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

Optimal. Leaf size=22 \[ \frac{\tanh ^{-1}(\sin (x))}{2 a^2}+\frac{\tan (x) \sec (x)}{2 a^2} \]

[Out]

ArcTanh[Sin[x]]/(2*a^2) + (Sec[x]*Tan[x])/(2*a^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0316159, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3175, 3768, 3770} \[ \frac{\tanh ^{-1}(\sin (x))}{2 a^2}+\frac{\tan (x) \sec (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(a - a*Sin[x]^2)^2,x]

[Out]

ArcTanh[Sin[x]]/(2*a^2) + (Sec[x]*Tan[x])/(2*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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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

Mathematica [B]  time = 0.0056328, size = 45, normalized size = 2.05 \[ \frac{\tan (x) \sec (x)-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.036, size = 44, normalized size = 2. \begin{align*} -{\frac{1}{4\,{a}^{2} \left ( -1+\sin \left ( x \right ) \right ) }}-{\frac{\ln \left ( -1+\sin \left ( x \right ) \right ) }{4\,{a}^{2}}}-{\frac{1}{4\,{a}^{2} \left ( 1+\sin \left ( x \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{4\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/a^2/(-1+sin(x))-1/4/a^2*ln(-1+sin(x))-1/4/a^2/(1+sin(x))+1/4/a^2*ln(1+sin(x))

________________________________________________________________________________________

Maxima [B]  time = 1.00151, size = 55, normalized size = 2.5 \begin{align*} -\frac{\sin \left (x\right )}{2 \,{\left (a^{2} \sin \left (x\right )^{2} - a^{2}\right )}} + \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, a^{2}} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sin(x)/(a^2*sin(x)^2 - a^2) + 1/4*log(sin(x) + 1)/a^2 - 1/4*log(sin(x) - 1)/a^2

________________________________________________________________________________________

Fricas [B]  time = 1.79149, size = 116, normalized size = 5.27 \begin{align*} \frac{\cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, \sin \left (x\right )}{4 \, a^{2} \cos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(cos(x)^2*log(sin(x) + 1) - cos(x)^2*log(-sin(x) + 1) + 2*sin(x))/(a^2*cos(x)^2)

________________________________________________________________________________________

Sympy [B]  time = 1.25807, size = 117, normalized size = 5.32 \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )} \sin ^{2}{\left (x \right )}}{4 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2}} + \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2}} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )} \sin ^{2}{\left (x \right )}}{4 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2}} - \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2}} - \frac{2 \sin{\left (x \right )}}{4 a^{2} \sin ^{2}{\left (x \right )} - 4 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sin(x) - 1)*sin(x)**2/(4*a**2*sin(x)**2 - 4*a**2) + log(sin(x) - 1)/(4*a**2*sin(x)**2 - 4*a**2) + log(sin
(x) + 1)*sin(x)**2/(4*a**2*sin(x)**2 - 4*a**2) - log(sin(x) + 1)/(4*a**2*sin(x)**2 - 4*a**2) - 2*sin(x)/(4*a**
2*sin(x)**2 - 4*a**2)

________________________________________________________________________________________

Giac [B]  time = 1.10988, size = 51, normalized size = 2.32 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, a^{2}} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{4 \, a^{2}} - \frac{\sin \left (x\right )}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(sin(x) + 1)/a^2 - 1/4*log(-sin(x) + 1)/a^2 - 1/2*sin(x)/((sin(x)^2 - 1)*a^2)