3.523 \(\int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx\)

Optimal. Leaf size=112 \[ -\frac {2 a \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {4 d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]

[Out]

-2/5*d^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/a/f-2/15*a*(15*c^2+10*c*d+7*d^2)*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2
)-4/15*(5*c-d)*d*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2761, 2751, 2646} \[ -\frac {2 a \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {4 d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2,x]

[Out]

(-2*a*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (4*(5*c - d)*d*Cos[e + f*x]*Sq
rt[a + a*Sin[e + f*x]])/(15*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*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)]

Rule 2761

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> -Simp[(
d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x]
)^m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], 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, -1]

Rubi steps

\begin {align*} \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx &=-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}+\frac {2 \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{5 a}\\ &=-\frac {4 (5 c-d) d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}+\frac {1}{15} \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {2 a \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {4 (5 c-d) d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.29, size = 111, normalized size = 0.99 \[ -\frac {\sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (30 c^2+4 d (5 c+2 d) \sin (e+f x)+40 c d-3 d^2 \cos (2 (e+f x))+19 d^2\right )}{15 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2,x]

[Out]

-1/15*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(30*c^2 + 40*c*d + 19*d^2 - 3*d^2*Cos[
2*(e + f*x)] + 4*d*(5*c + 2*d)*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 157, normalized size = 1.40 \[ \frac {2 \, {\left (3 \, d^{2} \cos \left (f x + e\right )^{3} - {\left (10 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, c^{2} - 10 \, c d - 7 \, d^{2} - {\left (15 \, c^{2} + 20 \, c d + 11 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, d^{2} \cos \left (f x + e\right )^{2} - 15 \, c^{2} - 10 \, c d - 7 \, d^{2} + 2 \, {\left (5 \, c d + 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

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))^2,x, algorithm="fricas")

[Out]

2/15*(3*d^2*cos(f*x + e)^3 - (10*c*d + d^2)*cos(f*x + e)^2 - 15*c^2 - 10*c*d - 7*d^2 - (15*c^2 + 20*c*d + 11*d
^2)*cos(f*x + e) - (3*d^2*cos(f*x + e)^2 - 15*c^2 - 10*c*d - 7*d^2 + 2*(5*c*d + 2*d^2)*cos(f*x + e))*sin(f*x +
e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)

________________________________________________________________________________________

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))^2,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)Unable to check si
gn: (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*f*(4*c^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+2*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi)))*sin(1/4
*(2*f*x-pi)+1/2*exp(1))/(2*f)^2-2*c*d*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(1/4*(2*f*x+2*exp(1)+pi))/f-24*d^2
*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(1/4*(6*f*x+6*exp(1)+pi))/(12*f)^2-40*d^2*f*sign(cos(1/2*(f*x+exp(1))
-1/4*pi))*sin(1/4*(10*f*x+10*exp(1)-pi))/(20*f)^2-6*c*d*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(1/4*(6*f*x+6*
exp(1)-pi))/(3*f)^2)

________________________________________________________________________________________

maple [A]  time = 0.81, size = 92, normalized size = 0.82 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (3 d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+10 c d \sin \left (f x +e \right )+4 d^{2} \sin \left (f x +e \right )+15 c^{2}+20 c d +8 d^{2}\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]

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))^2,x)

[Out]

2/15*(1+sin(f*x+e))*a*(sin(f*x+e)-1)*(3*d^2*sin(f*x+e)^2+10*c*d*sin(f*x+e)+4*d^2*sin(f*x+e)+15*c^2+20*c*d+8*d^
2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{2}\,{d x} \]

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))^2,x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]

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))^2,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \]

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))**2,x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*(c + d*sin(e + f*x))**2, x)

________________________________________________________________________________________