Integrand size = 33, antiderivative size = 373 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=-\frac {a^3 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n) (4+n)}+\frac {a^3 \left (B \left (15+19 n+4 n^2\right )+A \left (20+21 n+4 n^2\right )\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) (4+n) \sqrt {\cos ^2(e+f x)}}+\frac {a^3 (B (9+4 n)+A (11+4 n)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) (3+n) \sqrt {\cos ^2(e+f x)}}-\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}-\frac {(A (4+n)+B (6+n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^3+a^3 \sin (e+f x)\right )}{d f (3+n) (4+n)} \] Output:
-a^3*(B*(2*n^2+14*n+27)+A*(2*n^2+15*n+28))*cos(f*x+e)*(d*sin(f*x+e))^(1+n) /d/f/(2+n)/(3+n)/(4+n)+a^3*(B*(4*n^2+19*n+15)+A*(4*n^2+21*n+20))*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 )/d/f/(1+n)/(2+n)/(4+n)/(cos(f*x+e)^2)^(1/2)+a^3*(B*(9+4*n)+A*(11+4*n))*co s(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)/d^2/f/(2+n)/(3+n)/(cos(f*x+e)^2)^(1/2)-a*B*cos(f*x+e)*(d*sin(f*x+e))^ (1+n)*(a+a*sin(f*x+e))^2/d/f/(4+n)-(A*(4+n)+B*(6+n))*cos(f*x+e)*(d*sin(f*x +e))^(1+n)*(a^3+a^3*sin(f*x+e))/d/f/(3+n)/(4+n)
Time = 2.72 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.66 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\frac {a^3 \cos (e+f x) \sin (e+f x) (d \sin (e+f x))^n \left (\frac {A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right )}{1+n}+\sin (e+f x) \left (\frac {(3 A+B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right )}{2+n}+\sin (e+f x) \left (\frac {3 (A+B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(e+f x)\right )}{3+n}+\sin (e+f x) \left (\frac {(A+3 B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+n}{2},\frac {6+n}{2},\sin ^2(e+f x)\right )}{4+n}+\frac {B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+n}{2},\frac {7+n}{2},\sin ^2(e+f x)\right ) \sin (e+f x)}{5+n}\right )\right )\right )\right )}{f \sqrt {\cos ^2(e+f x)}} \] Input:
Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]),x ]
Output:
(a^3*Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n*((A*Hypergeometric2F1[1/ 2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2])/(1 + n) + Sin[e + f*x]*(((3*A + B)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2])/(2 + n) + Sin[e + f*x]*((3*(A + B)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Sin [e + f*x]^2])/(3 + n) + Sin[e + f*x]*(((A + 3*B)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Sin[e + f*x]^2])/(4 + n) + (B*Hypergeometric2F1[1/2, ( 5 + n)/2, (7 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/(5 + n))))))/(f*Sqrt[Co s[e + f*x]^2])
Time = 1.72 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3455, 3042, 3455, 3042, 3447, 3042, 3502, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (A+B \sin (e+f x)) (d \sin (e+f x))^n \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (A+B \sin (e+f x)) (d \sin (e+f x))^ndx\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\int (d \sin (e+f x))^n (\sin (e+f x) a+a)^2 (a d (B (n+1)+A (n+4))+a d (A (n+4)+B (n+6)) \sin (e+f x))dx}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (d \sin (e+f x))^n (\sin (e+f x) a+a)^2 (a d (B (n+1)+A (n+4))+a d (A (n+4)+B (n+6)) \sin (e+f x))dx}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {\int (d \sin (e+f x))^n (\sin (e+f x) a+a) \left (a^2 \left (2 A \left (n^2+6 n+8\right )+B \left (2 n^2+11 n+9\right )\right ) d^2+a^2 \left (B \left (2 n^2+14 n+27\right )+A \left (2 n^2+15 n+28\right )\right ) \sin (e+f x) d^2\right )dx}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int (d \sin (e+f x))^n (\sin (e+f x) a+a) \left (a^2 \left (2 A \left (n^2+6 n+8\right )+B \left (2 n^2+11 n+9\right )\right ) d^2+a^2 \left (B \left (2 n^2+14 n+27\right )+A \left (2 n^2+15 n+28\right )\right ) \sin (e+f x) d^2\right )dx}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\int (d \sin (e+f x))^n \left (d^2 \left (B \left (2 n^2+14 n+27\right )+A \left (2 n^2+15 n+28\right )\right ) \sin ^2(e+f x) a^3+d^2 \left (2 A \left (n^2+6 n+8\right )+B \left (2 n^2+11 n+9\right )\right ) a^3+\left (d^2 \left (2 A \left (n^2+6 n+8\right )+B \left (2 n^2+11 n+9\right )\right ) a^3+d^2 \left (B \left (2 n^2+14 n+27\right )+A \left (2 n^2+15 n+28\right )\right ) a^3\right ) \sin (e+f x)\right )dx}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int (d \sin (e+f x))^n \left (d^2 \left (B \left (2 n^2+14 n+27\right )+A \left (2 n^2+15 n+28\right )\right ) \sin (e+f x)^2 a^3+d^2 \left (2 A \left (n^2+6 n+8\right )+B \left (2 n^2+11 n+9\right )\right ) a^3+\left (d^2 \left (2 A \left (n^2+6 n+8\right )+B \left (2 n^2+11 n+9\right )\right ) a^3+d^2 \left (B \left (2 n^2+14 n+27\right )+A \left (2 n^2+15 n+28\right )\right ) a^3\right ) \sin (e+f x)\right )dx}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {\int (d \sin (e+f x))^n \left (a^3 (n+3) \left (B \left (4 n^2+19 n+15\right )+A \left (4 n^2+21 n+20\right )\right ) d^3+a^3 (n+2) (n+4) (B (4 n+9)+A (4 n+11)) \sin (e+f x) d^3\right )dx}{d (n+2)}-\frac {a^3 d \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (n+2)}}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int (d \sin (e+f x))^n \left (a^3 (n+3) \left (B \left (4 n^2+19 n+15\right )+A \left (4 n^2+21 n+20\right )\right ) d^3+a^3 (n+2) (n+4) (B (4 n+9)+A (4 n+11)) \sin (e+f x) d^3\right )dx}{d (n+2)}-\frac {a^3 d \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (n+2)}}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {a^3 d^3 (n+3) \left (A \left (4 n^2+21 n+20\right )+B \left (4 n^2+19 n+15\right )\right ) \int (d \sin (e+f x))^ndx+a^3 d^2 (n+2) (n+4) (A (4 n+11)+B (4 n+9)) \int (d \sin (e+f x))^{n+1}dx}{d (n+2)}-\frac {a^3 d \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (n+2)}}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^3 d^3 (n+3) \left (A \left (4 n^2+21 n+20\right )+B \left (4 n^2+19 n+15\right )\right ) \int (d \sin (e+f x))^ndx+a^3 d^2 (n+2) (n+4) (A (4 n+11)+B (4 n+9)) \int (d \sin (e+f x))^{n+1}dx}{d (n+2)}-\frac {a^3 d \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (n+2)}}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
| \(\Big \downarrow \) 3122 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 d^2 (n+3) \left (A \left (4 n^2+21 n+20\right )+B \left (4 n^2+19 n+15\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{f (n+1) \sqrt {\cos ^2(e+f x)}}+\frac {a^3 d (n+4) (A (4 n+11)+B (4 n+9)) \cos (e+f x) (d \sin (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(e+f x)\right )}{f \sqrt {\cos ^2(e+f x)}}}{d (n+2)}-\frac {a^3 d \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{f (n+2)}}{d (n+3)}-\frac {(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{f (n+3)}}{d (n+4)}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}\) |
Input:
Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]),x]
Output:
-((a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a + a*Sin[e + f*x])^2)/(d*f* (4 + n))) + (-(((A*(4 + n) + B*(6 + n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a^3 + a^3*Sin[e + f*x]))/(f*(3 + n))) + (-((a^3*d*(B*(27 + 14*n + 2*n ^2) + A*(28 + 15*n + 2*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(f*(2 + n))) + ((a^3*d^2*(3 + n)*(B*(15 + 19*n + 4*n^2) + A*(20 + 21*n + 4*n^2)) *Cos[e + f*x]*Hypergeometric2F1[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*(4 + n)*(B*(9 + 4*n) + A*(11 + 4*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*Sqrt[Cos[e + f*x]^2]))/(d*(2 + n)))/(d*(3 + n)))/(d*(4 + n))
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]
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*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] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
\[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{3} \left (A +B \sin \left (f x +e \right )\right )d x\]
Input:
int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(A+B*sin(f*x+e)),x)
Output:
int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(A+B*sin(f*x+e)),x)
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(A+B*sin(f*x+e)),x, algorith m="fricas")
Output:
integral((B*a^3*cos(f*x + e)^4 - (3*A + 5*B)*a^3*cos(f*x + e)^2 + 4*(A + B )*a^3 - ((A + 3*B)*a^3*cos(f*x + e)^2 - 4*(A + B)*a^3)*sin(f*x + e))*(d*si n(f*x + e))^n, x)
Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\text {Timed out} \] Input:
integrate((d*sin(f*x+e))**n*(a+a*sin(f*x+e))**3*(A+B*sin(f*x+e)),x)
Output:
Timed out
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(A+B*sin(f*x+e)),x, algorith m="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(d*sin(f*x + e))^n, x)
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(A+B*sin(f*x+e)),x, algorith m="giac")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(d*sin(f*x + e))^n, x)
Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3 \,d x \] 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)
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx=d^{n} a^{3} \left (\left (\int \sin \left (f x +e \right )^{n}d x \right ) a +\left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )^{4}d x \right ) b +\left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )^{3}d x \right ) a +3 \left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )^{3}d x \right ) b +3 \left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )^{2}d x \right ) a +3 \left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )^{2}d x \right ) b +3 \left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )d x \right ) a +\left (\int \sin \left (f x +e \right )^{n} \sin \left (f x +e \right )d x \right ) b \right ) \] Input:
int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(A+B*sin(f*x+e)),x)
Output:
d**n*a**3*(int(sin(e + f*x)**n,x)*a + int(sin(e + f*x)**n*sin(e + f*x)**4, x)*b + int(sin(e + f*x)**n*sin(e + f*x)**3,x)*a + 3*int(sin(e + f*x)**n*si n(e + f*x)**3,x)*b + 3*int(sin(e + f*x)**n*sin(e + f*x)**2,x)*a + 3*int(si n(e + f*x)**n*sin(e + f*x)**2,x)*b + 3*int(sin(e + f*x)**n*sin(e + f*x),x) *a + int(sin(e + f*x)**n*sin(e + f*x),x)*b)