\(\int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [35]

Optimal result

Integrand size = 21, antiderivative size = 56 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2830, 2725} \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}} \]

[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 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 \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] (verified)

Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {3}{2} (c+d x)\right )-4 \sin ^3\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a (1+\sin (c+d x))}}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right ) \left (\sin \left (d x +c \right )+2\right )}{3 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(51\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]

[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)

Sympy [F]

\[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sin {\left (c + d x \right )}\, dx \]

[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)

Maxima [F]

\[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \sin \left (d x + c\right ) \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (3 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )} \sqrt {a}}{3 \, d} \]

[In]

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

[Out]

1/3*sqrt(2)*(3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c) + sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*sin(-3/4*pi + 3/2*d*x + 3/2*c))*sqrt(a)/d

Mupad [F(-1)]

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

[In]

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

[Out]

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