\(\int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx\) [289]

Optimal result

Integrand size = 25, antiderivative size = 62 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=-\frac {2 a (3 A+B) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2830, 2725} \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=-\frac {2 a (3 A+B) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 B \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f} \]

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]),x]

[Out]

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

Rule 2725

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 2830

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*S
in[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*} \text {integral}& = -\frac {2 B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {1}{3} (3 A+B) \int \sqrt {a+a \sin (e+f x)} \, dx \\ & = -\frac {2 a (3 A+B) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (3 A+2 B+B \sin (e+f x))}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*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*A + 2*B + B*Sin[e + f*x]))/(3*f*(Cos[(
e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (B \sin \left (f x +e \right )+3 A +2 B \right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(58\)
parts	\(\frac {2 A \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) a}{\cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 B \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+2\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(96\)

[In]

int((A+B*sin(f*x+e))*(a+a*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.37 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=-\frac {2 \, {\left (B \cos \left (f x + e\right )^{2} + {\left (3 \, A + 2 \, B\right )} \cos \left (f x + e\right ) + {\left (B \cos \left (f x + e\right ) - 3 \, A - B\right )} \sin \left (f x + e\right ) + 3 \, A + B\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=\int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]

[In]

integrate((A+B*sin(f*x+e))*(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*(A + B*sin(e + f*x)), x)

Maxima [F]

\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((B*sin(f*x + e) + A)*sqrt(a*sin(f*x + e) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.37 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=\frac {\sqrt {2} {\left (B \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 3 \, {\left (2 \, A \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \sqrt {a}}{3 \, f} \]

[In]

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

[Out]

1/3*sqrt(2)*(B*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 3*(2*A*sgn(cos(-1/4*pi + 1
/2*f*x + 1/2*e)) + B*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e))*sqrt(a)/f

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )} \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(1/2),x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(1/2), x)