3.413 \(\int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \, dx\)

Optimal. Leaf size=108 \[ \frac {2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3127, 2661} \[ \frac {2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]],x]

[Out]

(2*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] +
c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])

Rule 2661

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

Rule 3127

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \, dx &=\frac {\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}} \, dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\\ &=\frac {2 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.56, size = 285, normalized size = 2.64 \[ \frac {2 \sec \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right ) \sqrt {-\frac {c \sqrt {\frac {b^2}{c^2}+1} \left (\sin \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right )-1\right )}{a+c \sqrt {\frac {b^2}{c^2}+1}}} \sqrt {\frac {c \sqrt {\frac {b^2}{c^2}+1} \left (\sin \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right )+1\right )}{c \sqrt {\frac {b^2}{c^2}+1}-a}} \sqrt {a+c \sqrt {\frac {b^2}{c^2}+1} \sin \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right )} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {a+\sqrt {\frac {b^2}{c^2}+1} c \sin \left (d+e x+\tan ^{-1}\left (\frac {b}{c}\right )\right )}{a-\sqrt {\frac {b^2}{c^2}+1} c},\frac {a+\sqrt {\frac {b^2}{c^2}+1} c \sin \left (d+e x+\tan ^{-1}\left (\frac {b}{c}\right )\right )}{a+\sqrt {\frac {b^2}{c^2}+1} c}\right )}{c e \sqrt {\frac {b^2}{c^2}+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*AppellF1[1/2, 1/2, 1/2, 3/2, (a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(a - Sqrt[1 + b^2/c^2]*c)
, (a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(a + Sqrt[1 + b^2/c^2]*c)]*Sec[d + e*x + ArcTan[b/c]]*S
qrt[-((Sqrt[1 + b^2/c^2]*c*(-1 + Sin[d + e*x + ArcTan[b/c]]))/(a + Sqrt[1 + b^2/c^2]*c))]*Sqrt[(Sqrt[1 + b^2/c
^2]*c*(1 + Sin[d + e*x + ArcTan[b/c]]))/(-a + Sqrt[1 + b^2/c^2]*c)]*Sqrt[a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x +
ArcTan[b/c]]])/(Sqrt[1 + b^2/c^2]*c*e)

________________________________________________________________________________________

fricas [F]  time = 3.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(b*cos(e*x + d) + c*sin(e*x + d) + a), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.37, size = 303, normalized size = 2.81 \[ -\frac {2 \left (-a +\sqrt {b^{2}+c^{2}}\right ) \sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {-\frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right ) \sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\, \EllipticF \left (\sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a}{-a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {-a +\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right )}{\sqrt {b^{2}+c^{2}}\, \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a \sqrt {b^{2}+c^{2}}}{\sqrt {b^{2}+c^{2}}}}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-a+(b^2+c^2)^(1/2))*(-((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(-a+(b^2+c^2)^(1/2)))^(1/2)*(-(sin(e*x+d
-arctan(-b,c))-1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)/(-a+
(b^2+c^2)^(1/2)))^(1/2)*EllipticF((-((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(-a+(b^2+c^2)^(1/2)))^(1/2),(-
(-a+(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1/2))/(b^2+c^2)^(1/2)/cos(e*x+d-arctan(-b,c))/((b^2*sin(e*x+d-arcta
n(-b,c))+c^2*sin(e*x+d-arctan(-b,c))+a*(b^2+c^2)^(1/2))/(b^2+c^2)^(1/2))^(1/2)/e

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________