size = 0.68 \[ \frac {-\frac {\left (\sin ^{7}\relax (x )\right )}{7}+\frac {3 \left (\sin ^{5}\relax (x )\right )}{5}-\left (\sin ^{3}\relax (x )\right )+\sin \relax (x )}{a} \]

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 = 0.36, size = 28, normalized size = 0.74 \[ -\frac {5 \, \sin \relax (x)^{7} - 21 \, \sin \relax (x)^{5} + 35 \, \sin \relax (x)^{3} - 35 \, \sin \relax (x)}{35 \, a} \]

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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mupad [B]  time = 0.10, size = 34, normalized size = 0.89 \[ \frac {\sin \relax (x)}{a}-\frac {{\sin \relax (x)}^3}{a}+\frac {3\,{\sin \relax (x)}^5}{5\,a}-\frac {{\sin \relax (x)}^7}{7\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 41.26, size = 580, normalized size = 15.26 \[ \frac {70 \tan ^{13}{\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} + \frac {140 \tan ^{11}{\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} + \frac {602 \tan ^{9}{\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} + \frac {424 \tan ^{7}{\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} + \frac {602 \tan ^{5}{\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} + \frac {140 \tan ^{3}{\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} + \frac {70 \tan {\left (\frac {x}{2} \right )}}{35 a \tan ^{14}{\left (\frac {x}{2} \right )} + 245 a \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a \tan ^{10}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a \tan ^{6}{\left (\frac {x}{2} \right )} + 735 a \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a \tan ^{2}{\left (\frac {x}{2} \right )} + 35 a} \]

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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