\(\int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx\) [594]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\frac {\operatorname {EllipticF}\left (e+f x+\tan ^{-1}(b,c),-\frac {b^2+c^2}{a}\right ) \sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}}}{f \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3320, 3319, 3261} \[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\frac {\sqrt {\frac {(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1} \operatorname {EllipticF}\left (e+f x+\tan ^{-1}(b,c),-\frac {b^2+c^2}{a}\right )}{f \sqrt {a+(b \sin (e+f x)+c \cos (e+f x))^2}} \]

[In]

Int[1/Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]

[Out]

(EllipticF[e + f*x + ArcTan[b, c], -((b^2 + c^2)/a)]*Sqrt[1 + (c*Cos[e + f*x] + b*Sin[e + f*x])^2/a])/(f*Sqrt[
a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2])

Rule 3261

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(Sqrt[a]*f))*EllipticF[e + f*x, -b/a]
, x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3319

Int[((a_) + (b_.)*(cos[(e_.) + (f_.)*(x_)]*(d_.) + (c_.)*sin[(e_.) + (f_.)*(x_)])^2)^(p_), x_Symbol] :> Int[(a
+ b*(Sqrt[c^2 + d^2]*Sin[ArcTan[c, d] + e + f*x])^2)^p, x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[p^2, 1/4] &
& GtQ[a, 0]

Rule 3320

Int[((a_) + (b_.)*(cos[(e_.) + (f_.)*(x_)]*(d_.) + (c_.)*sin[(e_.) + (f_.)*(x_)])^2)^(p_), x_Symbol] :> Dist[(
a + b*(c*Sin[e + f*x] + d*Cos[e + f*x])^2)^p/(1 + (b*(c*Sin[e + f*x] + d*Cos[e + f*x])^2)/a)^p, Int[(1 + (b*(c
*Sin[e + f*x] + d*Cos[e + f*x])^2)/a)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[p^2, 1/4] &&  !GtQ[a, 0
]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}} \int \frac {1}{\sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}}} \, dx}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \\ & = \frac {\sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}} \int \frac {1}{\sqrt {1+\frac {\left (b^2+c^2\right ) \sin ^2\left (e+f x+\tan ^{-1}(b,c)\right )}{a}}} \, dx}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \\ & = \frac {\operatorname {EllipticF}\left (e+f x+\tan ^{-1}(b,c),-\frac {b^2+c^2}{a}\right ) \sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}}}{f \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 1.86 (sec) , antiderivative size = 529, normalized size of antiderivative = 6.70 \[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {2 a+b^2+c^2+b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}} \sin \left (2 (e+f x)+\arctan \left (\frac {-b^2+c^2}{2 b c}\right )\right )}{2 a+b^2+c^2-b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}}},\frac {2 a+b^2+c^2+b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}} \sin \left (2 (e+f x)+\arctan \left (\frac {-b^2+c^2}{2 b c}\right )\right )}{2 a+b^2+c^2+b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}}}\right ) \sec \left (2 (e+f x)+\arctan \left (\frac {-b^2+c^2}{2 b c}\right )\right ) \sqrt {-\frac {b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}} \left (-1+\sin \left (2 (e+f x)+\arctan \left (\frac {-b^2+c^2}{2 b c}\right )\right )\right )}{2 a+b^2+c^2+b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}}}} \sqrt {-\frac {b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}} \left (1+\sin \left (2 (e+f x)+\arctan \left (\frac {-b^2+c^2}{2 b c}\right )\right )\right )}{2 a+b^2+c^2-b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}}}} \sqrt {2 a+b^2+c^2+b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}} \sin \left (2 (e+f x)+\arctan \left (\frac {-b^2+c^2}{2 b c}\right )\right )}}{b c \sqrt {\frac {\left (b^2+c^2\right )^2}{b^2 c^2}} f} \]

[In]

Integrate[1/Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]

[Out]

