\(\int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx\) [412]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3198, 2732} \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\frac {2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \]

[In]

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

[Out]

(2*EllipticE[(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]])/(e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/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 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{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*} \text {integral}& = \frac {\sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \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 {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \\ & = \frac {2 E\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 {a+b \cos (d+e x)+c \sin (d+e x)}}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.31 (sec) , antiderivative size = 1408, normalized size of antiderivative = 13.04 \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\frac {2 b \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{c e}+\frac {2 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {a+\sqrt {1+\frac {b^2}{c^2}} c \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{\sqrt {1+\frac {b^2}{c^2}} \left (1-\frac {a}{\sqrt {1+\frac {b^2}{c^2}} c}\right ) c},-\frac {a+\sqrt {1+\frac {b^2}{c^2}} c \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{\sqrt {1+\frac {b^2}{c^2}} \left (-1-\frac {a}{\sqrt {1+\frac {b^2}{c^2}} c}\right ) c}\right ) \sec \left (d+e x+\arctan \left (\frac {b}{c}\right )\right ) \sqrt {\frac {c \sqrt {\frac {b^2+c^2}{c^2}}-c \sqrt {\frac {b^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{a+c \sqrt {\frac {b^2+c^2}{c^2}}}} \sqrt {a+c \sqrt {\frac {b^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {b^2+c^2}{c^2}}+c \sqrt {\frac {b^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{-a+c \sqrt {\frac {b^2+c^2}{c^2}}}}}{\sqrt {1+\frac {b^2}{c^2}} c e}+\frac {b^2 \left (-\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (-1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )}\right ) \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}-b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2+c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2+c^2}{b^2}}}}}-\frac {\frac {2 b \left (a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )\right )}{b^2+c^2}-\frac {c \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}}\right )}{c e}+\frac {c \left (-\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (-1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )}\right ) \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}-b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2+c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2+c^2}{b^2}}}}}-\frac {\frac {2 b \left (a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )\right )}{b^2+c^2}-\frac {c \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}}\right )}{e} \]

[In]

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

[Out]

(2*b*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(c*e) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/
c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c
^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[
b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c
^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt
[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (b^2*
(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b
^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^
2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)
/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^
2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x
- ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])
)/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x
- ArcTan[c/b]]]))/(c*e) + (c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - A
rcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - Ar
cTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2
/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c
^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt
[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^
2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b
*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/e

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(690\) vs. \(2(137)=274\).

Time = 5.31 (sec) , antiderivative size = 691, normalized size of antiderivative = 6.40

method	result	size
default	\(\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 (\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 {\sqrt {b^{2}+c^{2}}\, \cos \left (e x +d -\arctan \left (-b , c\right )\right )^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+\cos \left (e x +d -\arctan \left (-b , c\right )\right )^{2} a}\, \left (\sqrt {b^{2}+c^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{a +\sqrt {b^{2}+c^{2}}}+\frac {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}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{a +\sqrt {b^{2}+c^{2}}}+\frac {a}{a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {a +\sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\right )-\operatorname {EllipticE}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{a +\sqrt {b^{2}+c^{2}}}+\frac {a}{a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {a +\sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\right ) a +\operatorname {EllipticF}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{a +\sqrt {b^{2}+c^{2}}}+\frac {a}{a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {a +\sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\right ) a \right )}{\sqrt {b^{2}+c^{2}}\, \sqrt {\left (\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a \right ) \cos \left (e x +d -\arctan \left (-b , c\right )\right )^{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}\)	\(691\)
risch	\(\text {Expression too large to display}\)	\(2150\)

[In]

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

[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-arct
an(-b,c))+1)*(b^2+c^2)^(1/2)/(-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)/(b^2+c^2)^(1/2)*((b^2+c^2)^(1/2)*cos(e*x+d-arctan(-b,c))^2*sin(e*x+d-arctan(-b,c))+cos(e*x
+d-arctan(-b,c))^2*a)^(1/2)*((b^2+c^2)^(1/2)*EllipticE((1/(a+(b^2+c^2)^(1/2))*(b^2+c^2)^(1/2)*sin(e*x+d-arctan
(-b,c))+1/(a+(b^2+c^2)^(1/2))*a)^(1/2),(-(a+(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-(b^2+c^2)^(1/2)*Elli
pticF((1/(a+(b^2+c^2)^(1/2))*(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+1/(a+(b^2+c^2)^(1/2))*a)^(1/2),(-(a+(b^2+
c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticE((1/(a+(b^2+c^2)^(1/2))*(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,
c))+1/(a+(b^2+c^2)^(1/2))*a)^(1/2),(-(a+(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))*a+EllipticF((1/(a+(b^2+c
^2)^(1/2))*(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+1/(a+(b^2+c^2)^(1/2))*a)^(1/2),(-(a+(b^2+c^2)^(1/2))/(-a+(b
^2+c^2)^(1/2)))^(1/2))*a)/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)/cos(e*
x+d-arctan(-b,c))/((b^2*sin(e*x+d-arctan(-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

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 1371, normalized size of antiderivative = 12.69 \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*(sqrt(2)*(I*a*b + a*c)*sqrt(b + I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 +
3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5
+ 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^
2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2))
+ sqrt(2)*(-I*a*b + a*c)*sqrt(b - I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3
*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 -
2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2
*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(-I*b^2 - I*c^2)*sin(e*x + d))/(b^2 + c^2))
- 3*sqrt(2)*(I*b^2 + I*c^2)*sqrt(b + I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*
c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 +
2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*
c^4 + c^6), weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)
*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 -
6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*
I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2))) - 3*sqrt(2)*(-I*b^2 - I*c^2
)*sqrt(b - I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)
*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 -
6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrassPInv
erse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4)
, -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2
+ 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(e
*x + d) - 3*(-I*b^2 - I*c^2)*sin(e*x + d))/(b^2 + c^2))))/((b^2 + c^2)*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\int \sqrt {a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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