Integrand size = 27, antiderivative size = 250 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\frac {2 (d-a e) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{\left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 e \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}} \]
2*(c*(-a*e+d)*cos(x)-b*(-a*e+d)*sin(x))/(a^2-b^2-c^2)/(a+b*cos(x)+c*sin(x) )^(1/2)+2*(-a*e+d)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arct an(b,c))*EllipticE(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^2-b^2-c^2)/((a+b*c os(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)+2*e*(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 = 6.45 (sec) , antiderivative size = 3176, normalized size of antiderivative = 12.70 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\text {Result too large to show} \]
Sqrt[a + b*Cos[x] + c*Sin[x]]*((2*(b^2 + c^2)*(-d + a*e))/(b*c*(-a^2 + b^2 + c^2)) - (2*(-(a*c*d) + a^2*c*e - b^2*d*Sin[x] - c^2*d*Sin[x] + a*b^2*e* Sin[x] + a*c^2*e*Sin[x]))/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x])) ) - (2*a*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[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[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sq rt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2 ] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/ c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt [(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c* Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)) + (2*b^2 *e*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[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[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b ^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sq rt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*S qrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^ 2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)) + (2*c*e*AppellF 1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/...
Time = 1.12 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3635, 27, 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.
| \(\displaystyle \int \frac {b e \cos (x)+c e \sin (x)+d}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {b e \cos (x)+c e \sin (x)+d}{(a+b \cos (x)+c \sin (x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2 \int -\frac {a d-\left (b^2+c^2\right ) e+b (d-a e) \cos (x)+c (d-a e) \sin (x)}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a d-\left (b^2+c^2\right ) e+b (d-a e) \cos (x)+c (d-a e) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a d-\left (b^2+c^2\right ) e+b (d-a e) \cos (x)+c (d-a e) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+(d-a e) \int \sqrt {a+b \cos (x)+c \sin (x)}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+(d-a e) \int \sqrt {a+b \cos (x)+c \sin (x)}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {(d-a e) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {(d-a e) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\frac {e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {2 e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\) |
(2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/((a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]]) + ((2*(d - a*e)*EllipticE[(x - ArcTan[b, c])/2, (2*S qrt[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 + Sqrt[b^2 + c^2])] + (2*(a^2 - b^2 - c^2)*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]])/(a^2 - b^2 - c^2)
3.6.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) *Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(2910\) vs. \(2(288)=576\).
Time = 28.98 (sec) , antiderivative size = 2911, normalized size of antiderivative = 11.64
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2911\) |
| parts | \(\text {Expression too large to display}\) | \(113315\) |
(-(-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*(b^2+c^2)^(1/ 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)*((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))+(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2 )*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*(sin(x-arctan(-b,c))^2*b^2+sin(x -arctan(-b,c))^2*c^2-a^2)*(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)*sin(x-arctan(-b,c))+a)*cos(x-arctan (-b,c))^2)^(1/2)*a*b^2+(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arct an(-b,c))^2)^(1/2)*a*c^2-(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arc tan(-b,c))+a)*(b^2+c^2))^(1/2)*sin(x-arctan(-b,c))*b^2-(cos(x-arctan(-b,c) )^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*sin(x-arctan( -b,c))*c^2)/(sin(x-arctan(-b,c))^2*b^2+sin(x-arctan(-b,c))^2*c^2+2*(b^2+c^ 2)^(1/2)*sin(x-arctan(-b,c))*a+a^2)*((b^2+c^2)*cos(x-arctan(-b,c))^2*(a*e- d)/(a^2-b^2-c^2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b, c))^2)^(1/2)+a*(b^2+c^2)^(1/2)*(a*e-d)/(a^2-b^2-c^2)*(1/(b^2+c^2)^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 2086, normalized size of antiderivative = 8.34 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\text {Too large to display} \]
-1/3*((sqrt(2)*(-I*a*b^2*d - a*b*c*d - I*(2*a^2*b^2 - 3*b^4 - 3*b^2*c^2)*e + (3*b*c^3 - (2*a^2*b - 3*b^3)*c)*e)*cos(x) + sqrt(2)*(-I*a*b*c*d - a*c^2 *d + I*(3*b*c^3 - (2*a^2*b - 3*b^3)*c)*e + (3*c^4 - (2*a^2 - 3*b^2)*c^2)*e )*sin(x) + sqrt(2)*(-I*a^2*b*d - a^2*c*d - I*(2*a^3*b - 3*a*b^3 - 3*a*b*c^ 2)*e + (3*a*c^3 - (2*a^3 - 3*a*b^2)*c)*e))*sqrt(b + I*c)*weierstrassPInver se(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)*(I*a*b^2*d - a*b*c*d + I*(2*a^2*b^2 - 3*b^4 - 3*b^2*c^2)*e + ( 3*b*c^3 - (2*a^2*b - 3*b^3)*c)*e)*cos(x) + sqrt(2)*(I*a*b*c*d - a*c^2*d - I*(3*b*c^3 - (2*a^2*b - 3*b^3)*c)*e + (3*c^4 - (2*a^2 - 3*b^2)*c^2)*e)*sin (x) + sqrt(2)*(I*a^2*b*d - a^2*c*d + I*(2*a^3*b - 3*a*b^3 - 3*a*b*c^2)*e + (3*a*c^3 - (2*a^3 - 3*a*b^2)*c)*e))*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(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2)) -...
Timed out. \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\int { \frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{{\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\int { \frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{{\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\int \frac {d+b\,e\,\cos \left (x\right )+c\,e\,\sin \left (x\right )}{{\left (a+b\,\cos \left (x\right )+c\,\sin \left (x\right )\right )}^{3/2}} \,d x \]