\(\int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx\) [339]

Optimal result

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

2*EllipticE(sin(1/2*d+1/2*e*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(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)
 

Mathematica [C] (warning: unable to verify)

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

Time = 6.27 (sec) , antiderivative size = 1408, normalized size of antiderivative = 13.04 \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(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]])/(Sq 
rt[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 - 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 - 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...
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 3598, 3042, 3132}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3598

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

Input:

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

Output:

(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])])
 

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(690\) vs. \(2(103)=206\).

Time = 2.06 (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 {\sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}{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 )+a \cos \left (e x +d -\arctan \left (-b , c\right )\right )^{2}}\, \left (\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 ) \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 ) a -\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 )+\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 \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	\(\text {Expression too large to display}\)	\(2150\)

Input:

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

Output:

-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-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(e*x+d-arctan(-b,c))-1)/(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))+a*cos(e*x+d-arctan(-b,c))^2)^(1/2)*(EllipticF( 
((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2))*sin(e*x+d-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))*(b^2+c^ 
2)^(1/2)-EllipticF(((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2))*sin(e*x+d-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))*a-(b^2+c^2)^(1/2)*EllipticE(((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2 
))*sin(e*x+d-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))+EllipticE(((b^2+c^2)^(1/2)/(a+(b^2+c^2)^( 
1/2))*sin(e*x+d-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))*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*s 
in(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 complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 1369, normalized size of antiderivative = 12.68 \[ \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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