\(\int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx\) [6]

Optimal result

Integrand size = 33, antiderivative size = 362 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=-\frac {n \left (B \left (3-n-4 n^2\right )+A \left (2-9 n+4 n^2\right )\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{15 a^3 d f (1+n) \sqrt {\cos ^2(e+f x)}}+\frac {(1-n) (1+n) (7 A+3 B-4 A n+4 B n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{15 a^3 d^2 f (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{5 d f (a+a \sin (e+f x))^3}+\frac {(A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{1+n}}{15 a d f (a+a \sin (e+f x))^2}+\frac {(1-n) (7 A+3 B-4 A n+4 B n) \cos (e+f x) (d \sin (e+f x))^{1+n}}{15 d f \left (a^3+a^3 \sin (e+f x)\right )} \] Output:

-1/15*n*(B*(-4*n^2-n+3)+A*(4*n^2-9*n+2))*cos(f*x+e)*hypergeom([1/2, 1/2+1/ 
2*n],[3/2+1/2*n],sin(f*x+e)^2)*(d*sin(f*x+e))^(1+n)/a^3/d/f/(1+n)/(cos(f*x 
+e)^2)^(1/2)+1/15*(1-n)*(1+n)*(-4*A*n+4*B*n+7*A+3*B)*cos(f*x+e)*hypergeom( 
[1/2, 1+1/2*n],[2+1/2*n],sin(f*x+e)^2)*(d*sin(f*x+e))^(2+n)/a^3/d^2/f/(2+n 
)/(cos(f*x+e)^2)^(1/2)+1/5*(A-B)*cos(f*x+e)*(d*sin(f*x+e))^(1+n)/d/f/(a+a* 
sin(f*x+e))^3+1/15*(A*(5-2*n)+2*B*n)*cos(f*x+e)*(d*sin(f*x+e))^(1+n)/a/d/f 
/(a+a*sin(f*x+e))^2+1/15*(1-n)*(-4*A*n+4*B*n+7*A+3*B)*cos(f*x+e)*(d*sin(f* 
x+e))^(1+n)/d/f/(a^3+a^3*sin(f*x+e))
 

Mathematica [A] (verified)

Time = 6.20 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.18 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\frac {(a A-a B) \cos (e+f x) \sin (e+f x) (d \sin (e+f x))^n}{5 a f (a+a \sin (e+f x))^3}+\frac {\sin ^{-n}(e+f x) (d \sin (e+f x))^n \left (\frac {\left (a^2 (A-B) (1-n)+a^2 (4 A+B-A n+B n)\right ) \cos (e+f x) \sin ^{1+n}(e+f x)}{3 a f (a+a \sin (e+f x))^2}+\frac {\frac {\left (-a^3 n (A (5-2 n)+2 B n)+a^3 \left (B \left (3+n-2 n^2\right )+A \left (7-6 n+2 n^2\right )\right )\right ) \cos (e+f x) \sin ^{1+n}(e+f x)}{a f (a+a \sin (e+f x))}+\frac {-\frac {a^3 n \left (B \left (3-n-4 n^2\right )+A \left (2-9 n+4 n^2\right )\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) \sin ^{1+n}(e+f x)}{f (1+n)}+\frac {a^3 (1-n) (1+n) (7 A+3 B-4 A n+4 B n) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) \sin ^{2+n}(e+f x)}{f (2+n)}}{a^2}}{3 a^2}\right )}{5 a^2} \] Input:

Integrate[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/(a + a*Sin[e + f*x])^3 
,x]
 

Output:

