3.276 \(\int \frac{\cos ^3(x)}{(a-a \sin ^2(x))^2} \, dx\)

Optimal. Leaf size=7 \[ \frac{\tanh ^{-1}(\sin (x))}{a^2} \]

[Out]

ArcTanh[Sin[x]]/a^2

________________________________________________________________________________________

Rubi [A]  time = 0.0393184, antiderivative size = 7, normalized size of antiderivative = 1., 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 = {3175, 3770} \[ \frac{\tanh ^{-1}(\sin (x))}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3/(a - a*Sin[x]^2)^2,x]

[Out]

ArcTanh[Sin[x]]/a^2

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 ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \sec (x) \, dx}{a^2}\\ &=\frac{\tanh ^{-1}(\sin (x))}{a^2}\\ \end{align*}

Mathematica [B]  time = 0.0035549, size = 37, normalized size = 5.29 \[ \frac{\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3/(a - a*Sin[x]^2)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 8, normalized size = 1.1 \begin{align*}{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(sin(x))/a^2

________________________________________________________________________________________

Maxima [B]  time = 0.975028, size = 28, normalized size = 4. \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a^{2}} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.92245, size = 62, normalized size = 8.86 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right ) - \log \left (-\sin \left (x\right ) + 1\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 11.15, size = 22, normalized size = 3.14 \begin{align*} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - 1 \right )}}{a^{2}} + \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.14169, size = 31, normalized size = 4.43 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a^{2}} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

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