Integrand size = 27, antiderivative size = 180 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\frac {2 e E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 (d-a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{\sqrt {a+b \cos (x)+c \sin (x)}} \]
2*e*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arctan(b,c))*Ellipt icE(sin(1/2*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(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)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/ 2*arctan(b,c))*EllipticF(sin(1/2*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(x)+c*sin(x))/(a+(b^2+c^2)^(1/2))) ^(1/2)/(a+b*cos(x)+c*sin(x))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 5.97 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.17 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\frac {2 d \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {a+b \cos (x)+c \sin (x)}{a-\sqrt {1+\frac {b^2}{c^2}} c},\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {1+\frac {b^2}{c^2}} c}\right ) \sec \left (x+\arctan \left (\frac {b}{c}\right )\right ) \sqrt {\frac {\sqrt {1+\frac {b^2}{c^2}} c-b \cos (x)-c \sin (x)}{a+\sqrt {1+\frac {b^2}{c^2}} c}} \sqrt {a+b \cos (x)+c \sin (x)} \sqrt {\frac {b \cos (x)+c \left (\sqrt {1+\frac {b^2}{c^2}}+\sin (x)\right )}{-a+\sqrt {1+\frac {b^2}{c^2}} c}}}{\sqrt {1+\frac {b^2}{c^2}} c}-\frac {\left (b^2+c^2\right ) e \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \cos (x)+c \sin (x)}{a-b \sqrt {1+\frac {c^2}{b^2}}},\frac {a+b \cos (x)+c \sin (x)}{a+b \sqrt {1+\frac {c^2}{b^2}}}\right ) \sin \left (x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1+\frac {c^2}{b^2}}-b \cos (x)-c \sin (x)}{a+b \sqrt {1+\frac {c^2}{b^2}}}} \sqrt {a+b \cos (x)+c \sin (x)} \sqrt {\frac {b \sqrt {1+\frac {c^2}{b^2}}+b \cos (x)+c \sin (x)}{-a+b \sqrt {1+\frac {c^2}{b^2}}}}}+\frac {e \left (2 b^3 \sqrt {1+\frac {c^2}{b^2}} \cos (x)-2 b \left (b^2+c^2\right ) \cos \left (x-\arctan \left (\frac {c}{b}\right )\right )+2 b^2 c \sqrt {1+\frac {c^2}{b^2}} \sin (x)+b^2 c \sin \left (x-\arctan \left (\frac {c}{b}\right )\right )+c^3 \sin \left (x-\arctan \left (\frac {c}{b}\right )\right )\right )}{b c \sqrt {1+\frac {c^2}{b^2}} \sqrt {a+b \cos (x)+c \sin (x)}} \]
(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]])
Time = 0.73 (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 |
| \(\displaystyle \frac {2 (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}\) |
(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]]
3.6.59.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(220)=440\).
Time = 8.37 (sec) , antiderivative size = 766, normalized size of antiderivative = 4.26
| 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 {\left (-\sin \left (x -\arctan \left (-b , c\right )\right )+1\right ) \sqrt {b^{2}+c^{2}}}{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 \left (b^{2} e +c^{2} e \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 {\left (-\sin \left (x -\arctan \left (-b , c\right )\right )+1\right ) \sqrt {b^{2}+c^{2}}}{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 {\left (\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )+a \right ) \cos \left (x -\arctan \left (-b , c\right )\right )^{2}}}\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}}}}}\) | \(766\) |
| parts | \(\text {Expression too large to display}\) | \(1423\) |
| risch | \(\text {Expression too large to display}\) | \(1804\) |
(-(-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)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(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*(b^2*e+c^2*e)*(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)*((si n(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-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))+a)*cos(x-arctan(-b,c))^2)^(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*si n(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)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1352, normalized size of antiderivative = 7.51 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\text {Too large to display} \]
1/3*(-3*I*sqrt(2)*(b^2 + c^2)*sqrt(b + 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* sqrt(2)*(b^2 + c^2)*sqrt(b - 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))) + sqrt(2)*(-...
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\int \frac {b e \cos {\left (x \right )} + c e \sin {\left (x \right )} + d}{\sqrt {a + b \cos {\left (x \right )} + c \sin {\left (x \right )}}}\, dx \]
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\int { \frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{\sqrt {b \cos \left (x\right ) + c \sin \left (x\right ) + a}} \,d x } \]
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\int { \frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{\sqrt {b \cos \left (x\right ) + c \sin \left (x\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx=\int \frac {d+b\,e\,\cos \left (x\right )+c\,e\,\sin \left (x\right )}{\sqrt {a+b\,\cos \left (x\right )+c\,\sin \left (x\right )}} \,d x \]