\(\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [285]

Optimal result

Integrand size = 29, antiderivative size = 132 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 a^3 x}{16}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{d}-\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \cos ^3(c+d x) \sin ^3(c+d x)}{2 d} \] Output:

5/16*a^3*x-4/3*a^3*cos(d*x+c)^3/d+a^3*cos(d*x+c)^5/d-1/7*a^3*cos(d*x+c)^7/ 
d+5/16*a^3*cos(d*x+c)*sin(d*x+c)/d-5/8*a^3*cos(d*x+c)^3*sin(d*x+c)/d-1/2*a 
^3*cos(d*x+c)^3*sin(d*x+c)^3/d
 

Mathematica [A] (verified)

Time = 7.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (420 c+420 d x-609 \cos (c+d x)-91 \cos (3 (c+d x))+63 \cos (5 (c+d x))-3 \cos (7 (c+d x))-63 \sin (2 (c+d x))-105 \sin (4 (c+d x))+21 \sin (6 (c+d x)))}{1344 d} \] Input:

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

Output:

(a^3*(420*c + 420*d*x - 609*Cos[c + d*x] - 91*Cos[3*(c + d*x)] + 63*Cos[5* 
(c + d*x)] - 3*Cos[7*(c + d*x)] - 63*Sin[2*(c + d*x)] - 105*Sin[4*(c + d*x 
)] + 21*Sin[6*(c + d*x)]))/(1344*d)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 2009}

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.

\(\displaystyle \int \sin ^2(c+d x) \cos ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^2 (a \sin (c+d x)+a)^3dx\)

\(\Big \downarrow \) 3352

\(\displaystyle \int \left (a^3 \sin ^5(c+d x) \cos ^2(c+d x)+3 a^3 \sin ^4(c+d x) \cos ^2(c+d x)+3 a^3 \sin ^3(c+d x) \cos ^2(c+d x)+a^3 \sin ^2(c+d x) \cos ^2(c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^5(c+d x)}{d}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}-\frac {a^3 \sin ^3(c+d x) \cos ^3(c+d x)}{2 d}-\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a^3 x}{16}\)

Input:

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

Output:

(5*a^3*x)/16 - (4*a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/d - (a^ 
3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (5*a^ 
3*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]^3*Sin[c + d*x]^3) 
/(2*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3352

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig 
[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F 
reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 239.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(-\frac {a^{3} \left (-420 d x +91 \cos \left (3 d x +3 c \right )-63 \cos \left (5 d x +5 c \right )-21 \sin \left (6 d x +6 c \right )+105 \sin \left (4 d x +4 c \right )+63 \sin \left (2 d x +2 c \right )+609 \cos \left (d x +c \right )+3 \cos \left (7 d x +7 c \right )+640\right )}{1344 d}\)	\(89\)
risch	\(\frac {5 a^{3} x}{16}-\frac {29 a^{3} \cos \left (d x +c \right )}{64 d}-\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{64 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{64 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {13 a^{3} \cos \left (3 d x +3 c \right )}{192 d}-\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\)	\(124\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{3}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\)	\(194\)
default	\(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{3}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\)	\(194\)
norman	\(\frac {\frac {5 a^{3} x}{16}-\frac {20 a^{3}}{21 d}-\frac {5 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 d}+\frac {119 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 d}-\frac {119 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{8 d}+\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{2 d}+\frac {5 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}+\frac {35 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {105 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {175 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}+\frac {175 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {105 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}+\frac {35 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}+\frac {5 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}-\frac {12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}-\frac {92 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\)	\(358\)
orering	\(\text {Expression too large to display}\)	\(4512\)

Input:

int(cos(d*x+c)^2*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

-1/1344*a^3*(-420*d*x+91*cos(3*d*x+3*c)-63*cos(5*d*x+5*c)-21*sin(6*d*x+6*c 
)+105*sin(4*d*x+4*c)+63*sin(2*d*x+2*c)+609*cos(d*x+c)+3*cos(7*d*x+7*c)+640 
)/d
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {48 \, a^{3} \cos \left (d x + c\right )^{7} - 336 \, a^{3} \cos \left (d x + c\right )^{5} + 448 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} d x - 21 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 18 \, a^{3} \cos \left (d x + c\right )^{3} + 5 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{336 \, d} \] Input:

integrate(cos(d*x+c)^2*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

-1/336*(48*a^3*cos(d*x + c)^7 - 336*a^3*cos(d*x + c)^5 + 448*a^3*cos(d*x + 
 c)^3 - 105*a^3*d*x - 21*(8*a^3*cos(d*x + c)^5 - 18*a^3*cos(d*x + c)^3 + 5 
*a^3*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (122) = 244\).

