3.4 \(\int \frac{\sin ^3(x)}{a-a \cos ^2(x)} \, dx\)

Optimal. Leaf size=7 \[ -\frac{\cos (x)}{a} \]

[Out]

-(Cos[x]/a)

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Rubi [A]  time = 0.0413498, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2638} \[ -\frac{\cos (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(a - a*Cos[x]^2),x]

[Out]

-(Cos[x]/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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0020658, size = 7, normalized size = 1. \[ -\frac{\cos (x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(a - a*Cos[x]^2),x]

[Out]

-(Cos[x]/a)

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Maple [A]  time = 0.015, size = 8, normalized size = 1.1 \begin{align*} -{\frac{\cos \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a-a*cos(x)^2),x)

[Out]

-cos(x)/a

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Maxima [A]  time = 0.945899, size = 9, normalized size = 1.29 \begin{align*} -\frac{\cos \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a-a*cos(x)^2),x, algorithm="maxima")

[Out]

-cos(x)/a

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Fricas [A]  time = 1.85319, size = 15, normalized size = 2.14 \begin{align*} -\frac{\cos \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a-a*cos(x)^2),x, algorithm="fricas")

[Out]

-cos(x)/a

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Sympy [B]  time = 2.00868, size = 36, normalized size = 5.14 \begin{align*} \frac{2 \tan ^{2}{\left (\frac{x}{2} \right )}}{3 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} - \frac{4}{3 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/(a-a*cos(x)**2),x)

[Out]

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

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Giac [A]  time = 1.12301, size = 9, normalized size = 1.29 \begin{align*} -\frac{\cos \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a-a*cos(x)^2),x, algorithm="giac")

[Out]

-cos(x)/a