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

Optimal. Leaf size=5 \[ \frac {x}{a} \]

[Out]

x/a

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Rubi [A]
time = 0.03, antiderivative size = 5, 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, 8} \begin {gather*} \frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a - a*Cos[x]^2),x]

[Out]

x/a

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 ^2(x)}{a-a \cos ^2(x)} \, dx &=\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/a

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.06, size = 8, normalized size = 1.60

method	result	size
risch	\(\frac {x}{a}\)	\(6\)
default	\(\frac {\arctan \left (\tan \left (x \right )\right )}{a}\)	\(8\)
norman	\(\frac {\frac {x \tan \left (\frac {x}{2}\right )}{a}+\frac {x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2} \tan \left (\frac {x}{2}\right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*arctan(tan(x))

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Maxima [A]
time = 0.48, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Fricas [A]
time = 0.38, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Sympy [A]
time = 0.26, size = 2, normalized size = 0.40 \begin {gather*} \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Giac [A]
time = 0.42, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Mupad [B]
time = 2.05, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a - a*cos(x)^2),x)

[Out]

x/a

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