\(\int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [298]

Optimal result

Integrand size = 29, antiderivative size = 87 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 x}{8 a}-\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(87)=174\).

Time = 1.45 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.11 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {24 (c-3 d x) \cos \left (\frac {c}{2}\right )-72 \cos \left (\frac {c}{2}+d x\right )-72 \cos \left (\frac {3 c}{2}+d x\right )+24 \cos \left (\frac {3 c}{2}+2 d x\right )-24 \cos \left (\frac {5 c}{2}+2 d x\right )+8 \cos \left (\frac {5 c}{2}+3 d x\right )+8 \cos \left (\frac {7 c}{2}+3 d x\right )-3 \cos \left (\frac {7 c}{2}+4 d x\right )+3 \cos \left (\frac {9 c}{2}+4 d x\right )-48 \sin \left (\frac {c}{2}\right )+24 c \sin \left (\frac {c}{2}\right )-72 d x \sin \left (\frac {c}{2}\right )+72 \sin \left (\frac {c}{2}+d x\right )-72 \sin \left (\frac {3 c}{2}+d x\right )+24 \sin \left (\frac {3 c}{2}+2 d x\right )+24 \sin \left (\frac {5 c}{2}+2 d x\right )-8 \sin \left (\frac {5 c}{2}+3 d x\right )+8 \sin \left (\frac {7 c}{2}+3 d x\right )-3 \sin \left (\frac {7 c}{2}+4 d x\right )-3 \sin \left (\frac {9 c}{2}+4 d x\right )}{192 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:

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

Output:

(24*(c - 3*d*x)*Cos[c/2] - 72*Cos[c/2 + d*x] - 72*Cos[(3*c)/2 + d*x] + 24* 
Cos[(3*c)/2 + 2*d*x] - 24*Cos[(5*c)/2 + 2*d*x] + 8*Cos[(5*c)/2 + 3*d*x] + 
8*Cos[(7*c)/2 + 3*d*x] - 3*Cos[(7*c)/2 + 4*d*x] + 3*Cos[(9*c)/2 + 4*d*x] - 
 48*Sin[c/2] + 24*c*Sin[c/2] - 72*d*x*Sin[c/2] + 72*Sin[c/2 + d*x] - 72*Si 
n[(3*c)/2 + d*x] + 24*Sin[(3*c)/2 + 2*d*x] + 24*Sin[(5*c)/2 + 2*d*x] - 8*S 
in[(5*c)/2 + 3*d*x] + 8*Sin[(7*c)/2 + 3*d*x] - 3*Sin[(7*c)/2 + 4*d*x] - 3* 
Sin[(9*c)/2 + 4*d*x])/(192*a*d*(Cos[c/2] + Sin[c/2]))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3042, 3318, 3042, 3113, 2009, 3115, 3042, 3115, 24}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3113

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\int \sin (c+d x)^4dx}{a}-\frac {\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)}{a d}\)

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

Input:

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

Output:

-((Cos[c + d*x] - Cos[c + d*x]^3/3)/(a*d)) - (-1/4*(Cos[c + d*x]*Sin[c + d 
*x]^3)/d + (3*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4)/a
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3318

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {-36 d x -72 \cos \left (d x +c \right )-3 \sin \left (4 d x +4 c \right )+8 \cos \left (3 d x +3 c \right )+24 \sin \left (2 d x +2 c \right )-64}{96 d a}\)	\(56\)
risch	\(-\frac {3 x}{8 a}-\frac {3 \cos \left (d x +c \right )}{4 a d}-\frac {\sin \left (4 d x +4 c \right )}{32 d a}+\frac {\cos \left (3 d x +3 c \right )}{12 a d}+\frac {\sin \left (2 d x +2 c \right )}{4 d a}\)	\(73\)
derivativedivides	\(\frac {\frac {16 \left (-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{64}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{64}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {1}{12}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\)	\(116\)
default	\(\frac {\frac {16 \left (-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{64}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{64}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {1}{12}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\)	\(116\)
norman	\(\frac {\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d a}-\frac {3 x}{8 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{8 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 a}-\frac {7}{12 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4 d a}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 d a}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{6 d a}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 d a}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(399\)