Time = 0.58 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.87 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {8 a^{3} \cos ^{7}{\left (c + d x \right )}}{105 d} - \frac {2 a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((3*a**3*x*sin(c + d*x)**6/16 + 9*a**3*x*sin(c + d*x)**4*cos(c + 
d*x)**2/16 + a**3*x*sin(c + d*x)**4/8 + 9*a**3*x*sin(c + d*x)**2*cos(c + d 
*x)**4/16 + a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*a**3*x*cos(c + d* 
x)**6/16 + a**3*x*cos(c + d*x)**4/8 + 3*a**3*sin(c + d*x)**5*cos(c + d*x)/ 
(16*d) - a**3*sin(c + d*x)**4*cos(c + d*x)**3/(3*d) - a**3*sin(c + d*x)**3 
*cos(c + d*x)**3/(2*d) + a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) - 4*a**3* 
sin(c + d*x)**2*cos(c + d*x)**5/(15*d) - a**3*sin(c + d*x)**2*cos(c + d*x) 
**3/d - 3*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) - a**3*sin(c + d*x)*cos 
(c + d*x)**3/(8*d) - 8*a**3*cos(c + d*x)**7/(105*d) - 2*a**3*cos(c + d*x)* 
*5/(5*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**2*cos(c)**2, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {64 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{3} - 1344 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{3} + 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 210 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{6720 \, d} \] Input:

integrate(cos(d*x+c)^2*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

Output:

-1/6720*(64*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^ 
3 - 1344*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^3 + 105*(4*sin(2*d*x + 2* 
c)^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^3 - 210*(4*d*x + 4*c - sin(4* 
d*x + 4*c))*a^3)/d
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5}{16} \, a^{3} x - \frac {a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {13 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {29 \, a^{3} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:

integrate(cos(d*x+c)^2*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

5/16*a^3*x - 1/448*a^3*cos(7*d*x + 7*c)/d + 3/64*a^3*cos(5*d*x + 5*c)/d - 
13/192*a^3*cos(3*d*x + 3*c)/d - 29/64*a^3*cos(d*x + c)/d + 1/64*a^3*sin(6* 
d*x + 6*c)/d - 5/64*a^3*sin(4*d*x + 4*c)/d - 3/64*a^3*sin(2*d*x + 2*c)/d
 

Mupad [B] (verification not implemented)

Time = 19.98 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.51 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5\,a^3\,x}{16}-\frac {\frac {5\,a^3\,\left (c+d\,x\right )}{16}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {119\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}+\frac {119\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{2}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {a^3\,\left (105\,c+105\,d\,x-320\right )}{336}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {35\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (735\,c+735\,d\,x-2240\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {105\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (2205\,c+2205\,d\,x-2688\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {175\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (3675\,c+3675\,d\,x-896\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {105\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (2205\,c+2205\,d\,x-4032\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {175\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (3675\,c+3675\,d\,x-10304\right )}{336}\right )+\frac {5\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \] Input:

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

Output:

(5*a^3*x)/16 - ((5*a^3*(c + d*x))/16 + (3*a^3*tan(c/2 + (d*x)/2)^3)/2 - (1 
19*a^3*tan(c/2 + (d*x)/2)^5)/8 + (119*a^3*tan(c/2 + (d*x)/2)^9)/8 - (3*a^3 
*tan(c/2 + (d*x)/2)^11)/2 - (5*a^3*tan(c/2 + (d*x)/2)^13)/8 - (a^3*(105*c 
+ 105*d*x - 320))/336 + tan(c/2 + (d*x)/2)^2*((35*a^3*(c + d*x))/16 - (a^3 
*(735*c + 735*d*x - 2240))/336) + tan(c/2 + (d*x)/2)^4*((105*a^3*(c + d*x) 
)/16 - (a^3*(2205*c + 2205*d*x - 2688))/336) + tan(c/2 + (d*x)/2)^6*((175* 
a^3*(c + d*x))/16 - (a^3*(3675*c + 3675*d*x - 896))/336) + tan(c/2 + (d*x) 
/2)^10*((105*a^3*(c + d*x))/16 - (a^3*(2205*c + 2205*d*x - 4032))/336) + t 
an(c/2 + (d*x)/2)^8*((175*a^3*(c + d*x))/16 - (a^3*(3675*c + 3675*d*x - 10 
304))/336) + (5*a^3*tan(c/2 + (d*x)/2))/8)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^7 
)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.88 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+42 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )-160 \cos \left (d x +c \right )+105 d x +160\right )}{336 d} \] Input:

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

Output:

(a**3*(48*cos(c + d*x)*sin(c + d*x)**6 + 168*cos(c + d*x)*sin(c + d*x)**5 
+ 192*cos(c + d*x)*sin(c + d*x)**4 + 42*cos(c + d*x)*sin(c + d*x)**3 - 80* 
cos(c + d*x)*sin(c + d*x)**2 - 105*cos(c + d*x)*sin(c + d*x) - 160*cos(c + 
 d*x) + 105*d*x + 160))/(336*d)