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]

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

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Rubi [A]  time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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)]

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*}

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Mathematica [A]  time = 0.12, 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]))

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fricas [A]  time = 0.46, size = 85, normalized size = 1.37 \[ -\frac {2 \, {\left (d \cos \left (f x + e\right )^{2} + {\left (3 \, c + 2 \, d\right )} \cos \left (f x + e\right ) + {\left (d \cos \left (f x + e\right ) - 3 \, c - d\right )} \sin \left (f x + e\right ) + 3 \, c + d\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)sqrt(2*a)*(2*c*sig
n(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(1/4*(2*f*x-pi)+1/2*exp(1))/f-4*d*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(
1/4*(2*f*x+2*exp(1)+pi))/(2*f)^2-12*d*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(1/4*(6*f*x+6*exp(1)-pi))/(6*f)^
2)

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maple [A]  time = 0.64, size = 58, normalized size = 0.94 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (d \sin \left (f x +e \right )+3 c +2 d \right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]

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)

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