\(\int \sqrt {a+b \sin ^2(e+f x)} \, dx\) [129]

Optimal result

Integrand size = 16, antiderivative size = 51 \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3257, 3256} \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]

[In]

Int[Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

(EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (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 3257

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {a \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]

[In]

Integrate[Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

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

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )}{\cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\)	\(71\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\left \{\begin {array}{cl} \frac {\sqrt {a}\,\mathrm {E}\left (e+f\,x\middle |-\frac {b}{a}\right )}{f} & \text {\ if\ \ }0<a\\ \int \sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x & \text {\ if\ \ }\neg 0<a \end {array}\right . \]

[In]

int((a + b*sin(e + f*x)^2)^(1/2),x)

[Out]

piecewise(0 < a, (a^(1/2)*ellipticE(e + f*x, -b/a))/f, ~0 < a, int((a + b*sin(e + f*x)^2)^(1/2), x))