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 = 0.0197121, antiderivative size = 11, 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} \[ \frac{\tan ^{-1}\left (\frac{\tan (x)}{b}\right )}{b} \]

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*}

Mathematica [A]  time = 0.0318574, size = 11, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\tan (x)}{b}\right )}{b} \]

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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Maple [A]  time = 0.033, size = 12, normalized size = 1.1 \begin{align*}{\frac{1}{b}\arctan \left ({\frac{\tan \left ( x \right ) }{b}} \right ) } \end{align*}

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 = 1.4628, size = 15, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\frac{\tan \left (x\right )}{b}\right )}{b} \end{align*}

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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Fricas [B]  time = 1.91518, size = 85, normalized size = 7.73 \begin{align*} -\frac{\arctan \left (\frac{{\left (b^{2} + 1\right )} \cos \left (x\right )^{2} - 1}{2 \, b \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, b} \end{align*}

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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Sympy [A]  time = 32.779, size = 2118, normalized size = 192.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**4*sqrt(1 - b**2)*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 - 2*sqrt(1 - b**2)/b**2 -
2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - b**4*sqrt(
1 - b**2)*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2
*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - b**4*sqrt(1 - b**2)*sqrt(1 + 2*s
qrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1
- b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + b**4*sqrt(1 - b**2)*sqrt(1 + 2*sqrt(1 - b**2)/b**2 -
2/b**2)*log(sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 1
6*b**2*sqrt(1 - b**2) + 16*b**2) - 5*b**4*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 - 2*sqrt(1 - b*
*2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) +
5*b**4*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b
**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + 3*b**4*sqrt(1 + 2*sqrt(1 - b**2)/b
**2 - 2/b**2)*log(-sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b
**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - 3*b**4*sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 + 2*sqrt(
1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b
**2) - 12*b**2*sqrt(1 - b**2)*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2
/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + 12*b**2*sqr
t(1 - b**2)*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/
(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + 4*b**2*sqrt(1 - b**2)*sqrt(1 +
2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sq
rt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - 4*b**2*sqrt(1 - b**2)*sqrt(1 + 2*sqrt(1 - b**2)/b
**2 - 2/b**2)*log(sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b*
*4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + 20*b**2*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 - 2*sqrt
(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*
b**2) - 20*b**2*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/
2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - 4*b**2*sqrt(1 + 2*sqrt(1 -
b**2)/b**2 - 2/b**2)*log(-sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2
) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + 4*b**2*sqrt(1 + 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 +
2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2
) + 16*b**2) + 16*sqrt(1 - b**2)*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(-sqrt(1 - 2*sqrt(1 - b**2)/b**2
- 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - 16*sqrt(
1 - b**2)*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2
*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) - 16*sqrt(1 - 2*sqrt(1 - b**2)/b**
2 - 2/b**2)*log(-sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**
4 - 16*b**2*sqrt(1 - b**2) + 16*b**2) + 16*sqrt(1 - 2*sqrt(1 - b**2)/b**2 - 2/b**2)*log(sqrt(1 - 2*sqrt(1 - b*
*2)/b**2 - 2/b**2) + tan(x/2))/(2*b**6 + 8*b**4*sqrt(1 - b**2) - 16*b**4 - 16*b**2*sqrt(1 - b**2) + 16*b**2),
Ne(b, 0)), (tan(x/2)/2 - 1/(2*tan(x/2)), True))

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Giac [A]  time = 1.11093, size = 30, normalized size = 2.73 \begin{align*} \frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{\tan \left (x\right )}{b}\right )}{b} \end{align*}

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