\(\int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [324]

Optimal result

Integrand size = 29, antiderivative size = 92 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {8 a^2 \cos ^3(c+d x)}{105 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^3(c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d} \] Output:

-8/105*a^2*cos(d*x+c)^3/d/(a+a*sin(d*x+c))^(3/2)-2/35*a*cos(d*x+c)^3/d/(a+ 
a*sin(d*x+c))^(1/2)-2/7*cos(d*x+c)^3*(a+a*sin(d*x+c))^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \sqrt {a (1+\sin (c+d x))} (59-15 \cos (2 (c+d x))+66 \sin (c+d x))}{105 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:

Integrate[Cos[c + d*x]^2*Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

-1/105*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])] 
*(59 - 15*Cos[2*(c + d*x)] + 66*Sin[c + d*x]))/(d*(Cos[(c + d*x)/2] + Sin[ 
(c + d*x)/2]))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3042, 3335, 3042, 3153, 3042, 3152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x) \cos (c+d x)^2 \sqrt {a \sin (c+d x)+a}dx\)

\(\Big \downarrow \) 3335

\(\displaystyle \frac {1}{7} \int \cos ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \cos (c+d x)^2 \sqrt {\sin (c+d x) a+a}dx-\frac {2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\)

\(\Big \downarrow \) 3153

\(\displaystyle \frac {1}{7} \left (\frac {4}{5} a \int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^3(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {4}{5} a \int \frac {\cos (c+d x)^2}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^3(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\)

\(\Big \downarrow \) 3152

\(\displaystyle \frac {1}{7} \left (-\frac {8 a^2 \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac {2 a \cos ^3(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\)

Input:

Int[Cos[c + d*x]^2*Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(-2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*d) + ((-8*a^2*Cos[c + d*x] 
^3)/(15*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^3)/(5*d*Sqrt[a + 
 a*Sin[c + d*x]]))/7
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3152

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x 
])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 
 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
 

rule 3153

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p))   Int[(g* 
Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, 
g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && 
NeQ[m + p, 0]
 

rule 3335

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* 
(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S 
imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1))   Int[(g*Cos[e + f*x])^p*(a + 
 b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ 
a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p + 1, 0]
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71

method	result	size
default	\(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{2} \left (15 \sin \left (d x +c \right )^{2}+33 \sin \left (d x +c \right )+22\right )}{105 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(65\)

Input:

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

Output:

-2/105*(1+sin(d*x+c))*a*(sin(d*x+c)-1)^2*(15*sin(d*x+c)^2+33*sin(d*x+c)+22 
)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 18 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + {\left (15 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right ) + 8\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{105 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \] Input:

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

Output:

-2/105*(15*cos(d*x + c)^4 + 18*cos(d*x + c)^3 - cos(d*x + c)^2 + (15*cos(d 
*x + c)^3 - 3*cos(d*x + c)^2 - 4*cos(d*x + c) - 8)*sin(d*x + c) + 4*cos(d* 
x + c) + 8)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)
 

Sympy [F]

\[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {8 \, \sqrt {2} {\left (30 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )} \sqrt {a}}{105 \, d} \] Input:

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

Output:

8/105*sqrt(2)*(30*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d* 
x + 1/2*c)^7 - 63*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d* 
x + 1/2*c)^5 + 35*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d* 
x + 1/2*c)^3)*sqrt(a)/d
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )d x \right ) \] Input:

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

Output:

sqrt(a)*int(sqrt(sin(c + d*x) + 1)*cos(c + d*x)**2*sin(c + d*x),x)