(Sqrt[2]*AppellF1[1/2, 1/2, 1/2, 3/2, (2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*Sin[2*(e + f*x) + A
rcTan[(-b^2 + c^2)/(2*b*c)]])/(2*a + b^2 + c^2 - b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]), (2*a + b^2 + c^2 + b*c*Sq
rt[(b^2 + c^2)^2/(b^2*c^2)]*Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]])/(2*a + b^2 + c^2 + b*c*Sqrt[(b^2
+ c^2)^2/(b^2*c^2)])]*Sec[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]*Sqrt[-((b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)
]*(-1 + Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]))/(2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]
))]*Sqrt[-((b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*(1 + Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]))/(2*a + b^
2 + c^2 - b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]))]*Sqrt[2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*Sin[2*
(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]])/(b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*f)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 6.64 (sec) , antiderivative size = 2174, normalized size of antiderivative = 27.52

method	result	size
default	\(\text {Expression too large to display}\)	\(2174\)

[In]

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

[Out]

-4/f*((RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1)*sin(f*x+e)+cos(f*x+e)-1)/
(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)*sin(f*x+e)+cos(f*x+e)-1)*(RootO
f((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2
-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4))/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+
a,index=1)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4)))^(1/2)*(RootOf((c^2+
a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)*sin(f*x+e)+cos(f*x+e)-1)^2*(-(RootOf((c^2+a)
*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*
a)*_Z^2+4*_Z*b*c+c^2+a,index=1))*(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=3
)*sin(f*x+e)+cos(f*x+e)-1)/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1)-Root
Of((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=3))/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*
b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)*sin(f*x+e)+cos(f*x+e)-1))^(1/2)*(-(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+
(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+
c^2+a,index=1))*(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4)*sin(f*x+e)+cos(
f*x+e)-1)/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1)-RootOf((c^2+a)*_Z^4-4
*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4))/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z
^2+4*_Z*b*c+c^2+a,index=2)*sin(f*x+e)+cos(f*x+e)-1))^(1/2)*EllipticF(((RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2
*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1)*sin(f*x+e)+cos(f*x+e)-1)/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2
*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)*sin(f*x+e)+cos(f*x+e)-1)*(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z
^2+4*_Z*b*c+c^2+a,index=2)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4))/(Roo
tOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b
^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4)))^(1/2),((RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_
Z*b*c+c^2+a,index=4)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1))*(RootOf((c
^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=3)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c
^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2))/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,in
dex=3)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1))/(RootOf((c^2+a)*_Z^4-4*b
*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4
*_Z*b*c+c^2+a,index=2)))^(1/2))/(cos(f*x+e)^2*c^2+2*cos(f*x+e)*sin(f*x+e)*b*c+sin(f*x+e)^2*b^2+a)^(1/2)/(cos(f
*x+e)^2+sin(f*x+e)^2-2*cos(f*x+e)+1)*(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,ind
ex=1)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=4))/(RootOf((c^2+a)*_Z^4-4*b*
