3.1.4 \(\int \frac {\sin ^3(x)}{a-a \cos ^2(x)} \, dx\) [4]

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

[Out]

-cos(x)/a

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Rubi [A]
time = 0.03, antiderivative size = 7, normalized size of antiderivative = 1.00, 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 = {3254, 2718} \begin {gather*} -\frac {\cos (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(a - a*Cos[x]^2),x]

[Out]

-(Cos[x]/a)

Rule 2718

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

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

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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cos (x)}{a} \end {gather*}

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 = 0.05, size = 8, normalized size = 1.14

method	result	size
derivativedivides	\(-\frac {\cos \left (x \right )}{a}\)	\(8\)
default	\(-\frac {\cos \left (x \right )}{a}\)	\(8\)
risch	\(-\frac {\cos \left (x \right )}{a}\)	\(8\)
norman	\(\frac {\frac {2 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {4 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3} \tan \left (\frac {x}{2}\right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)/a

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Maxima [A]
time = 0.26, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (x\right )}{a} \end {gather*}

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 = 0.38, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (x\right )}{a} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (5) = 10\).
time = 0.46, size = 12, normalized size = 1.71 \begin {gather*} - \frac {2}{a \tan ^{2}{\left (\frac {x}{2} \right )} + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*tan(x/2)**2 + a)

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Giac [A]
time = 0.41, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (x\right )}{a} \end {gather*}

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

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Mupad [B]
time = 0.02, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (x\right )}{a} \end {gather*}

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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