### 3.53 $$\int \sqrt{a+b \sin (c+d x)} \, dx$$

Optimal. Leaf size=62 $\frac{2 \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}$

[Out]

(2*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)
])

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Rubi [A]  time = 0.0353472, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {2655, 2653} $\frac{2 \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(2*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)
])

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \sqrt{a+b \sin (c+d x)} \, dx &=\frac{\sqrt{a+b \sin (c+d x)} \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{\sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{2 E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ \end{align*}

Mathematica [A]  time = 2.17287, size = 61, normalized size = 0.98 $-\frac{2 \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(-2*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a
+ b)])

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Maple [B]  time = 0.135, size = 239, normalized size = 3.9 \begin{align*} -2\,{\frac{a-b}{b\cos \left ( dx+c \right ) \sqrt{a+b\sin \left ( dx+c \right ) }d}\sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}}\sqrt{-{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) b}{a-b}}} \left ({\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) a+{\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) b-a{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) -{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) b \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b*(a-b)*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*(Ell
ipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a+EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(
a+b))^(1/2))*b-a*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))-EllipticF(((a+b*sin(d*x+c))/(a-
b))^(1/2),((a-b)/(a+b))^(1/2))*b)/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sin \left (d x + c\right ) + a}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sin(d*x + c) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sin(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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