3.42 \(\int \frac{1}{(a+b \cos ^2(x))^2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac{b \sin (x) \cos (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )} \]

[Out]

-((2*a + b)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)) - (b*Cos[x]*Sin[x])/(2*a*(a + b)*(
a + b*Cos[x]^2))

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Rubi [A]  time = 0.0493253, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3184, 12, 3181, 205} \[ -\frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac{b \sin (x) \cos (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[x]^2)^(-2),x]

[Out]

-((2*a + b)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)) - (b*Cos[x]*Sin[x])/(2*a*(a + b)*(
a + b*Cos[x]^2))

Rule 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
e + f*x]^2)^(p + 1))/(2*a*f*(p + 1)*(a + b)), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^(p
+ 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ
[a + b, 0] && LtQ[p, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \cos ^2(x)\right )^2} \, dx &=-\frac{b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}-\frac{\int \frac{-2 a-b}{a+b \cos ^2(x)} \, dx}{2 a (a+b)}\\ &=-\frac{b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}+\frac{(2 a+b) \int \frac{1}{a+b \cos ^2(x)} \, dx}{2 a (a+b)}\\ &=-\frac{b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}-\frac{(2 a+b) \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{2 a (a+b)}\\ &=-\frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac{b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.221658, size = 70, normalized size = 1.08 \[ -\frac{(-2 a-b) \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac{b \sin (2 x)}{2 a (a+b) (2 a+b \cos (2 x)+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[x]^2)^(-2),x]

[Out]

-((-2*a - b)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)) - (b*Sin[2*x])/(2*a*(a + b)*(2*a
+ b + b*Cos[2*x]))

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Maple [A]  time = 0.02, size = 81, normalized size = 1.3 \begin{align*} -{\frac{b\tan \left ( x \right ) }{ \left ( 2\,a+2\,b \right ) a \left ( \left ( \tan \left ( x \right ) \right ) ^{2}a+a+b \right ) }}+{\frac{1}{a+b}\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}}+{\frac{b}{ \left ( 2\,a+2\,b \right ) a}\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cos(x)^2)^2,x)

[Out]

-1/2*b/(a+b)/a*tan(x)/(tan(x)^2*a+a+b)+1/(a+b)/((a+b)*a)^(1/2)*arctan(a*tan(x)/((a+b)*a)^(1/2))+1/2/(a+b)/a/((
a+b)*a)^(1/2)*arctan(a*tan(x)/((a+b)*a)^(1/2))*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(x)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.91031, size = 774, normalized size = 11.91 \begin{align*} \left [-\frac{4 \,{\left (a^{2} b + a b^{2}\right )} \cos \left (x\right ) \sin \left (x\right ) +{\left ({\left (2 \, a b + b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a^{2} + a b\right )} \sqrt{-a^{2} - a b} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt{-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right )}{8 \,{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} +{\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cos \left (x\right )^{2}\right )}}, -\frac{2 \,{\left (a^{2} b + a b^{2}\right )} \cos \left (x\right ) \sin \left (x\right ) +{\left ({\left (2 \, a b + b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a^{2} + a b\right )} \sqrt{a^{2} + a b} \arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt{a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right )}{4 \,{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} +{\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cos \left (x\right )^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(x)^2)^2,x, algorithm="fricas")

[Out]

[-1/8*(4*(a^2*b + a*b^2)*cos(x)*sin(x) + ((2*a*b + b^2)*cos(x)^2 + 2*a^2 + a*b)*sqrt(-a^2 - a*b)*log(((8*a^2 +
8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 3*a*b)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - a*cos(x))*sqrt(-a^2 - a*b)*sin(x
) + a^2)/(b^2*cos(x)^4 + 2*a*b*cos(x)^2 + a^2)))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*cos(
x)^2), -1/4*(2*(a^2*b + a*b^2)*cos(x)*sin(x) + ((2*a*b + b^2)*cos(x)^2 + 2*a^2 + a*b)*sqrt(a^2 + a*b)*arctan(1
/2*((2*a + b)*cos(x)^2 - a)/(sqrt(a^2 + a*b)*cos(x)*sin(x))))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 +
a^2*b^3)*cos(x)^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(x)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.16169, size = 93, normalized size = 1.43 \begin{align*} \frac{{\left (\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )\right )}{\left (2 \, a + b\right )}}{2 \,{\left (a^{2} + a b\right )}^{\frac{3}{2}}} - \frac{b \tan \left (x\right )}{2 \,{\left (a \tan \left (x\right )^{2} + a + b\right )}{\left (a^{2} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(x)^2)^2,x, algorithm="giac")

[Out]

1/2*(pi*floor(x/pi + 1/2)*sgn(a) + arctan(a*tan(x)/sqrt(a^2 + a*b)))*(2*a + b)/(a^2 + a*b)^(3/2) - 1/2*b*tan(x
)/((a*tan(x)^2 + a + b)*(a^2 + a*b))