\(\int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [473]

Optimal result

Integrand size = 31, antiderivative size = 92 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {46 a \cos ^5(c+d x)}{315 d (a+a \sin (c+d x))^{5/2}}+\frac {20 \cos ^5(c+d x)}{63 d (a+a \sin (c+d x))^{3/2}}-\frac {2 \cos ^5(c+d x)}{9 a d \sqrt {a+a \sin (c+d x)}} \] Output:

-46/315*a*cos(d*x+c)^5/d/(a+a*sin(d*x+c))^(5/2)+20/63*cos(d*x+c)^5/d/(a+a* 
sin(d*x+c))^(3/2)-2/9*cos(d*x+c)^5/a/d/(a+a*sin(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 7.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 \sqrt {a (1+\sin (c+d x))} (51-35 \cos (2 (c+d x))+40 \sin (c+d x))}{315 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]
 

Output:

-1/315*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])] 
*(51 - 35*Cos[2*(c + d*x)] + 40*Sin[c + d*x]))/(a^2*d*(Cos[(c + d*x)/2] + 
Sin[(c + d*x)/2]))
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.48, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3042, 3356, 27, 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.

\(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{(a \sin (c+d x)+a)^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{(a \sin (c+d x)+a)^{3/2}}dx\)

\(\Big \downarrow \) 3356

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\cos ^4(c+d x) (4 \sin (c+d x) a+3 a)}{\sqrt {\sin (c+d x) a+a}}dx}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\cos (c+d x)^4 (4 \sin (c+d x) a+3 a)}{\sqrt {\sin (c+d x) a+a}}dx}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3335

\(\displaystyle \frac {\frac {23}{9} a \int \frac {\cos ^4(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx-\frac {8 a \cos ^5(c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {23}{9} a \int \frac {\cos (c+d x)^4}{\sqrt {\sin (c+d x) a+a}}dx-\frac {8 a \cos ^5(c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3153

\(\displaystyle \frac {\frac {23}{9} a \left (\frac {4}{7} a \int \frac {\cos ^4(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {8 a \cos ^5(c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {23}{9} a \left (\frac {4}{7} a \int \frac {\cos (c+d x)^4}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {8 a \cos ^5(c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3152

\(\displaystyle \frac {\frac {23}{9} a \left (-\frac {8 a^2 \cos ^5(c+d x)}{35 d (a \sin (c+d x)+a)^{5/2}}-\frac {2 a \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {8 a \cos ^5(c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {\cos ^5(c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\)

Input:

Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]
 

Output:

Cos[c + d*x]^5/(2*d*(a + a*Sin[c + d*x])^(3/2)) + ((-8*a*Cos[c + d*x]^5)/( 
9*d*Sqrt[a + a*Sin[c + d*x]]) + (23*a*((-8*a^2*Cos[c + d*x]^5)/(35*d*(a + 
a*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(3 
/2))))/9)/(4*a^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 3356

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{3} \left (35 \sin \left (d x +c \right )^{2}+20 \sin \left (d x +c \right )+8\right )}{315 a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(67\)

Input:

int(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBO 
SE)
 

Output:

2/315/a*(1+sin(d*x+c))*(sin(d*x+c)-1)^3*(35*sin(d*x+c)^2+20*sin(d*x+c)+8)/ 
cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.54 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (35 \, \cos \left (d x + c\right )^{5} + 85 \, \cos \left (d x + c\right )^{4} - 73 \, \cos \left (d x + c\right )^{3} - 169 \, \cos \left (d x + c\right )^{2} - {\left (35 \, \cos \left (d x + c\right )^{4} - 50 \, \cos \left (d x + c\right )^{3} - 123 \, \cos \left (d x + c\right )^{2} + 46 \, \cos \left (d x + c\right ) + 92\right )} \sin \left (d x + c\right ) + 46 \, \cos \left (d x + c\right ) + 92\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{315 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm="f 
ricas")
                                                                                    
                                                                                    
 

Output:

-2/315*(35*cos(d*x + c)^5 + 85*cos(d*x + c)^4 - 73*cos(d*x + c)^3 - 169*co 
s(d*x + c)^2 - (35*cos(d*x + c)^4 - 50*cos(d*x + c)^3 - 123*cos(d*x + c)^2 
 + 46*cos(d*x + c) + 92)*sin(d*x + c) + 46*cos(d*x + c) + 92)*sqrt(a*sin(d 
*x + c) + a)/(a^2*d*cos(d*x + c) + a^2*d*sin(d*x + c) + a^2*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**4*sin(d*x+c)**2/(a+a*sin(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm="m 
axima")
 

Output:

integrate(cos(d*x + c)^4*sin(d*x + c)^2/(a*sin(d*x + c) + a)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {8 \, \sqrt {2} {\left (140 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 180 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}\right )}}{315 \, a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm="g 
iac")
 

Output:

8/315*sqrt(2)*(140*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 180*sqrt(a)* 
sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 + 63*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2* 
c)^5)/(a^2*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + a*sin(c + d*x))^(3/2),x)
 

Output:

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + a*sin(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2}}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:

int(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x)
 

Output:

(sqrt(a)*int((sqrt(sin(c + d*x) + 1)*cos(c + d*x)**4*sin(c + d*x)**2)/(sin 
(c + d*x)**2 + 2*sin(c + d*x) + 1),x))/a**2