3.1.6 \(\int \frac {\sin (x)}{a-a \cos ^2(x)} \, dx\) [6]

Optimal. Leaf size=8 \[ -\frac {\tanh ^{-1}(\cos (x))}{a} \]

[Out]

-arctanh(cos(x))/a

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Rubi [A]
time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3254, 3855} \begin {gather*} -\frac {\tanh ^{-1}(\cos (x))}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a - a*Cos[x]^2),x]

[Out]

-(ArcTanh[Cos[x]]/a)

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]

Rule 3855

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

Rubi steps

\begin {align*} \int \frac {\sin (x)}{a-a \cos ^2(x)} \, dx &=\frac {\int \csc (x) \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(8)=16\).
time = 0.01, size = 21, normalized size = 2.62 \begin {gather*} \frac {-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a - a*Cos[x]^2),x]

[Out]

(-Log[Cos[x/2]] + Log[Sin[x/2]])/a

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Maple [A]
time = 0.06, size = 9, normalized size = 1.12

method	result	size
derivativedivides	\(-\frac {\arctanh \left (\cos \left (x \right )\right )}{a}\)	\(9\)
default	\(-\frac {\arctanh \left (\cos \left (x \right )\right )}{a}\)	\(9\)
norman	\(\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a}\)	\(10\)
risch	\(\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(cos(x))/a

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (8) = 16\).
time = 0.26, size = 21, normalized size = 2.62 \begin {gather*} -\frac {\log \left (\cos \left (x\right ) + 1\right )}{2 \, a} + \frac {\log \left (\cos \left (x\right ) - 1\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(cos(x) + 1)/a + 1/2*log(cos(x) - 1)/a

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).
time = 0.41, size = 22, normalized size = 2.75 \begin {gather*} -\frac {\log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(1/2*cos(x) + 1/2) - log(-1/2*cos(x) + 1/2))/a

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
time = 0.09, size = 19, normalized size = 2.38 \begin {gather*} \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{2 a} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{2 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(cos(x) - 1)/(2*a) - log(cos(x) + 1)/(2*a)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (8) = 16\).
time = 0.42, size = 23, normalized size = 2.88 \begin {gather*} -\frac {\log \left (\cos \left (x\right ) + 1\right )}{2 \, a} + \frac {\log \left (-\cos \left (x\right ) + 1\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(cos(x) + 1)/a + 1/2*log(-cos(x) + 1)/a

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Mupad [B]
time = 2.11, size = 8, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {atanh}\left (\cos \left (x\right )\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a - a*cos(x)^2),x)

[Out]

-atanh(cos(x))/a

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