\(\int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx\) [502]

Optimal result

Integrand size = 27, antiderivative size = 180 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\frac {2 e E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 (d-a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{\sqrt {a+b \cos (x)+c \sin (x)}} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 5.47 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.17 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\frac {2 d \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {a+b \cos (x)+c \sin (x)}{a-\sqrt {1+\frac {b^2}{c^2}} c},\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {1+\frac {b^2}{c^2}} c}\right ) \sec \left (x+\arctan \left (\frac {b}{c}\right )\right ) \sqrt {\frac {\sqrt {1+\frac {b^2}{c^2}} c-b \cos (x)-c \sin (x)}{a+\sqrt {1+\frac {b^2}{c^2}} c}} \sqrt {a+b \cos (x)+c \sin (x)} \sqrt {\frac {b \cos (x)+c \left (\sqrt {1+\frac {b^2}{c^2}}+\sin (x)\right )}{-a+\sqrt {1+\frac {b^2}{c^2}} c}}}{\sqrt {1+\frac {b^2}{c^2}} c}-\frac {\left (b^2+c^2\right ) e \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \cos (x)+c \sin (x)}{a-b \sqrt {1+\frac {c^2}{b^2}}},\frac {a+b \cos (x)+c \sin (x)}{a+b \sqrt {1+\frac {c^2}{b^2}}}\right ) \sin \left (x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1+\frac {c^2}{b^2}}-b \cos (x)-c \sin (x)}{a+b \sqrt {1+\frac {c^2}{b^2}}}} \sqrt {a+b \cos (x)+c \sin (x)} \sqrt {\frac {b \sqrt {1+\frac {c^2}{b^2}}+b \cos (x)+c \sin (x)}{-a+b \sqrt {1+\frac {c^2}{b^2}}}}}+\frac {e \left (2 b^3 \sqrt {1+\frac {c^2}{b^2}} \cos (x)-2 b \left (b^2+c^2\right ) \cos \left (x-\arctan \left (\frac {c}{b}\right )\right )+2 b^2 c \sqrt {1+\frac {c^2}{b^2}} \sin (x)+b^2 c \sin \left (x-\arctan \left (\frac {c}{b}\right )\right )+c^3 \sin \left (x-\arctan \left (\frac {c}{b}\right )\right )\right )}{b c \sqrt {1+\frac {c^2}{b^2}} \sqrt {a+b \cos (x)+c \sin (x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3628

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3598

\(\displaystyle \frac {e \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+(d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+(d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx\)

\(\Big \downarrow \) 3132

\(\displaystyle (d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {(d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3598

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[Sqrt[a + b*Cos[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 + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[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]
 

rule 3606

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( 
x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq 
rt[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]/(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]
 

rule 3628

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] 
, x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] 
, x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ 
A*b - a*B, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(760\) vs. \(2(167)=334\).

Time = 1.44 (sec) , antiderivative size = 761, normalized size of antiderivative = 4.23

method	result	size
default	\(\frac {\sqrt {-\frac {\left (-b^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-c^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-a \sqrt {b^{2}+c^{2}}\right ) \cos \left (x -\arctan \left (-b , c\right )\right )^{2}}{\sqrt {b^{2}+c^{2}}}}\, \left (\frac {2 d \sqrt {b^{2}+c^{2}}\, \left (\frac {a}{\sqrt {b^{2}+c^{2}}}+1\right ) \sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )+a}{a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (\sin \left (x -\arctan \left (-b , c\right )\right )+1\right ) \sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \left (\sin \left (x -\arctan \left (-b , c\right )\right )-1\right )}{a +\sqrt {b^{2}+c^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (x -\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 {-\frac {\left (-b^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-c^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-a \sqrt {b^{2}+c^{2}}\right ) \cos \left (x -\arctan \left (-b , c\right )\right )^{2}}{\sqrt {b^{2}+c^{2}}}}}+\frac {2 e \left (b^{2}+c^{2}\right ) \left (\frac {a}{\sqrt {b^{2}+c^{2}}}+1\right ) \sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )+a}{a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (\sin \left (x -\arctan \left (-b , c\right )\right )+1\right ) \sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \left (\sin \left (x -\arctan \left (-b , c\right )\right )-1\right )}{a +\sqrt {b^{2}+c^{2}}}}\, \left (\left (-\frac {a}{\sqrt {b^{2}+c^{2}}}+1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (x -\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 )-\operatorname {EllipticF}\left (\sqrt {\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (x -\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 )\right )}{\sqrt {\cos \left (x -\arctan \left (-b , c\right )\right )^{2} \left (\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )+a \right )}}\right )}{\sqrt {b^{2}+c^{2}}\, \cos \left (x -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (x -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (x -\arctan \left (-b , c\right )\right )+a \sqrt {b^{2}+c^{2}}}{\sqrt {b^{2}+c^{2}}}}}\)	\(761\)
parts	\(\text {Expression too large to display}\)	\(1423\)
risch	\(\text {Expression too large to display}\)	\(1804\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 1348, normalized size of antiderivative = 7.49 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*(3*I*(b^2 + c^2)*sqrt(1/2*b + 1/2*I*c)*e*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(x) - 3*(I*b^2 + I*c^2)*sin(x))/(b^2 + c^2))) - 3*I* 
(b^2 + c^2)*sqrt(1/2*b - 1/2*I*c)*e*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(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2))) + (2*I*a*b*e...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\frac {-2 \sqrt {\cos \left (x \right ) b +\sin \left (x \right ) c +a}\, c e +\left (\int \frac {\sqrt {\cos \left (x \right ) b +\sin \left (x \right ) c +a}}{\cos \left (x \right ) b +\sin \left (x \right ) c +a}d x \right ) b d +\left (\int \frac {\sqrt {\cos \left (x \right ) b +\sin \left (x \right ) c +a}\, \cos \left (x \right )}{\cos \left (x \right ) b +\sin \left (x \right ) c +a}d x \right ) b^{2} e +\left (\int \frac {\sqrt {\cos \left (x \right ) b +\sin \left (x \right ) c +a}\, \cos \left (x \right )}{\cos \left (x \right ) b +\sin \left (x \right ) c +a}d x \right ) c^{2} e}{b} \] Input:

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

Output:

( - 2*sqrt(cos(x)*b + sin(x)*c + a)*c*e + int(sqrt(cos(x)*b + sin(x)*c + a 
)/(cos(x)*b + sin(x)*c + a),x)*b*d + int((sqrt(cos(x)*b + sin(x)*c + a)*co 
s(x))/(cos(x)*b + sin(x)*c + a),x)*b**2*e + int((sqrt(cos(x)*b + sin(x)*c 
+ a)*cos(x))/(cos(x)*b + sin(x)*c + a),x)*c**2*e)/b