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

Optimal. Leaf size=20 \[ \frac{x}{2 a}-\frac{\sin (x) \cos (x)}{2 a} \]

[Out]

x/(2*a) - (Cos[x]*Sin[x])/(2*a)

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Rubi [A]  time = 0.0452092, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3175, 2635, 8} \[ \frac{x}{2 a}-\frac{\sin (x) \cos (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a - a*Cos[x]^2),x]

[Out]

x/(2*a) - (Cos[x]*Sin[x])/(2*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 ^4(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \sin ^2(x) \, dx}{a}\\ &=-\frac{\cos (x) \sin (x)}{2 a}+\frac{\int 1 \, dx}{2 a}\\ &=\frac{x}{2 a}-\frac{\cos (x) \sin (x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0027754, size = 18, normalized size = 0.9 \[ \frac{\frac{x}{2}-\frac{1}{4} \sin (2 x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x/2 - Sin[2*x]/4)/a

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Maple [A]  time = 0.022, size = 25, normalized size = 1.3 \begin{align*} -{\frac{\tan \left ( x \right ) }{2\,a \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ) }{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45167, size = 28, normalized size = 1.4 \begin{align*} \frac{x}{2 \, a} - \frac{\tan \left (x\right )}{2 \,{\left (a \tan \left (x\right )^{2} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/a - 1/2*tan(x)/(a*tan(x)^2 + a)

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Fricas [A]  time = 1.7447, size = 38, normalized size = 1.9 \begin{align*} -\frac{\cos \left (x\right ) \sin \left (x\right ) - x}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(cos(x)*sin(x) - x)/a

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Sympy [B]  time = 4.43492, size = 153, normalized size = 7.65 \begin{align*} \frac{x \tan ^{4}{\left (\frac{x}{2} \right )}}{2 a \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a} + \frac{2 x \tan ^{2}{\left (\frac{x}{2} \right )}}{2 a \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a} + \frac{x}{2 a \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a} + \frac{2 \tan ^{3}{\left (\frac{x}{2} \right )}}{2 a \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{2 a \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*tan(x/2)**4/(2*a*tan(x/2)**4 + 4*a*tan(x/2)**2 + 2*a) + 2*x*tan(x/2)**2/(2*a*tan(x/2)**4 + 4*a*tan(x/2)**2 +
2*a) + x/(2*a*tan(x/2)**4 + 4*a*tan(x/2)**2 + 2*a) + 2*tan(x/2)**3/(2*a*tan(x/2)**4 + 4*a*tan(x/2)**2 + 2*a)
- 2*tan(x/2)/(2*a*tan(x/2)**4 + 4*a*tan(x/2)**2 + 2*a)

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Giac [A]  time = 1.13313, size = 30, normalized size = 1.5 \begin{align*} \frac{x}{2 \, a} - \frac{\tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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