3.1.31 \(\int \frac {\cos (x)}{a+b \cos ^2(x)} \, dx\) [31]
Optimal. Leaf size=29 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b}} \]
[Out]
arctanh(sin(x)*b^(1/2)/(a+b)^(1/2))/b^(1/2)/(a+b)^(1/2)
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3265, 214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cos[x]/(a + b*Cos[x]^2),x]
[Out]
ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 3265
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \frac {\cos (x)}{a+b \cos ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Cos[x]/(a + b*Cos[x]^2),x]
[Out]
ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b])
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Maple [A]
time = 0.07, size = 21, normalized size = 0.72
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\arctanh \left (\frac {b \sin \left (x \right )}{\sqrt {\left (a +b \right ) b}}\right )}{\sqrt {\left (a +b \right ) b}}\) |
\(21\) |
| risch |
\(\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i x}}{\sqrt {a b +b^{2}}}-1\right )}{2 \sqrt {a b +b^{2}}}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i x}}{\sqrt {a b +b^{2}}}-1\right )}{2 \sqrt {a b +b^{2}}}\) |
\(80\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(x)/(a+b*cos(x)^2),x,method=_RETURNVERBOSE)
[Out]
1/((a+b)*b)^(1/2)*arctanh(b*sin(x)/((a+b)*b)^(1/2))
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Maxima [A]
time = 0.48, size = 39, normalized size = 1.34 \begin {gather*} -\frac {\log \left (\frac {b \sin \left (x\right ) - \sqrt {{\left (a + b\right )} b}}{b \sin \left (x\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)/(a+b*cos(x)^2),x, algorithm="maxima")
[Out]
-1/2*log((b*sin(x) - sqrt((a + b)*b))/(b*sin(x) + sqrt((a + b)*b)))/sqrt((a + b)*b)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (21) = 42\).
time = 0.43, size = 95, normalized size = 3.28 \begin {gather*} \left [\frac {\log \left (-\frac {b \cos \left (x\right )^{2} - 2 \, \sqrt {a b + b^{2}} \sin \left (x\right ) - a - 2 \, b}{b \cos \left (x\right )^{2} + a}\right )}{2 \, \sqrt {a b + b^{2}}}, -\frac {\sqrt {-a b - b^{2}} \arctan \left (\frac {\sqrt {-a b - b^{2}} \sin \left (x\right )}{a + b}\right )}{a b + b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)/(a+b*cos(x)^2),x, algorithm="fricas")
[Out]
[1/2*log(-(b*cos(x)^2 - 2*sqrt(a*b + b^2)*sin(x) - a - 2*b)/(b*cos(x)^2 + a))/sqrt(a*b + b^2), -sqrt(-a*b - b^
2)*arctan(sqrt(-a*b - b^2)*sin(x)/(a + b))/(a*b + b^2)]
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 55508 vs.
\(2 (27) = 54\).
time = 69.00, size = 55508, normalized size = 1914.07 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)/(a+b*cos(x)**2),x)
[Out]
Piecewise((zoo*(-log(tan(x/2) - 1) + log(tan(x/2) + 1)), Eq(a, 0) & Eq(b, 0)), (sin(x)/a, Eq(b, 0)), (tan(x/2)
/(2*b) + 1/(2*b*tan(x/2)), Eq(a, -b)), (-13*a**6*b*sqrt(-a/(a + b) + b/(a + b) - 2*sqrt(-a*b)/(a + b))*log(-sq
rt(-a/(a + b) + b/(a + b) + 2*sqrt(-a*b)/(a + b)) + tan(x/2))/(2*a**7*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)) - 130*a**6*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)) - 24*a**6*b*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)) + 858*a**5*b**3*s
qrt(-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)) + 416*a
**5*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)) - 858*a**4*b**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)) - 1144*a**4*b**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)) - 858*a**3*b**5*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)) + 858*a**2*b**6*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)) + 1144*a**2*b**5*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)) - 130*a*b**7*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)) - 416*a*b**6*sq
rt(-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
)) + 2*b**8*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)) + 24*b**7*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))) + 13*a**6*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**7*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)) - 130*a**6*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)) - 24*a**6*b*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)) + 858*a**5*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)) + 416*a**5*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)) - 85
8*a**4*b**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)) - 1144*a**4*b**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)) - 858*a**3*b**5*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)) + 858*a**2*b**6*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)) + 1144*a**2*b**5*sqrt(-a*b)*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)) - 130*a*b**7*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)) - 416*a*b**6*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)) + 2*b**8*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)) + 24*b**7*sq
rt(-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
))) + 11*a**6*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**7*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)) - 130*a**6*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)) - 24*a**6*b*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)) + 858*a**5*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)) + 416*a**5*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)) - 858*a**4*b**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)) - 1144*a**4
*b**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)) - 858*a**3*b**5*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)) + 858*a**2*b**6*sqrt(-a/(a...
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Giac [A]
time = 0.40, size = 31, normalized size = 1.07 \begin {gather*} -\frac {\arctan \left (\frac {b \sin \left (x\right )}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(x)/(a+b*cos(x)^2),x, algorithm="giac")
[Out]
-arctan(b*sin(x)/sqrt(-a*b - b^2))/sqrt(-a*b - b^2)
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Mupad [B]
time = 0.09, size = 21, normalized size = 0.72 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sin \left (x\right )}{\sqrt {a+b}}\right )}{\sqrt {b}\,\sqrt {a+b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(x)/(a + b*cos(x)^2),x)
[Out]
atanh((b^(1/2)*sin(x))/(a + b)^(1/2))/(b^(1/2)*(a + b)^(1/2))
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