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

Optimal. Leaf size=11 \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b} \]

[Out]

arctan(tan(x)/b)/b

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, 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} \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[(b^2*Cos[x]^2 + Sin[x]^2)^(-1),x]

[Out]

ArcTan[Tan[x]/b]/b

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

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Mathematica [A]  time = 0.03, size = 11, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Tan[x]/b]/b

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fricas [B]  time = 2.04, size = 31, normalized size = 2.82 \[ -\frac {\arctan \left (\frac {{\left (b^{2} + 1\right )} \cos \relax (x)^{2} - 1}{2 \, b \cos \relax (x) \sin \relax (x)}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.13, size = 22, normalized size = 2.00 \[ \frac {\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \relax (x)}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 12, normalized size = 1.09 \[ \frac {\arctan \left (\frac {\tan \relax (x )}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(tan(x)/b)/b

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maxima [A]  time = 0.41, size = 11, normalized size = 1.00 \[ \frac {\arctan \left (\frac {\tan \relax (x)}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(tan(x)/b)/b

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mupad [B]  time = 2.83, size = 11, normalized size = 1.00 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\relax (x)}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^2 + b^2*cos(x)^2),x)

[Out]

atan(tan(x)/b)/b

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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