3.1.20 \(\int \frac {1}{a+b \cos ^2(x)} \, dx\) [20]
Optimal. Leaf size=30 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a+b}} \]
[Out]
-arctan(cot(x)*(a+b)^(1/2)/a^(1/2))/a^(1/2)/(a+b)^(1/2)
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3260, 211}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*Cos[x]^2)^(-1),x]
[Out]
-(ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]))
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 3260
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Rubi steps
\begin {align*} \int \frac {1}{a+b \cos ^2(x)} \, dx &=-\text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a+b}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 29, normalized size = 0.97 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Cos[x]^2)^(-1),x]
[Out]
ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])
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Maple [A]
time = 0.07, size = 21, normalized size = 0.70
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\arctan \left (\frac {a \tan \left (x \right )}{\sqrt {\left (a +b \right ) a}}\right )}{\sqrt {\left (a +b \right ) a}}\) |
\(21\) |
| risch |
\(-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-a b}-b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}}\) |
\(160\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*cos(x)^2),x,method=_RETURNVERBOSE)
[Out]
1/((a+b)*a)^(1/2)*arctan(a*tan(x)/((a+b)*a)^(1/2))
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Maxima [A]
time = 0.47, size = 20, normalized size = 0.67 \begin {gather*} \frac {\arctan \left (\frac {a \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(x)^2),x, algorithm="maxima")
[Out]
arctan(a*tan(x)/sqrt((a + b)*a))/sqrt((a + b)*a)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (22) = 44\).
time = 0.44, size = 163, normalized size = 5.43 \begin {gather*} \left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt {-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right )}{4 \, {\left (a^{2} + a b\right )}}, -\frac {\arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt {a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, \sqrt {a^{2} + a b}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(x)^2),x, algorithm="fricas")
[Out]
[-1/4*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 3*a*b)*cos(x)^2 + 4*((2*a + b)*cos(x)^
3 - a*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2)/(b^2*cos(x)^4 + 2*a*b*cos(x)^2 + a^2))/(a^2 + a*b), -1/2*arctan(1
/2*((2*a + b)*cos(x)^2 - a)/(sqrt(a^2 + a*b)*cos(x)*sin(x)))/sqrt(a^2 + a*b)]
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10924 vs.
\(2 (29) = 58\).
time = 19.41, size = 10924, normalized size = 364.13 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(x)**2),x)
[Out]
Piecewise((zoo*tan(x/2)/(tan(x/2)**2 - 1), Eq(a, 0) & Eq(b, 0)), (-2*tan(x/2)/(b*(tan(x/2)**2 - 1)), Eq(a, 0))
, (-tan(x/2)/(2*b) + 1/(2*b*tan(x/2)), Eq(a, -b)), (a**3*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*l
og(-sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + tan(x/2))/(2*a**4*sqrt(-a/(a + b) + b/(a + b) - 2*sq
rt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(-a/(a + b) + b/(a + b)
- 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a**3*sqrt(-a*b)*sqrt(-a/(a + b
) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(
-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*
sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a*
b**2*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)
/(a + b))) - a**3*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*log(sqrt(-a/(a + b) + b/(a + b) + 2*sqrt
(-a*b)/(a + b)) + tan(x/2))/(2*a**4*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a
+ b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b)
+ b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a**3*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sq
rt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a
+ b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*
b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a*b**2*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a +
b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b))) + a**3*sqrt(-a/(a + b) + b/(a
+ b) + 2*sqrt(-a*b)/(a + b))*log(-sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b)) + tan(x/2))/(2*a**4*sqr
t(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3
*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8
*a**3*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b
)/(a + b)) - 10*a**2*b**2*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*
sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b)
+ 2*sqrt(-a*b)/(a + b)) + 8*a*b**2*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a
+ b) + b/(a + b) + 2*sqrt(-a*b)/(a + b))) - a**3*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b))*log(sqrt(
-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b)) + tan(x/2))/(2*a**4*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/
(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(
-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a**3*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a
+ b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(-a/(a + b
) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(-a/(
a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a*b**2*sqrt
(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b))
) - 10*a**2*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*log(-sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a
*b)/(a + b)) + tan(x/2))/(2*a**4*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a +
b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b
/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a**3*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(
-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a +
b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/
(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a*b**2*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b)
- 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b))) + 10*a**2*b*sqrt(-a/(a + b) + b/
(a + b) - 2*sqrt(-a*b)/(a + b))*log(sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + tan(x/2))/(2*a**4*sq
rt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**
3*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) -
8*a**3*sqrt(-a*b)*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*
b)/(a + b)) - 10*a**2*b**2*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(-a/(a + b) + b/(a + b) + 2
*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(-a/(a + b)...
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Giac [A]
time = 0.42, size = 37, normalized size = 1.23 \begin {gather*} \frac {\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )}{\sqrt {a^{2} + a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(x)^2),x, algorithm="giac")
[Out]
(pi*floor(x/pi + 1/2)*sgn(a) + arctan(a*tan(x)/sqrt(a^2 + a*b)))/sqrt(a^2 + a*b)
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Mupad [B]
time = 2.38, size = 24, normalized size = 0.80 \begin {gather*} \frac {\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (x\right )}{\sqrt {a^2+b\,a}}\right )}{\sqrt {a^2+b\,a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b*cos(x)^2),x)
[Out]
atan((a*tan(x))/(a*b + a^2)^(1/2))/(a*b + a^2)^(1/2)
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