\(\int \frac {\cos ^5(x)}{(a-a \sin ^2(x))^2} \, dx\) [275]

Optimal result

Integrand size = 16, antiderivative size = 6 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)}{a^2} \]

[Out]

sin(x)/a^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2717} \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)}{a^2} \]

[In]

Int[Cos[x]^5/(a - a*Sin[x]^2)^2,x]

[Out]

Sin[x]/a^2

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 (x) \, dx}{a^2} \\ & = \frac {\sin (x)}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)}{a^2} \]

[In]

Integrate[Cos[x]^5/(a - a*Sin[x]^2)^2,x]

[Out]

Sin[x]/a^2

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(\frac {\sin \left (x \right )}{a^{2}}\)	\(7\)
default	\(\frac {\sin \left (x \right )}{a^{2}}\)	\(7\)
risch	\(\frac {\sin \left (x \right )}{a^{2}}\)	\(7\)
parallelrisch	\(\frac {\sin \left (x \right )}{a^{2}}\)	\(7\)
norman	\(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5} a \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\)	\(112\)

[In]

int(cos(x)^5/(a-a*sin(x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

sin(x)/a^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )}{a^{2}} \]

[In]

integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

sin(x)/a^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (5) = 10\).

Time = 6.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 3.17 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {2 \tan {\left (\frac {x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2}} \]

[In]

integrate(cos(x)**5/(a-a*sin(x)**2)**2,x)

[Out]

2*tan(x/2)/(a**2*tan(x/2)**2 + a**2)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )}{a^{2}} \]

[In]

integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

sin(x)/a^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )}{a^{2}} \]

[In]

integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

sin(x)/a^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )}{a^2} \]

[In]

int(cos(x)^5/(a - a*sin(x)^2)^2,x)

[Out]

sin(x)/a^2