\(\int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx\) [526]

Optimal result

Integrand size = 14, antiderivative size = 57 \[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}} \]

[Out]

2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*((a+b*
cos(d*x+c))/(a+b))^(1/2)/d/(a+b*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2742, 2740} \[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}} \]

[In]

Int[1/Sqrt[a + b*Cos[c + d*x]],x]

[Out]

(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{\sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}} \]

[In]

Integrate[1/Sqrt[a + b*Cos[c + d*x]],x]

[Out]

(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.92 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {2 \sqrt {-\frac {2 b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b}{a +b}}\, \operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}\bigg | \frac {\sqrt {2}\, \sqrt {b}}{\sqrt {a +b}}\right )}{d \sqrt {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}}\)	\(78\)

[In]

int(1/(a+cos(d*x+c)*b)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d/(2*b*cos(1/2*d*x+1/2*c)^2+a-b)^(1/2)*(-(2*b*sin(1/2*d*x+1/2*c)^2-a-b)/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x
+1/2*c,2^(1/2)/(a+b)^(1/2)*b^(1/2))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {-i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )}{b d} \]

[In]

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

[Out]

(-I*sqrt(2)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x
+ c) + 3*I*b*sin(d*x + c) + 2*a)/b) + I*sqrt(2)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8
*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b))/(b*d)

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \cos {\left (c + d x \right )}}}\, dx \]

[In]

integrate(1/(a+b*cos(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(a + b*cos(c + d*x)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(b*cos(d*x + c) + a), x)

Giac [F]

\[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(b*cos(d*x + c) + a), x)

Mupad [B] (verification not implemented)

Time = 14.56 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {2\,b}{a+b}\right )\,\sqrt {\frac {a+b\,\cos \left (c+d\,x\right )}{a+b}}}{d\,\sqrt {a+b\,\cos \left (c+d\,x\right )}} \]

[In]

int(1/(a + b*cos(c + d*x))^(1/2),x)

[Out]

(2*ellipticF(c/2 + (d*x)/2, (2*b)/(a + b))*((a + b*cos(c + d*x))/(a + b))^(1/2))/(d*(a + b*cos(c + d*x))^(1/2)
)