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

Optimal. Leaf size=33 \[ \frac{3 x}{8 a}-\frac{\sin ^3(x) \cos (x)}{4 a}-\frac{3 \sin (x) \cos (x)}{8 a} \]

[Out]

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

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Rubi [A]  time = 0.0524986, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3175, 2635, 8} \[ \frac{3 x}{8 a}-\frac{\sin ^3(x) \cos (x)}{4 a}-\frac{3 \sin (x) \cos (x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^6/(a - a*Cos[x]^2),x]

[Out]

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

Rule 3175

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]

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\sin ^6(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \sin ^4(x) \, dx}{a}\\ &=-\frac{\cos (x) \sin ^3(x)}{4 a}+\frac{3 \int \sin ^2(x) \, dx}{4 a}\\ &=-\frac{3 \cos (x) \sin (x)}{8 a}-\frac{\cos (x) \sin ^3(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 (x) \sin ^3(x)}{4 a}\\ \end{align*}

Mathematica [A]  time = 0.0042435, size = 26, normalized size = 0.79 \[ \frac{\frac{3 x}{8}-\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.083, size = 42, normalized size = 1.3 \begin{align*} -{\frac{5\, \left ( \tan \left ( x \right ) \right ) ^{3}}{8\,a \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{3\,\tan \left ( x \right ) }{8\,a \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{3\,\arctan \left ( \tan \left ( x \right ) \right ) }{8\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42433, size = 50, normalized size = 1.52 \begin{align*} -\frac{5 \, \tan \left (x\right )^{3} + 3 \, \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.89114, size = 62, normalized size = 1.88 \begin{align*} \frac{{\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 3 \, x}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((2*cos(x)^3 - 5*cos(x))*sin(x) + 3*x)/a

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Sympy [B]  time = 16.2315, size = 473, normalized size = 14.33 \begin{align*} \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{6 \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{22 \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{22 \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{6 \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**6/(a-a*cos(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) + 6*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) + 22*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) - 22*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) - 6*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)

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Giac [A]  time = 1.1531, size = 42, normalized size = 1.27 \begin{align*} \frac{3 \, x}{8 \, a} - \frac{5 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{8 \,{\left (\tan \left (x\right )^{2} + 1\right )}^{2} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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