3.524 \(\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x)) \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0554464, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2751, 2646} \[ -\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]),x]

[Out]

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

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

Mathematica [A]  time = 0.123096, size = 82, normalized size = 1.32 \[ -\frac{2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (3 c+d \sin (e+f x)+2 d)}{3 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.614, size = 58, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ) a \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( d\sin \left ( fx+e \right ) +3\,c+2\,d \right ) }{3\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(1+sin(f*x+e))*a*(-1+sin(f*x+e))*(d*sin(f*x+e)+3*c+2*d)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.83614, size = 225, normalized size = 3.63 \begin{align*} -\frac{2 \,{\left (d \cos \left (f x + e\right )^{2} +{\left (3 \, c + 2 \, d\right )} \cos \left (f x + e\right ) +{\left (d \cos \left (f x + e\right ) - 3 \, c - d\right )} \sin \left (f x + e\right ) + 3 \, c + d\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{3 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(d*cos(f*x + e)^2 + (3*c + 2*d)*cos(f*x + e) + (d*cos(f*x + e) - 3*c - d)*sin(f*x + e) + 3*c + d)*sqrt(a*
sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (c + d \sin{\left (e + f x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*(c + d*sin(e + f*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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