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

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

[Out]

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

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Rubi [A]  time = 0.0249527, antiderivative size = 8, normalized size of antiderivative = 1., 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 = {3175, 3770} \[ -\frac{\tanh ^{-1}(\cos (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a - a*Cos[x]^2),x]

[Out]

-(ArcTanh[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 3770

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*}

Mathematica [B]  time = 0.0067421, size = 21, normalized size = 2.62 \[ \frac{\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )}{a} \]

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 = 0.014, size = 9, normalized size = 1.1 \begin{align*} -{\frac{{\it Artanh} \left ( \cos \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(cos(x))/a

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Maxima [B]  time = 0.952486, size = 28, normalized size = 3.5 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{2 \, a} + \frac{\log \left (\cos \left (x\right ) - 1\right )}{2 \, a} \end{align*}

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]  time = 2.01169, size = 77, normalized size = 9.62 \begin{align*} -\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{align*}

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]  time = 0.264378, size = 19, normalized size = 2.38 \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2 a} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2 a} \end{align*}

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]  time = 1.16478, size = 31, normalized size = 3.88 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{2 \, a} + \frac{\log \left (-\cos \left (x\right ) + 1\right )}{2 \, a} \end{align*}

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