### 3.4 $$\int \sqrt{a+a \sin (c+d x)} \, dx$$

Optimal. Leaf size=26 $-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}$

[Out]

(-2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])

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Rubi [A]  time = 0.0133678, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {2646} $-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])

Rule 2646

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

Rubi steps

\begin{align*} \int \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{2 a \cos (c+d x)}{d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [B]  time = 0.0400557, size = 65, normalized size = 2.5 $\frac{2 \sqrt{a (\sin (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(2*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])
)

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Maple [A]  time = 0.071, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a \left ( 1+\sin \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) }{\cos \left ( dx+c \right ) \sqrt{a+a\sin \left ( dx+c \right ) }d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1+sin(d*x+c))*a*(sin(d*x+c)-1)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.52851, size = 136, normalized size = 5.23 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/(d*cos(d*x + c) + d*sin(d*x + c) + d)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*sin(c + d*x) + a), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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