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

Optimal. Leaf size=19 \[ \frac{\cos ^3(x)}{3 a}-\frac{\cos (x)}{a} \]

[Out]

-(Cos[x]/a) + Cos[x]^3/(3*a)

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Rubi [A]  time = 0.0451872, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2633} \[ \frac{\cos ^3(x)}{3 a}-\frac{\cos (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^5/(a - a*Cos[x]^2),x]

[Out]

-(Cos[x]/a) + Cos[x]^3/(3*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 2633

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]

Rubi steps

\begin{align*} \int \frac{\sin ^5(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \sin ^3(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a}\\ &=-\frac{\cos (x)}{a}+\frac{\cos ^3(x)}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0031018, size = 19, normalized size = 1. \[ \frac{\frac{1}{12} \cos (3 x)-\frac{3 \cos (x)}{4}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*Cos[x])/4 + Cos[3*x]/12)/a

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Maple [A]  time = 0.018, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{3}}-\cos \left ( x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.946388, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(cos(x)^3 - 3*cos(x))/a

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Fricas [A]  time = 1.73446, size = 39, normalized size = 2.05 \begin{align*} \frac{\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(cos(x)^3 - 3*cos(x))/a

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Sympy [B]  time = 18.3349, size = 162, normalized size = 8.53 \begin{align*} \frac{14 \tan ^{6}{\left (\frac{x}{2} \right )}}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{42 \tan ^{4}{\left (\frac{x}{2} \right )}}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} - \frac{18 \tan ^{2}{\left (\frac{x}{2} \right )}}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} - \frac{6}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14*tan(x/2)**6/(15*a*tan(x/2)**6 + 45*a*tan(x/2)**4 + 45*a*tan(x/2)**2 + 15*a) + 42*tan(x/2)**4/(15*a*tan(x/2)
**6 + 45*a*tan(x/2)**4 + 45*a*tan(x/2)**2 + 15*a) - 18*tan(x/2)**2/(15*a*tan(x/2)**6 + 45*a*tan(x/2)**4 + 45*a
*tan(x/2)**2 + 15*a) - 6/(15*a*tan(x/2)**6 + 45*a*tan(x/2)**4 + 45*a*tan(x/2)**2 + 15*a)

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Giac [A]  time = 1.11841, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(cos(x)^3 - 3*cos(x))/a