\(\int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx\) [263]

Optimal result

Integrand size = 16, antiderivative size = 18 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin (x)}{a}-\frac {\sin ^3(x)}{3 a} \]

[Out]

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

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} \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin (x)}{a}-\frac {\sin ^3(x)}{3 a} \]

[In]

Int[Cos[x]^5/(a - a*Sin[x]^2),x]

[Out]

Sin[x]/a - Sin[x]^3/(3*a)

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin (x)-\frac {\sin ^3(x)}{3}}{a} \]

[In]

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

[Out]

(Sin[x] - Sin[x]^3/3)/a

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {-\frac {\left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a}\)	\(14\)
default	\(\frac {-\frac {\left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a}\)	\(14\)
parallelrisch	\(\frac {9 \sin \left (x \right )+\sin \left (3 x \right )}{12 a}\)	\(15\)
risch	\(\frac {3 \sin \left (x \right )}{4 a}+\frac {\sin \left (3 x \right )}{12 a}\)	\(18\)
norman	\(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {10 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {4 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {4 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {10 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {2 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\)	\(87\)

[In]

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

[Out]

1/a*(-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 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=\frac {{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{3 \, a} \]

[In]

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

[Out]

1/3*(cos(x)^2 + 2)*sin(x)/a

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (12) = 24\).

Time = 2.59 (sec) , antiderivative size = 124, normalized size of antiderivative = 6.89 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=\frac {6 \tan ^{5}{\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} + \frac {4 \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} + \frac {6 \tan {\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=-\frac {\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{3 \, a} \]

[In]

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

[Out]

-1/3*(sin(x)^3 - 3*sin(x))/a

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=-\frac {\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{3 \, a} \]

[In]

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

[Out]

-1/3*(sin(x)^3 - 3*sin(x))/a

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx=\frac {3\,\sin \left (x\right )-{\sin \left (x\right )}^3}{3\,a} \]

[In]

int(cos(x)^5/(a - a*sin(x)^2),x)

[Out]

(3*sin(x) - sin(x)^3)/(3*a)