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

Optimal. Leaf size=7 \[ \frac{\tanh ^{-1}(\sin (x))}{a} \]

[Out]

ArcTanh[Sin[x]]/a

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Rubi [A]  time = 0.0260759, antiderivative size = 7, 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} \[ \frac{\tanh ^{-1}(\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(a - a*Sin[x]^2),x]

[Out]

ArcTanh[Sin[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{\cos (x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \sec (x) \, dx}{a}\\ &=\frac{\tanh ^{-1}(\sin (x))}{a}\\ \end{align*}

Mathematica [B]  time = 0.004368, size = 37, normalized size = 5.29 \[ \frac{\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(a - a*Sin[x]^2),x]

[Out]

(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/a

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Maple [A]  time = 0.028, size = 8, normalized size = 1.1 \begin{align*}{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(sin(x))/a

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Maxima [B]  time = 1.01844, size = 28, normalized size = 4. \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.92961, size = 59, normalized size = 8.43 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right ) - \log \left (-\sin \left (x\right ) + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.430734, size = 19, normalized size = 2.71 \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{2 a} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13484, size = 31, normalized size = 4.43 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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