Integrand size = 36, antiderivative size = 145 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {2^{\frac {1}{2}+m} a^3 c^2 (B (2-m)-A (3+m)) \cos ^5(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-3+m}}{5 f (3+m)}-\frac {a^2 B c^2 \cos ^5(e+f x) (a+a \sin (e+f x))^{-2+m}}{f (3+m)} \]
1/5*2^(1/2+m)*a^3*c^2*(B*(2-m)-A*(3+m))*cos(f*x+e)^5*hypergeom([5/2, 1/2-m ],[7/2],1/2-1/2*sin(f*x+e))*(1+sin(f*x+e))^(1/2-m)*(a+a*sin(f*x+e))^(-3+m) /f/(3+m)-a^2*B*c^2*cos(f*x+e)^5*(a+a*sin(f*x+e))^(-2+m)/f/(3+m)
Result contains complex when optimal does not.
Time = 13.09 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.06 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {i c^2 (a (1+\sin (e+f x)))^m (\cos (e+f x)+i (1+\sin (e+f x))) \left (-\frac {4 i (3 A-2 B) \operatorname {Hypergeometric2F1}(1,1+m,1-m,i \cos (e+f x)-\sin (e+f x))}{m}-\frac {(8 A-7 B) \operatorname {Hypergeometric2F1}(1,m,-m,i \cos (e+f x)-\sin (e+f x)) (\cos (e+f x)-i \sin (e+f x))}{1+m}+\frac {(8 A-7 B) \operatorname {Hypergeometric2F1}(1,2+m,2-m,i \cos (e+f x)-\sin (e+f x)) (\cos (e+f x)+i \sin (e+f x))}{-1+m}+\frac {2 i (A-2 B) \operatorname {Hypergeometric2F1}(1,3+m,3-m,i \cos (e+f x)-\sin (e+f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))}{-2+m}+\frac {2 (A-2 B) \operatorname {Hypergeometric2F1}(1,-1+m,-1-m,i \cos (e+f x)-\sin (e+f x)) (i \cos (2 (e+f x))+\sin (2 (e+f x)))}{2+m}-\frac {B \operatorname {Hypergeometric2F1}(1,-2+m,-2-m,i \cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))-i \sin (3 (e+f x)))}{3+m}+\frac {B \operatorname {Hypergeometric2F1}(1,4+m,4-m,i \cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))+i \sin (3 (e+f x)))}{-3+m}\right )}{8 f} \]
((I/8)*c^2*(a*(1 + Sin[e + f*x]))^m*(Cos[e + f*x] + I*(1 + Sin[e + f*x]))* (((-4*I)*(3*A - 2*B)*Hypergeometric2F1[1, 1 + m, 1 - m, I*Cos[e + f*x] - S in[e + f*x]])/m - ((8*A - 7*B)*Hypergeometric2F1[1, m, -m, I*Cos[e + f*x] - Sin[e + f*x]]*(Cos[e + f*x] - I*Sin[e + f*x]))/(1 + m) + ((8*A - 7*B)*Hy pergeometric2F1[1, 2 + m, 2 - m, I*Cos[e + f*x] - Sin[e + f*x]]*(Cos[e + f *x] + I*Sin[e + f*x]))/(-1 + m) + ((2*I)*(A - 2*B)*Hypergeometric2F1[1, 3 + m, 3 - m, I*Cos[e + f*x] - Sin[e + f*x]]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]))/(-2 + m) + (2*(A - 2*B)*Hypergeometric2F1[1, -1 + m, -1 - m, I*C os[e + f*x] - Sin[e + f*x]]*(I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]))/(2 + m) - (B*Hypergeometric2F1[1, -2 + m, -2 - m, I*Cos[e + f*x] - Sin[e + f*x] ]*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)]))/(3 + m) + (B*Hypergeometric2F1[ 1, 4 + m, 4 - m, I*Cos[e + f*x] - Sin[e + f*x]]*(Cos[3*(e + f*x)] + I*Sin[ 3*(e + f*x)]))/(-3 + m)))/f
Time = 0.63 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 3446, 3042, 3339, 3042, 3168, 80, 79}
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.
| \(\displaystyle \int (c-c \sin (e+f x))^2 (a \sin (e+f x)+a)^m (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c-c \sin (e+f x))^2 (a \sin (e+f x)+a)^m (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^2 c^2 \int \cos ^4(e+f x) (\sin (e+f x) a+a)^{m-2} (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \cos (e+f x)^4 (\sin (e+f x) a+a)^{m-2} (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle a^2 c^2 \left (\left (A-\frac {B (2-m)}{m+3}\right ) \int \cos ^4(e+f x) (\sin (e+f x) a+a)^{m-2}dx-\frac {B \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\left (A-\frac {B (2-m)}{m+3}\right ) \int \cos (e+f x)^4 (\sin (e+f x) a+a)^{m-2}dx-\frac {B \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)}\right )\) |
| \(\Big \downarrow \) 3168 |
| \(\displaystyle a^2 c^2 \left (\frac {a^2 \left (A-\frac {B (2-m)}{m+3}\right ) \cos ^5(e+f x) \int (a-a \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^{m-\frac {1}{2}}d\sin (e+f x)}{f (a-a \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{5/2}}-\frac {B \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)}\right )\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle a^2 c^2 \left (\frac {a^2 2^{m-\frac {1}{2}} \left (A-\frac {B (2-m)}{m+3}\right ) \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-3} \int \left (\frac {1}{2} \sin (e+f x)+\frac {1}{2}\right )^{m-\frac {1}{2}} (a-a \sin (e+f x))^{3/2}d\sin (e+f x)}{f (a-a \sin (e+f x))^{5/2}}-\frac {B \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)}\right )\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle a^2 c^2 \left (-\frac {a 2^{m+\frac {1}{2}} \left (A-\frac {B (2-m)}{m+3}\right ) \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-3} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{5 f}-\frac {B \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)}\right )\) |
a^2*c^2*(-1/5*(2^(1/2 + m)*a*(A - (B*(2 - m))/(3 + m))*Cos[e + f*x]^5*Hype rgeometric2F1[5/2, 1/2 - m, 7/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^ (1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/f - (B*Cos[e + f*x]^5*(a + a*Sin[ e + f*x])^(-2 + m))/(f*(3 + m)))
3.2.97.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin [e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))) Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
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] && NeQ[m + p + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{2}d x\]
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
integral(-((A - 2*B)*c^2*cos(f*x + e)^2 - 2*(A - B)*c^2 + (B*c^2*cos(f*x + e)^2 + 2*(A - B)*c^2)*sin(f*x + e))*(a*sin(f*x + e) + a)^m, x)
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=c^{2} \left (\int A \left (a \sin {\left (e + f x \right )} + a\right )^{m}\, dx + \int \left (- 2 A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\right )\, dx + \int A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx + \int B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
c**2*(Integral(A*(a*sin(e + f*x) + a)**m, x) + Integral(-2*A*(a*sin(e + f* x) + a)**m*sin(e + f*x), x) + Integral(A*(a*sin(e + f*x) + a)**m*sin(e + f *x)**2, x) + Integral(B*(a*sin(e + f*x) + a)**m*sin(e + f*x), x) + Integra l(-2*B*(a*sin(e + f*x) + a)**m*sin(e + f*x)**2, x) + Integral(B*(a*sin(e + f*x) + a)**m*sin(e + f*x)**3, x))
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]