\(\int \sqrt {a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx\) [501]

Optimal result

Integrand size = 27, antiderivative size = 229 \[ \int \sqrt {a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx=\frac {2 (3 d+a e) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{3 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) e \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{3 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x)) \] Output:

2/3*(a*e+3*d)*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*s 
in(x))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/3*(a^2-b^2-c^2)*e*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) 
-2/3*(a+b*cos(x)+c*sin(x))^(1/2)*(c*e*cos(x)-b*e*sin(x))
 

Mathematica [C] (warning: unable to verify)

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

Time = 6.26 (sec) , antiderivative size = 3006, normalized size of antiderivative = 13.13 \[ \int \sqrt {a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Result too large to show} \] Input:

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

Output:

Sqrt[a + b*Cos[x] + c*Sin[x]]*((2*b*(3*d + a*e))/(3*c) - (2*c*e*Cos[x])/3 
+ (2*b*e*Sin[x])/3) + (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/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sq 
rt[(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 + ArcT 
an[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c) + (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*Sqr 
t[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sq 
rt[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])])/(3*Sqrt[1 + b^2/c^2]*c) + (2*c*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 + ...
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3625, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3625

\(\displaystyle \frac {2 \int \frac {a \left (3 a d+\left (b^2+c^2\right ) e\right )+a b (3 d+a e) \cos (x)+a c (3 d+a e) \sin (x)}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a \left (3 a d+\left (b^2+c^2\right ) e\right )+a b (3 d+a e) \cos (x)+a c (3 d+a e) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a \left (3 a d+\left (b^2+c^2\right ) e\right )+a b (3 d+a e) \cos (x)+a c (3 d+a e) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {a (a e+3 d) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (a e+3 d) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {\frac {a (a e+3 d) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a (a e+3 d) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {2 a (a e+3 d) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {\frac {2 a (a e+3 d) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 a (a e+3 d) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {2 a (a e+3 d) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 a e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))\)

Input:

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

Output:

(-2*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*e*Cos[x] - b*e*Sin[x]))/3 + ((2*a*(3* 
d + a*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 
 + Sqrt[b^2 + c^2])] - (2*a*(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*Si 
n[x])/(a + Sqrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3625

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_.)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*((a 
 + b*Cos[d + e*x] + c*Sin[d + e*x])^n/(a*e*(n + 1))), x] + Simp[1/(a*(n + 1 
))   Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n 
 + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] 
 + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; 
FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - 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. \(1414\) vs. \(2(208)=416\).

Time = 0.83 (sec) , antiderivative size = 1415, normalized size of antiderivative = 6.18

method	result	size
default	\(\text {Expression too large to display}\)	\(1415\)
parts	\(\text {Expression too large to display}\)	\(6201\)

Input:

int((a+b*cos(x)+c*sin(x))^(1/2)*(d+b*e*cos(x)+c*e*sin(x)),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)*(e*(b^2+c^2)^(3/ 
2)*(-2/3/(b^2+c^2)^(1/2)*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arc 
tan(-b,c))+a))^(1/2)+2/3*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-a 
rctan(-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)*s 
in(x-arctan(-b,c))+a))^(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))-4/3/(b^2+c^2)^(1/2)*a*(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-arct 
an(-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))))+2*(a*b^2*e+a*c^2*e+b^2*d+c^2*d)*(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)*...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 1496, normalized size of antiderivative = 6.53 \[ \int \sqrt {a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Too large to display} \] Input:

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

Output:

-2/9*((-3*I*a*b*d - 3*a*c*d + I*(2*a^2*b - 3*b^3 - 3*b*c^2)*e - (3*c^3 - ( 
2*a^2 - 3*b^2)*c)*e)*sqrt(1/2*b + 1/2*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)) + (3*I*a*b*d 
- 3*a*c*d - I*(2*a^2*b - 3*b^3 - 3*b*c^2)*e - (3*c^3 - (2*a^2 - 3*b^2)*c)* 
e)*sqrt(1/2*b - 1/2*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)) + 3*(3*I*(b^2 + c^2)*d + I*(a* 
b^2 + a*c^2)*e)*sqrt(1/2*b + 1/2*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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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

integrate((a+b*cos(x)+c*sin(x))^(1/2)*(d+b*e*cos(x)+c*e*sin(x)),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. \[ \int \sqrt {a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx=\int \sqrt {a+b\,\cos \left (x\right )+c\,\sin \left (x\right )}\,\left (d+b\,e\,\cos \left (x\right )+c\,e\,\sin \left (x\right )\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(cos(x)*b + sin(x)*c + a)*cos(x)*b**2*c**2*e + 6*sqrt(cos(x)*b + 
 sin(x)*c + a)*sin(x)*b**3*c*e + 4*sqrt(cos(x)*b + sin(x)*c + a)*sin(x)*b* 
c**3*e - 2*sqrt(cos(x)*b + sin(x)*c + a)*a*b*c**2*e + 12*sqrt(cos(x)*b + s 
in(x)*c + a)*b**3*d + 6*sqrt(cos(x)*b + sin(x)*c + a)*b*c**2*d + 12*int(sq 
rt(cos(x)*b + sin(x)*c + a)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2 
*c + sin(x)*c**3 + 2*a*b**2 + a*c**2),x)*a*b**4*c*d + 12*int(sqrt(cos(x)*b 
 + sin(x)*c + a)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x) 
*c**3 + 2*a*b**2 + a*c**2),x)*a*b**2*c**3*d + 3*int(sqrt(cos(x)*b + sin(x) 
*c + a)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x)*c**3 + 2 
*a*b**2 + a*c**2),x)*a*c**5*d + 6*int((sqrt(cos(x)*b + sin(x)*c + a)*sin(x 
)**2)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x)*c**3 + 2*a 
*b**2 + a*c**2),x)*b**6*c*e + 15*int((sqrt(cos(x)*b + sin(x)*c + a)*sin(x) 
**2)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x)*c**3 + 2*a* 
b**2 + a*c**2),x)*b**4*c**3*e + 12*int((sqrt(cos(x)*b + sin(x)*c + a)*sin( 
x)**2)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x)*c**3 + 2* 
a*b**2 + a*c**2),x)*b**2*c**5*e + 3*int((sqrt(cos(x)*b + sin(x)*c + a)*sin 
(x)**2)/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x)*c**3 + 2 
*a*b**2 + a*c**2),x)*c**7*e + 6*int((sqrt(cos(x)*b + sin(x)*c + a)*sin(x)) 
/(2*cos(x)*b**3 + cos(x)*b*c**2 + 2*sin(x)*b**2*c + sin(x)*c**3 + 2*a*b**2 
 + a*c**2),x)*a*b**4*c**2*e + 9*int((sqrt(cos(x)*b + sin(x)*c + a)*sin(...