\(\int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx\) [262]

Optimal result

Integrand size = 16, antiderivative size = 29 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin (x)}{a}-\frac {2 \sin ^3(x)}{3 a}+\frac {\sin ^5(x)}{5 a} \]

[Out]

sin(x)/a-2/3*sin(x)^3/a+1/5*sin(x)^5/a

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 29, 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} \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin ^5(x)}{5 a}-\frac {2 \sin ^3(x)}{3 a}+\frac {\sin (x)}{a} \]

[In]

Int[Cos[x]^7/(a - a*Sin[x]^2),x]

[Out]

Sin[x]/a - (2*Sin[x]^3)/(3*a) + Sin[x]^5/(5*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 ^5(x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )}{a} \\ & = \frac {\sin (x)}{a}-\frac {2 \sin ^3(x)}{3 a}+\frac {\sin ^5(x)}{5 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5}}{a} \]

[In]

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

[Out]

(Sin[x] - (2*Sin[x]^3)/3 + Sin[x]^5/5)/a

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a}\)	\(20\)
default	\(\frac {\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a}\)	\(20\)
parallelrisch	\(\frac {150 \sin \left (x \right )+3 \sin \left (5 x \right )+25 \sin \left (3 x \right )}{240 a}\)	\(23\)
risch	\(\frac {5 \sin \left (x \right )}{8 a}+\frac {\sin \left (5 x \right )}{80 a}+\frac {5 \sin \left (3 x \right )}{48 a}\)	\(27\)
norman	\(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {14 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {42 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {86 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{15 a}+\frac {86 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{15 a}+\frac {42 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {14 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {2 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\)	\(109\)

[In]

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

[Out]

1/a*(1/5*sin(x)^5-2/3*sin(x)^3+sin(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right )}{15 \, a} \]

[In]

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

[Out]

1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)/a

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (22) = 44\).

Time = 5.83 (sec) , antiderivative size = 311, normalized size of antiderivative = 10.72 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {30 \tan ^{9}{\left (\frac {x}{2} \right )}}{15 a \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a} + \frac {40 \tan ^{7}{\left (\frac {x}{2} \right )}}{15 a \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a} + \frac {116 \tan ^{5}{\left (\frac {x}{2} \right )}}{15 a \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a} + \frac {40 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a} + \frac {30 \tan {\left (\frac {x}{2} \right )}}{15 a \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a} \]

[In]

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

[Out]

30*tan(x/2)**9/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6 + 150*a*tan(x/2)**4 + 75*a*tan(x/2)**
2 + 15*a) + 40*tan(x/2)**7/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6 + 150*a*tan(x/2)**4 + 75*
a*tan(x/2)**2 + 15*a) + 116*tan(x/2)**5/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6 + 150*a*tan(
x/2)**4 + 75*a*tan(x/2)**2 + 15*a) + 40*tan(x/2)**3/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6
+ 150*a*tan(x/2)**4 + 75*a*tan(x/2)**2 + 15*a) + 30*tan(x/2)/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan
(x/2)**6 + 150*a*tan(x/2)**4 + 75*a*tan(x/2)**2 + 15*a)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a} \]

[In]

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

[Out]

1/15*(3*sin(x)^5 - 10*sin(x)^3 + 15*sin(x))/a

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a} \]

[In]

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

[Out]

1/15*(3*sin(x)^5 - 10*sin(x)^3 + 15*sin(x))/a

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\cos ^7(x)}{a-a \sin ^2(x)} \, dx=\frac {\frac {{\sin \left (x\right )}^5}{5}-\frac {2\,{\sin \left (x\right )}^3}{3}+\sin \left (x\right )}{a} \]

[In]

int(cos(x)^7/(a - a*sin(x)^2),x)

[Out]

(sin(x) - (2*sin(x)^3)/3 + sin(x)^5/5)/a