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

Optimal. Leaf size=9 \[ \frac{\tan ^{-1}(a \tan (x))}{a} \]

[Out]

ArcTan[a*Tan[x]]/a

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Rubi [A]  time = 0.0182644, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {203} \[ \frac{\tan ^{-1}(a \tan (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^2 + a^2*Sin[x]^2)^(-1),x]

[Out]

ArcTan[a*Tan[x]]/a

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0339063, size = 9, normalized size = 1. \[ \frac{\tan ^{-1}(a \tan (x))}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[a*Tan[x]]/a

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Maple [A]  time = 0.038, size = 10, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( a\tan \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(a*tan(x))/a

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Maxima [A]  time = 1.51675, size = 12, normalized size = 1.33 \begin{align*} \frac{\arctan \left (a \tan \left (x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(a*tan(x))/a

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Fricas [B]  time = 1.83198, size = 88, normalized size = 9.78 \begin{align*} -\frac{\arctan \left (\frac{{\left (a^{2} + 1\right )} \cos \left (x\right )^{2} - a^{2}}{2 \, a \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 29.4254, size = 2011, normalized size = 223.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-16*a**5*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x
/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + 16*a**5*sqrt(-2*a**2 - 2*a*s
qrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) -
16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + 16*a**4*sqrt(a**2 - 1)*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(-sq
rt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**
2 - 1) + 2*a) - 16*a**4*sqrt(a**2 - 1)*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 - 2*a*sqrt(a**2
- 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + 20*a**3*sq
rt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**
4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) - 20*a**3*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log
(sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(
a**2 - 1) + 2*a) - 4*a**3*sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1) +
tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + 4*a**3*sqrt(-2*a**2 +
2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 -
1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) - 12*a**2*sqrt(a**2 - 1)*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*lo
g(-sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqr
t(a**2 - 1) + 2*a) + 12*a**2*sqrt(a**2 - 1)*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 - 2*a*sqrt
(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + 4*a**
2*sqrt(a**2 - 1)*sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1) + tan(x/2)
)/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) - 4*a**2*sqrt(a**2 - 1)*sqrt(-2*a
**2 + 2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a
**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) - 5*a*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a*
*2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) +
2*a) + 5*a*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*
a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + 3*a*sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1)
+ 1)*log(-sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*
a**2*sqrt(a**2 - 1) + 2*a) - 3*a*sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1)
+ 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + sqrt(a**2 - 1)*s
qrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a*
*4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) - sqrt(a**2 - 1)*sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1)
+ 1)*log(sqrt(-2*a**2 - 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a*
*2*sqrt(a**2 - 1) + 2*a) - sqrt(a**2 - 1)*sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1)*log(-sqrt(-2*a**2 + 2*a*sqrt(
a**2 - 1) + 1) + tan(x/2))/(16*a**5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a) + sqrt(a
**2 - 1)*sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1)*log(sqrt(-2*a**2 + 2*a*sqrt(a**2 - 1) + 1) + tan(x/2))/(16*a**
5 - 16*a**4*sqrt(a**2 - 1) - 16*a**3 + 8*a**2*sqrt(a**2 - 1) + 2*a), Ne(a, 0)), (-2*tan(x/2)/(tan(x/2)**2 - 1)
, True))

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Giac [B]  time = 1.12955, size = 27, normalized size = 3. \begin{align*} \frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (a \tan \left (x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(pi*floor(x/pi + 1/2) + arctan(a*tan(x)))/a