\(\int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx\) [267]

Optimal result

Integrand size = 16, antiderivative size = 33 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {3 x}{8 a}+\frac {3 \cos (x) \sin (x)}{8 a}+\frac {\cos ^3(x) \sin (x)}{4 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3254, 2715, 8} \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {3 x}{8 a}+\frac {\sin (x) \cos ^3(x)}{4 a}+\frac {3 \sin (x) \cos (x)}{8 a} \]

[In]

Int[Cos[x]^6/(a - a*Sin[x]^2),x]

[Out]

(3*x)/(8*a) + (3*Cos[x]*Sin[x])/(8*a) + (Cos[x]^3*Sin[x])/(4*a)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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 ^4(x) \, dx}{a} \\ & = \frac {\cos ^3(x) \sin (x)}{4 a}+\frac {3 \int \cos ^2(x) \, dx}{4 a} \\ & = \frac {3 \cos (x) \sin (x)}{8 a}+\frac {\cos ^3(x) \sin (x)}{4 a}+\frac {3 \int 1 \, dx}{8 a} \\ & = \frac {3 x}{8 a}+\frac {3 \cos (x) \sin (x)}{8 a}+\frac {\cos ^3(x) \sin (x)}{4 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {\frac {3 x}{8}+\frac {1}{4} \sin (2 x)+\frac {1}{32} \sin (4 x)}{a} \]

[In]

Integrate[Cos[x]^6/(a - a*Sin[x]^2),x]

[Out]

((3*x)/8 + Sin[2*x]/4 + Sin[4*x]/32)/a

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61

method	result	size
parallelrisch	\(\frac {12 x +\sin \left (4 x \right )+8 \sin \left (2 x \right )}{32 a}\)	\(20\)
risch	\(\frac {3 x}{8 a}+\frac {\sin \left (4 x \right )}{32 a}+\frac {\sin \left (2 x \right )}{4 a}\)	\(26\)
default	\(\frac {\frac {\tan \left (x \right )}{4 \left (1+\tan ^{2}\left (x \right )\right )^{2}}+\frac {3 \tan \left (x \right )}{8 \left (1+\tan ^{2}\left (x \right )\right )}+\frac {3 \arctan \left (\tan \left (x \right )\right )}{8}}{a}\)	\(35\)
norman	\(\frac {\frac {\tan ^{7}\left (\frac {x}{2}\right )}{a}-\frac {5 \tan \left (\frac {x}{2}\right )}{4 a}-\frac {\tan ^{3}\left (\frac {x}{2}\right )}{2 a}+\frac {5 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {5 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{4 a}-\frac {\tan ^{11}\left (\frac {x}{2}\right )}{2 a}-\frac {5 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{4 a}-\frac {3 x}{8 a}-\frac {15 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8 a}-\frac {27 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 a}-\frac {15 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {27 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {3 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{8 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\)	\(187\)

[In]

int(cos(x)^6/(a-a*sin(x)^2),x,method=_RETURNVERBOSE)

[Out]

1/32*(12*x+sin(4*x)+8*sin(2*x))/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {{\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 3 \, x}{8 \, a} \]

[In]

integrate(cos(x)^6/(a-a*sin(x)^2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (29) = 58\).

Time = 3.75 (sec) , antiderivative size = 473, normalized size of antiderivative = 14.33 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {3 x \tan ^{8}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {12 x \tan ^{6}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {18 x \tan ^{4}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {12 x \tan ^{2}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {3 x}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} - \frac {10 \tan ^{7}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {6 \tan ^{5}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} - \frac {6 \tan ^{3}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {10 \tan {\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} \]

[In]

integrate(cos(x)**6/(a-a*sin(x)**2),x)

[Out]

3*x*tan(x/2)**8/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 12*x*tan(x/
2)**6/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 18*x*tan(x/2)**4/(8*a
*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 12*x*tan(x/2)**2/(8*a*tan(x/2)*
*8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 3*x/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 +
48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) - 10*tan(x/2)**7/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/
2)**4 + 32*a*tan(x/2)**2 + 8*a) + 6*tan(x/2)**5/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*
tan(x/2)**2 + 8*a) - 6*tan(x/2)**3/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 +
8*a) + 10*tan(x/2)/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {3 \, \tan \left (x\right )^{3} + 5 \, \tan \left (x\right )}{8 \, {\left (a \tan \left (x\right )^{4} + 2 \, a \tan \left (x\right )^{2} + a\right )}} + \frac {3 \, x}{8 \, a} \]

[In]

integrate(cos(x)^6/(a-a*sin(x)^2),x, algorithm="maxima")

[Out]

1/8*(3*tan(x)^3 + 5*tan(x))/(a*tan(x)^4 + 2*a*tan(x)^2 + a) + 3/8*x/a

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {3 \, \arctan \left (\tan \left (x\right )\right )}{8 \, a} + \frac {\frac {3 \, \tan \left (x\right )^{3}}{a} + \frac {5 \, \tan \left (x\right )}{a}}{8 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \]

[In]

integrate(cos(x)^6/(a-a*sin(x)^2),x, algorithm="giac")

[Out]

3/8*arctan(tan(x))/a + 1/8*(3*tan(x)^3/a + 5*tan(x)/a)/(tan(x)^2 + 1)^2

Mupad [B] (verification not implemented)

Time = 13.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^6(x)}{a-a \sin ^2(x)} \, dx=\frac {\sin \left (2\,x\right )}{4\,a}+\frac {\sin \left (4\,x\right )}{32\,a}+\frac {3\,x}{8\,a} \]

[In]

int(cos(x)^6/(a - a*sin(x)^2),x)

[Out]

sin(2*x)/(4*a) + sin(4*x)/(32*a) + (3*x)/(8*a)