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

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {3254, 2713} \begin {gather*} \frac {\cos ^3(x)}{3 a}-\frac {\cos (x)}{a} \end {gather*}

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 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*} \int \frac {\sin ^5(x)}{a-a \cos ^2(x)} \, dx &=\frac {\int \sin ^3(x) \, dx}{a}\\ &=-\frac {\text {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*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {-\frac {3 \cos (x)}{4}+\frac {1}{12} \cos (3 x)}{a} \end {gather*}

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 = 0.07, size = 16, normalized size = 0.84

method	result	size
default	\(\frac {\frac {\left (\cos ^{3}\left (x \right )\right )}{3}-\cos \left (x \right )}{a}\)	\(16\)
risch	\(-\frac {3 \cos \left (x \right )}{4 a}+\frac {\cos \left (3 x \right )}{12 a}\)	\(18\)
norman	\(\frac {-\frac {4 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {20 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {4 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {28 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5} \tan \left (\frac {x}{2}\right )}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 14, normalized size = 0.74 \begin {gather*} \frac {\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end {gather*}

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 = 0.38, size = 14, normalized size = 0.74 \begin {gather*} \frac {\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (12) = 24\).
time = 1.37, size = 78, normalized size = 4.11 \begin {gather*} - \frac {12 \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} - \frac {4}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*tan(x/2)**2/(3*a*tan(x/2)**6 + 9*a*tan(x/2)**4 + 9*a*tan(x/2)**2 + 3*a) - 4/(3*a*tan(x/2)**6 + 9*a*tan(x/2
)**4 + 9*a*tan(x/2)**2 + 3*a)

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Giac [A]
time = 0.42, size = 14, normalized size = 0.74 \begin {gather*} \frac {\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end {gather*}

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

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Mupad [B]
time = 2.04, size = 16, normalized size = 0.84 \begin {gather*} -\frac {3\,\cos \left (x\right )-{\cos \left (x\right )}^3}{3\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5/(a - a*cos(x)^2),x)

[Out]

-(3*cos(x) - cos(x)^3)/(3*a)

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