((a*A - a*B)*Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n)/(5*a*f*(a + a*S 
in[e + f*x])^3) + ((d*Sin[e + f*x])^n*(((a^2*(A - B)*(1 - n) + a^2*(4*A + 
B - A*n + B*n))*Cos[e + f*x]*Sin[e + f*x]^(1 + n))/(3*a*f*(a + a*Sin[e + f 
*x])^2) + (((-(a^3*n*(A*(5 - 2*n) + 2*B*n)) + a^3*(B*(3 + n - 2*n^2) + A*( 
7 - 6*n + 2*n^2)))*Cos[e + f*x]*Sin[e + f*x]^(1 + n))/(a*f*(a + a*Sin[e + 
f*x])) + (-((a^3*n*(B*(3 - n - 4*n^2) + A*(2 - 9*n + 4*n^2))*Sqrt[Cos[e + 
f*x]^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e 
 + f*x]*Sin[e + f*x]^(1 + n))/(f*(1 + n))) + (a^3*(1 - n)*(1 + n)*(7*A + 3 
*B - 4*A*n + 4*B*n)*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, (2 + n)/2, 
 (4 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*Sin[e + f*x]^(2 + n))/(f*(2 + n)) 
)/a^2)/(3*a^2)))/(5*a^2*Sin[e + f*x]^n)
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3457, 3042, 3457, 3042, 3457, 25, 3042, 3227, 3042, 3122}

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.

\(\displaystyle \int \frac {(A+B \sin (e+f x)) (d \sin (e+f x))^n}{(a \sin (e+f x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(A+B \sin (e+f x)) (d \sin (e+f x))^n}{(a \sin (e+f x)+a)^3}dx\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\int \frac {(d \sin (e+f x))^n (a d (-n A+4 A+B+B n)-a (A-B) d (1-n) \sin (e+f x))}{(\sin (e+f x) a+a)^2}dx}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(d \sin (e+f x))^n (a d (-n A+4 A+B+B n)-a (A-B) d (1-n) \sin (e+f x))}{(\sin (e+f x) a+a)^2}dx}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {\int \frac {(d \sin (e+f x))^n \left (a^2 \left (B \left (-2 n^2+n+3\right )+A \left (2 n^2-6 n+7\right )\right ) d^2+a^2 n (A (5-2 n)+2 B n) \sin (e+f x) d^2\right )}{\sin (e+f x) a+a}dx}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {(d \sin (e+f x))^n \left (a^2 \left (B \left (-2 n^2+n+3\right )+A \left (2 n^2-6 n+7\right )\right ) d^2+a^2 n (A (5-2 n)+2 B n) \sin (e+f x) d^2\right )}{\sin (e+f x) a+a}dx}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {\frac {\int -(d \sin (e+f x))^n \left (a^3 d^3 n \left (B \left (-4 n^2-n+3\right )+A \left (4 n^2-9 n+2\right )\right )-a^3 d^3 (1-n) (n+1) (-4 n A+7 A+3 B+4 B n) \sin (e+f x)\right )dx}{a^2 d}+\frac {a^2 d (1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (a \sin (e+f x)+a)}}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {a^2 d (1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (a \sin (e+f x)+a)}-\frac {\int (d \sin (e+f x))^n \left (a^3 d^3 n \left (B \left (-4 n^2-n+3\right )+A \left (4 n^2-9 n+2\right )\right )-a^3 d^3 (1-n) (n+1) (-4 n A+7 A+3 B+4 B n) \sin (e+f x)\right )dx}{a^2 d}}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {a^2 d (1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (a \sin (e+f x)+a)}-\frac {\int (d \sin (e+f x))^n \left (a^3 d^3 n \left (B \left (-4 n^2-n+3\right )+A \left (4 n^2-9 n+2\right )\right )-a^3 d^3 (1-n) (n+1) (-4 n A+7 A+3 B+4 B n) \sin (e+f x)\right )dx}{a^2 d}}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {\frac {a^2 d (1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (a \sin (e+f x)+a)}-\frac {a^3 d^3 n \left (A \left (4 n^2-9 n+2\right )+B \left (-4 n^2-n+3\right )\right ) \int (d \sin (e+f x))^ndx-a^3 d^2 (1-n) (n+1) (-4 A n+7 A+4 B n+3 B) \int (d \sin (e+f x))^{n+1}dx}{a^2 d}}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {a^2 d (1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (a \sin (e+f x)+a)}-\frac {a^3 d^3 n \left (A \left (4 n^2-9 n+2\right )+B \left (-4 n^2-n+3\right )\right ) \int (d \sin (e+f x))^ndx-a^3 d^2 (1-n) (n+1) (-4 A n+7 A+4 B n+3 B) \int (d \sin (e+f x))^{n+1}dx}{a^2 d}}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {\frac {\frac {a^2 d (1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (a \sin (e+f x)+a)}-\frac {\frac {a^3 d^2 n \left (A \left (4 n^2-9 n+2\right )+B \left (-4 n^2-n+3\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{f (n+1) \sqrt {\cos ^2(e+f x)}}-\frac {a^3 d (1-n) (n+1) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(e+f x)\right )}{f (n+2) \sqrt {\cos ^2(e+f x)}}}{a^2 d}}{3 a^2 d}+\frac {a (A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 f (a \sin (e+f x)+a)^2}}{5 a^2 d}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}\)

