\(\int \frac {\cos ^4(x)}{(a-a \sin ^2(x))^2} \, dx\) [281]

Optimal result

Integrand size = 16, antiderivative size = 5 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^2} \]

[Out]

x/a^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 5, 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, 8} \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^2} \]

[In]

Int[Cos[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

x/a^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 1 \, dx}{a^2} \\ & = \frac {x}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^2} \]

[In]

Integrate[Cos[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

x/a^2

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
risch	\(\frac {x}{a^{2}}\)	\(6\)
default	\(\frac {\arctan \left (\tan \left (x \right )\right )}{a^{2}}\)	\(8\)
norman	\(\frac {\frac {x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}+\frac {x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{a}-\frac {x}{a}-\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4} a \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\)	\(114\)

[In]

int(cos(x)^4/(a-a*sin(x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

x/a^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^{2}} \]

[In]

integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

x/a^2

Sympy [A] (verification not implemented)

Time = 3.78 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^{2}} \]

[In]

integrate(cos(x)**4/(a-a*sin(x)**2)**2,x)

[Out]

x/a**2

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^{2}} \]

[In]

integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

x/a^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^{2}} \]

[In]

integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

x/a^2

Mupad [B] (verification not implemented)

Time = 13.95 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {x}{a^2} \]

[In]

int(cos(x)^4/(a - a*sin(x)^2)^2,x)

[Out]

x/a^2