c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)-RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*
_Z*b*c+c^2+a,index=4))/(RootOf((c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=2)-RootOf((
c^2+a)*_Z^4-4*b*c*_Z^3+(4*b^2-2*c^2+2*a)*_Z^2+4*_Z*b*c+c^2+a,index=1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 1623, normalized size of antiderivative = 20.54 \[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-((b^3 + b*c^2 - I*c^3 + 2*a*b - I*(b^2 + 2*a)*c - 2*(b^3 - 3*I*b^2*c - 3*b*c^2 + I*c^3)*sqrt((a*b^6 - 4*I*a*b
*c^5 + a*c^6 + a^2*b^4 - 4*I*a^2*b*c^3 - (5*a*b^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2 + 4*I*(a*b^5 + a^2*b^
3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^6 + c^8)))*sqrt((b^4 + 2*I*b*c^3 - c^4 + 2*a*b^2 - 2*a*c^2 + 2*I*
(b^3 + 2*a*b)*c + 2*(b^4 + 2*b^2*c^2 + c^4)*sqrt((a*b^6 - 4*I*a*b*c^5 + a*c^6 + a^2*b^4 - 4*I*a^2*b*c^3 - (5*a
*b^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2 + 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^
6 + c^8)))/(b^4 + 2*b^2*c^2 + c^4))*elliptic_f(arcsin(sqrt((b^4 + 2*I*b*c^3 - c^4 + 2*a*b^2 - 2*a*c^2 + 2*I*(b
^3 + 2*a*b)*c + 2*(b^4 + 2*b^2*c^2 + c^4)*sqrt((a*b^6 - 4*I*a*b*c^5 + a*c^6 + a^2*b^4 - 4*I*a^2*b*c^3 - (5*a*b
^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2 + 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^6
+ c^8)))/(b^4 + 2*b^2*c^2 + c^4))*(cos(f*x + e) + I*sin(f*x + e))), (b^4 + c^4 + 8*a*b^2 + 2*(b^2 + 4*a)*c^2 +
8*a^2 - 4*(b^4 - 2*I*b*c^3 - c^4 + 2*a*b^2 - 2*a*c^2 - 2*I*(b^3 + 2*a*b)*c)*sqrt((a*b^6 - 4*I*a*b*c^5 + a*c^6
+ a^2*b^4 - 4*I*a^2*b*c^3 - (5*a*b^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2 + 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 +
4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^6 + c^8)))/(b^4 + 2*b^2*c^2 + c^4)) + (b^3 + b*c^2 + I*c^3 + 2*a*b + I*(b^2 +
2*a)*c - 2*(b^3 + 3*I*b^2*c - 3*b*c^2 - I*c^3)*sqrt((a*b^6 + 4*I*a*b*c^5 + a*c^6 + a^2*b^4 + 4*I*a^2*b*c^3 -
(5*a*b^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2 - 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^
2*c^6 + c^8)))*sqrt((b^4 - 2*I*b*c^3 - c^4 + 2*a*b^2 - 2*a*c^2 - 2*I*(b^3 + 2*a*b)*c + 2*(b^4 + 2*b^2*c^2 + c^
4)*sqrt((a*b^6 + 4*I*a*b*c^5 + a*c^6 + a^2*b^4 + 4*I*a^2*b*c^3 - (5*a*b^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c
^2 - 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^6 + c^8)))/(b^4 + 2*b^2*c^2 + c^4))*ellip
tic_f(arcsin(sqrt((b^4 - 2*I*b*c^3 - c^4 + 2*a*b^2 - 2*a*c^2 - 2*I*(b^3 + 2*a*b)*c + 2*(b^4 + 2*b^2*c^2 + c^4)
*sqrt((a*b^6 + 4*I*a*b*c^5 + a*c^6 + a^2*b^4 + 4*I*a^2*b*c^3 - (5*a*b^2 - a^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2
- 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^6 + c^8)))/(b^4 + 2*b^2*c^2 + c^4))*(cos(f*
x + e) - I*sin(f*x + e))), (b^4 + c^4 + 8*a*b^2 + 2*(b^2 + 4*a)*c^2 + 8*a^2 - 4*(b^4 + 2*I*b*c^3 - c^4 + 2*a*b
^2 - 2*a*c^2 + 2*I*(b^3 + 2*a*b)*c)*sqrt((a*b^6 + 4*I*a*b*c^5 + a*c^6 + a^2*b^4 + 4*I*a^2*b*c^3 - (5*a*b^2 - a
^2)*c^4 - (5*a*b^4 + 6*a^2*b^2)*c^2 - 4*I*(a*b^5 + a^2*b^3)*c)/(b^8 + 4*b^6*c^2 + 6*b^4*c^4 + 4*b^2*c^6 + c^8)
))/(b^4 + 2*b^2*c^2 + c^4)))/((b^4 + 2*b^2*c^2 + c^4)*f)

Sympy [F]

\[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\int \frac {1}{\sqrt {a + \left (b \sin {\left (e + f x \right )} + c \cos {\left (e + f x \right )}\right )^{2}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(a + (b*sin(e + f*x) + c*cos(e + f*x))**2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\int { \frac {1}{\sqrt {{\left (c \cos \left (f x + e\right ) + b \sin \left (f x + e\right )\right )}^{2} + a}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\int { \frac {1}{\sqrt {{\left (c \cos \left (f x + e\right ) + b \sin \left (f x + e\right )\right )}^{2} + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx=\int \frac {1}{\sqrt {a+{\left (c\,\cos \left (e+f\,x\right )+b\,\sin \left (e+f\,x\right )\right )}^2}} \,d x \]

[In]

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

[Out]

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