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

Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}} \]

[Out]

ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]/(Sqrt[a + b]*Sqrt[a + c])

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Rubi [A]  time = 0.0503087, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[x]^2 + c*Sin[x]^2)^(-1),x]

[Out]

ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]/(Sqrt[a + b]*Sqrt[a + c])

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

Mathematica [A]  time = 0.0625933, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[x]^2 + c*Sin[x]^2)^(-1),x]

[Out]

ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]/(Sqrt[a + b]*Sqrt[a + c])

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Maple [A]  time = 0.032, size = 27, normalized size = 0.8 \begin{align*}{\arctan \left ({ \left ( a+c \right ) \tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a+c \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+c*sin(x)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.14726, size = 660, normalized size = 20. \begin{align*} \left [-\frac{\sqrt{-a^{2} - a b -{\left (a + b\right )} c} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2} + 2 \,{\left (4 \, a + 3 \, b\right )} c + c^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b +{\left (5 \, a + 3 \, b\right )} c + c^{2}\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b + c\right )} \cos \left (x\right )^{3} -{\left (a + c\right )} \cos \left (x\right )\right )} \sqrt{-a^{2} - a b -{\left (a + b\right )} c} \sin \left (x\right ) + a^{2} + 2 \, a c + c^{2}}{{\left (b^{2} - 2 \, b c + c^{2}\right )} \cos \left (x\right )^{4} + 2 \,{\left (a b -{\left (a - b\right )} c - c^{2}\right )} \cos \left (x\right )^{2} + a^{2} + 2 \, a c + c^{2}}\right )}{4 \,{\left (a^{2} + a b +{\left (a + b\right )} c\right )}}, -\frac{\arctan \left (\frac{{\left (2 \, a + b + c\right )} \cos \left (x\right )^{2} - a - c}{2 \, \sqrt{a^{2} + a b +{\left (a + b\right )} c} \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, \sqrt{a^{2} + a b +{\left (a + b\right )} c}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \cos ^{2}{\left (x \right )} + c \sin ^{2}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a + b*cos(x)**2 + c*sin(x)**2), x)

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Giac [B]  time = 1.14046, size = 82, normalized size = 2.48 \begin{align*} \frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, c\right ) + \arctan \left (\frac{a \tan \left (x\right ) + c \tan \left (x\right )}{\sqrt{a^{2} + a b + a c + b c}}\right )}{\sqrt{a^{2} + a b + a c + b c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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