\(\int \frac {\cos ^7(x)}{(a-a \sin ^2(x))^2} \, dx\) [274]

Optimal result

Integrand size = 16, antiderivative size = 18 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)}{a^2}-\frac {\sin ^3(x)}{3 a^2} \]

[Out]

sin(x)/a^2-1/3*sin(x)^3/a^2

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2713} \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)}{a^2}-\frac {\sin ^3(x)}{3 a^2} \]

[In]

Int[Cos[x]^7/(a - a*Sin[x]^2)^2,x]

[Out]

Sin[x]/a^2 - Sin[x]^3/(3*a^2)

Rule 2713

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 3254

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)-\frac {\sin ^3(x)}{3}}{a^2} \]

[In]

Integrate[Cos[x]^7/(a - a*Sin[x]^2)^2,x]

[Out]

(Sin[x] - Sin[x]^3/3)/a^2

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {-\frac {\left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a^{2}}\)	\(14\)
default	\(\frac {-\frac {\left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a^{2}}\)	\(14\)
parallelrisch	\(\frac {9 \sin \left (x \right )+\sin \left (3 x \right )}{12 a^{2}}\)	\(15\)
risch	\(\frac {3 \sin \left (x \right )}{4 a^{2}}+\frac {\sin \left (3 x \right )}{12 a^{2}}\)	\(18\)

[In]

int(cos(x)^7/(a-a*sin(x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{3 \, a^{2}} \]

[In]

integrate(cos(x)^7/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

1/3*(cos(x)^2 + 2)*sin(x)/a^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (15) = 30\).

Time = 13.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 8.00 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {6 \tan ^{5}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {4 \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {6 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a^{2}} \]

[In]

integrate(cos(x)**7/(a-a*sin(x)**2)**2,x)

[Out]

6*tan(x/2)**5/(3*a**2*tan(x/2)**6 + 9*a**2*tan(x/2)**4 + 9*a**2*tan(x/2)**2 + 3*a**2) + 4*tan(x/2)**3/(3*a**2*
tan(x/2)**6 + 9*a**2*tan(x/2)**4 + 9*a**2*tan(x/2)**2 + 3*a**2) + 6*tan(x/2)/(3*a**2*tan(x/2)**6 + 9*a**2*tan(
x/2)**4 + 9*a**2*tan(x/2)**2 + 3*a**2)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=-\frac {\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{3 \, a^{2}} \]

[In]

integrate(cos(x)^7/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

-1/3*(sin(x)^3 - 3*sin(x))/a^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=-\frac {\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{3 \, a^{2}} \]

[In]

integrate(cos(x)^7/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

-1/3*(sin(x)^3 - 3*sin(x))/a^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3\,\sin \left (x\right )-{\sin \left (x\right )}^3}{3\,a^2} \]

[In]

int(cos(x)^7/(a - a*sin(x)^2)^2,x)

[Out]

(3*sin(x) - sin(x)^3)/(3*a^2)