Input:

Int[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/(a + a*Sin[e + f*x])^3,x]
 

Output:

((A - B)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(5*d*f*(a + a*Sin[e + f*x] 
)^3) + ((a*(A*(5 - 2*n) + 2*B*n)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(3 
*f*(a + a*Sin[e + f*x])^2) + ((a^2*d*(1 - n)*(7*A + 3*B - 4*A*n + 4*B*n)*C 
os[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(f*(a + a*Sin[e + f*x])) - ((a^3*d^2 
*n*(B*(3 - n - 4*n^2) + A*(2 - 9*n + 4*n^2))*Cos[e + f*x]*Hypergeometric2F 
1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(f* 
(1 + n)*Sqrt[Cos[e + f*x]^2]) - (a^3*d*(1 - n)*(1 + n)*(7*A + 3*B - 4*A*n 
+ 4*B*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + 
 f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(f*(2 + n)*Sqrt[Cos[e + f*x]^2]))/(a^2* 
d))/(3*a^2*d))/(5*a^2*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3122

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3227

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
 

rule 3457

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( 
n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b 
*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 
2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ 
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] 
 &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

Maple [F]

\[\int \frac {\left (d \sin \left (f x +e \right )\right )^{n} \left (A +B \sin \left (f x +e \right )\right )}{\left (a +a \sin \left (f x +e \right )\right )^{3}}d x\]

Input:

int((d*sin(f*x+e))^n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3,x)
 

Output:

int((d*sin(f*x+e))^n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3,x)
 

Fricas [F]

\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((d*sin(f*x+e))^n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorith 
m="fricas")
 

Output:

integral(-(B*sin(f*x + e) + A)*(d*sin(f*x + e))^n/(3*a^3*cos(f*x + e)^2 - 
4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((d*sin(f*x+e))**n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((d*sin(f*x+e))^n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorith 
m="maxima")
 

Output:

integrate((B*sin(f*x + e) + A)*(d*sin(f*x + e))^n/(a*sin(f*x + e) + a)^3, 
x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*sin(f*x+e))^n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorith 
m="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,0]%%%} / %%%{1,[0,0,3]%%%} Error: Bad Argument Valu 
e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:

int(((d*sin(e + f*x))^n*(A + B*sin(e + f*x)))/(a + a*sin(e + f*x))^3,x)
 

Output:

int(((d*sin(e + f*x))^n*(A + B*sin(e + f*x)))/(a + a*sin(e + f*x))^3, x)
 

Reduce [F]

\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx=\frac {d^{n} \left (\left (\int \frac {\sin \left (f x +e \right )^{n}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) a +\left (\int \frac {\sin \left (f x +e \right )^{n} \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) b \right )}{a^{3}} \] Input:

int((d*sin(f*x+e))^n*(A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3,x)
 

Output:

(d**n*(int(sin(e + f*x)**n/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e 
+ f*x) + 1),x)*a + int((sin(e + f*x)**n*sin(e + f*x))/(sin(e + f*x)**3 + 3 
*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*b))/a**3