3.35 \(\int \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)

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Rubi [A]  time = 0.0451658, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2751, 2646} \[ -\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

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 \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{1}{3} \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 a \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.117609, size = 81, normalized size = 1.45 \[ -\frac{\sqrt{a (\sin (c+d x)+1)} \left (-4 \sin ^3\left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )}{3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-((3*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] - 4*Sin[(c + d*x)/2]^3)*Sqrt[a*(1 + Sin[c + d*x])])/(3*d*(Cos[(c
+ d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 0.377, size = 51, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) a \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( \sin \left ( dx+c \right ) +2 \right ) }{3\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(1+sin(d*x+c))*a*(sin(d*x+c)-1)*(sin(d*x+c)+2)/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} \sin \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44562, size = 190, normalized size = 3.39 \begin{align*} -\frac{2 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(cos(d*x + c)^2 + (cos(d*x + c) - 1)*sin(d*x + c) + 2*cos(d*x + c) + 1)*sqrt(a*sin(d*x + c) + a)/(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 \left (\sin{\left (c + d x \right )} + 1\right )} \sin{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sin(c + d*x) + 1))*sin(c + d*x), 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} \sin \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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