\(\int \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx\) [593]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.08 (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, 3256} \[ \int \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx=\frac {\sqrt {a+(b \sin (e+f x)+c \cos (e+f x))^2} E\left (e+f x+\tan ^{-1}(b,c)|-\frac {b^2+c^2}{a}\right )}{f \sqrt {\frac {(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1}} \]

[In]

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

[Out]

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

Rule 3256

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]/f)*EllipticE[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 {a+(c \cos (e+f x)+b \sin (e+f x))^2} \int \sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}} \, dx}{\sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}}} \\ & = \frac {\sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2} \int \sqrt {1+\frac {\left (b^2+c^2\right ) \sin ^2\left (e+f x+\tan ^{-1}(b,c)\right )}{a}} \, dx}{\sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}}} \\ & = \frac {E\left (e+f x+\tan ^{-1}(b,c)|-\frac {b^2+c^2}{a}\right ) \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2}}{f \sqrt {1+\frac {(c \cos (e+f x)+b \sin (e+f x))^2}{a}}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(79)=158\).

Time = 2.14 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.11 \[ \int \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx=-\frac {E\left (\arcsin \left (\frac {\sqrt {\frac {\sqrt {\left (b^2+c^2\right )^2}+\left (b^2-c^2\right ) \cos (2 (e+f x))-2 b c \sin (2 (e+f x))}{\sqrt {\left (b^2+c^2\right )^2}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {\left (b^2+c^2\right )^2}}{2 a+b^2+c^2+\sqrt {\left (b^2+c^2\right )^2}}\right ) \sqrt {2 a+b^2+c^2+\left (-b^2+c^2\right ) \cos (2 (e+f x))+2 b c \sin (2 (e+f x))} \left (2 b c \cos (2 (e+f x))+\left (b^2-c^2\right ) \sin (2 (e+f x))\right )}{\sqrt {2} \sqrt {\left (b^2+c^2\right )^2} f \sqrt {\frac {2 a+b^2+c^2+\left (-b^2+c^2\right ) \cos (2 (e+f x))+2 b c \sin (2 (e+f x))}{2 a+b^2+c^2+\sqrt {\left (b^2+c^2\right )^2}}} \sqrt {\frac {\left (2 b c \cos (2 (e+f x))+\left (b^2-c^2\right ) \sin (2 (e+f x))\right )^2}{\left (b^2+c^2\right )^2}}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 7.42 (sec) , antiderivative size = 315013, normalized size of antiderivative = 3987.51

method	result	size
default	\(\text {Expression too large to display}\)	\(315013\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(2*b*c*cos(f*x + e)*sin(f*x + e) - (b^2 - c^2)*cos(f*x + e)^2 + b^2 + a), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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