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

Optimal. Leaf size=15 \[ \frac{\tan ^{-1}\left (\frac{a \tan (x)}{b}\right )}{a b} \]

[Out]

ArcTan[(a*Tan[x])/b]/(a*b)

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Rubi [A]  time = 0.0427766, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\tan ^{-1}\left (\frac{a \tan (x)}{b}\right )}{a b} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(a*Tan[x])/b]/(a*b)

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Rubi in Sympy [A]  time = 26.9054, size = 10, normalized size = 0.67 \[ \frac{\operatorname{atan}{\left (\frac{a \tan{\left (x \right )}}{b} \right )}}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**2*cos(x)**2+a**2*sin(x)**2),x)

[Out]

atan(a*tan(x)/b)/(a*b)

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Mathematica [A]  time = 0.0517988, size = 15, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{a \tan (x)}{b}\right )}{a b} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(a*Tan[x])/b]/(a*b)

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Maple [A]  time = 0.003, size = 16, normalized size = 1.1 \[{\frac{1}{ab}\arctan \left ({\frac{a\tan \left ( x \right ) }{b}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^2*cos(x)^2+a^2*sin(x)^2),x)

[Out]

arctan(a*tan(x)/b)/a/b

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Maxima [A]  time = 1.5373, size = 20, normalized size = 1.33 \[ \frac{\arctan \left (\frac{a \tan \left (x\right )}{b}\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(a*tan(x)/b)/(a*b)

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Fricas [A]  time = 0.233105, size = 58, normalized size = 3.87 \[ -\frac{\arctan \left (\frac{{\left (a^{2} + b^{2}\right )} \cos \left (x\right )^{2} - a^{2}}{2 \, a b \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**2*cos(x)**2+a**2*sin(x)**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.204, size = 35, normalized size = 2.33 \[ \frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{a \tan \left (x\right )}{b}\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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