Input:

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

Output:

1/96*(-36*d*x-72*cos(d*x+c)-3*sin(4*d*x+4*c)+8*cos(3*d*x+3*c)+24*sin(2*d*x 
+2*c)-64)/d/a
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {8 \, \cos \left (d x + c\right )^{3} - 9 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 24 \, \cos \left (d x + c\right )}{24 \, a d} \] Input:

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

Output:

1/24*(8*cos(d*x + c)^3 - 9*d*x - 3*(2*cos(d*x + c)^3 - 5*cos(d*x + c))*sin 
(d*x + c) - 24*cos(d*x + c))/(a*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1049 vs. \(2 (70) = 140\).

Time = 7.32 (sec) , antiderivative size = 1049, normalized size of antiderivative = 12.06 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((-9*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d 
*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/ 
2)**2 + 24*a*d) - 36*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/2)**8 + 
 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 
 + d*x/2)**2 + 24*a*d) - 54*d*x*tan(c/2 + d*x/2)**4/(24*a*d*tan(c/2 + d*x/ 
2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d* 
tan(c/2 + d*x/2)**2 + 24*a*d) - 36*d*x*tan(c/2 + d*x/2)**2/(24*a*d*tan(c/2 
 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 
96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 9*d*x/(24*a*d*tan(c/2 + d*x/2)**8 + 
 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 
 + d*x/2)**2 + 24*a*d) - 18*tan(c/2 + d*x/2)**7/(24*a*d*tan(c/2 + d*x/2)** 
8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan( 
c/2 + d*x/2)**2 + 24*a*d) - 66*tan(c/2 + d*x/2)**5/(24*a*d*tan(c/2 + d*x/2 
)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*t 
an(c/2 + d*x/2)**2 + 24*a*d) - 96*tan(c/2 + d*x/2)**4/(24*a*d*tan(c/2 + d* 
x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a* 
d*tan(c/2 + d*x/2)**2 + 24*a*d) + 66*tan(c/2 + d*x/2)**3/(24*a*d*tan(c/2 + 
 d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96 
*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 128*tan(c/2 + d*x/2)**2/(24*a*d*tan(c 
/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (79) = 158\).

Time = 0.11 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {64 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {33 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {33 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {9 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 16}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \] Input:

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

Output:

1/12*((9*sin(d*x + c)/(cos(d*x + c) + 1) - 64*sin(d*x + c)^2/(cos(d*x + c) 
 + 1)^2 + 33*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 48*sin(d*x + c)^4/(cos( 
d*x + c) + 1)^4 - 33*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 9*sin(d*x + c)^ 
7/(cos(d*x + c) + 1)^7 - 16)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 
+ 6*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + 
c) + 1)^6 + a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 9*arctan(sin(d*x + c) 
/(cos(d*x + c) + 1))/a)/d
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {9 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 64 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \] Input:

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

Output:

-1/24*(9*(d*x + c)/a + 2*(9*tan(1/2*d*x + 1/2*c)^7 + 33*tan(1/2*d*x + 1/2* 
c)^5 + 48*tan(1/2*d*x + 1/2*c)^4 - 33*tan(1/2*d*x + 1/2*c)^3 + 64*tan(1/2* 
d*x + 1/2*c)^2 - 9*tan(1/2*d*x + 1/2*c) + 16)/((tan(1/2*d*x + 1/2*c)^2 + 1 
)^4*a))/d
 

Mupad [B] (verification not implemented)

Time = 17.64 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^3}{3\,a\,d}-\frac {\cos \left (c+d\,x\right )}{a\,d}-\frac {3\,x}{8\,a}-\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a\,d}+\frac {5\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a\,d} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+9 \cos \left (d x +c \right ) \sin \left (d x +c \right )-16 \cos \left (d x +c \right )-9 d x +16}{24 a d} \] Input:

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

Output:

(6*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c + d*x)*sin(c + d*x)**2 + 9*cos(c 
 + d*x)*sin(c + d*x) - 16*cos(c + d*x) - 9*d*x + 16)/(24*a*d)