\(\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx\) [211]

Optimal result

Integrand size = 40, antiderivative size = 267 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac {(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac {2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c^2 f (7+2 m) \left (15+16 m+4 m^2\right )}+\frac {2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^3 f (5+2 m) (7+2 m) \left (3+8 m+4 m^2\right )} \]

[Out]

(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-4-m)/f/(7+2*m)+(3*A-2*B*(2+m))*cos(f*x+e)*(a+a*sin(f*x+
e))^m*(c-c*sin(f*x+e))^(-3-m)/c/f/(4*m^2+24*m+35)+2*(3*A-2*B*(2+m))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x
+e))^(-2-m)/c^2/f/(8*m^3+60*m^2+142*m+105)+2*(3*A-2*B*(2+m))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-
1-m)/c^3/f/(16*m^4+128*m^3+344*m^2+352*m+105)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {3051, 2822, 2821} \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\frac {2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^3 f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right )}+\frac {2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c^2 f (2 m+7) \left (4 m^2+16 m+15\right )}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4}}{f (2 m+7)}+\frac {(3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{c f (2 m+5) (2 m+7)} \]

[In]

Int[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - m),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m))/(f*(7 + 2*m)) + ((3*A - 2*B*(2 + m
))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(c*f*(5 + 2*m)*(7 + 2*m)) + (2*(3*A - 2*
B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c^2*f*(7 + 2*m)*(15 + 16*m + 4*
m^2)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^3*f*(5 +
2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2))

Rule 2821

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

Rule 2822

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] + Dist[(m + n + 1)/(a*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m
, 1] ||  !SumSimplerQ[n, 1])

Rule 3051

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

Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac {(3 A-2 B (2+m)) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx}{c (7+2 m)} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac {(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac {(2 (3 A-2 B (2+m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx}{c^2 (5+2 m) (7+2 m)} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac {(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac {2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c^2 f (3+2 m) (5+2 m) (7+2 m)}+\frac {(2 (3 A-2 B (2+m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx}{c^3 (3+2 m) (5+2 m) (7+2 m)} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac {(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac {2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c^2 f (3+2 m) (5+2 m) (7+2 m)}+\frac {2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^3 f (1+2 m) (3+2 m) (5+2 m) (7+2 m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.69 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\frac {\sec (e+f x) (a (1+\sin (e+f x)))^{1+m} (c-c \sin (e+f x))^{-m} \left (B \left (13+16 m+4 m^2\right )-2 A \left (18+41 m+24 m^2+4 m^3\right )+\left (13+16 m+4 m^2\right ) (3 A-2 B (2+m)) \sin (e+f x)+4 (2+m) (-3 A+2 B (2+m)) \sin ^2(e+f x)+(6 A-4 B (2+m)) \sin ^3(e+f x)\right )}{a c^4 f (1+2 m) (3+2 m) (5+2 m) (7+2 m) (-1+\sin (e+f x))^3} \]

[In]

Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - m),x]

[Out]

(Sec[e + f*x]*(a*(1 + Sin[e + f*x]))^(1 + m)*(B*(13 + 16*m + 4*m^2) - 2*A*(18 + 41*m + 24*m^2 + 4*m^3) + (13 +
16*m + 4*m^2)*(3*A - 2*B*(2 + m))*Sin[e + f*x] + 4*(2 + m)*(-3*A + 2*B*(2 + m))*Sin[e + f*x]^2 + (6*A - 4*B*(
2 + m))*Sin[e + f*x]^3))/(a*c^4*f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(-1 + Sin[e + f*x])^3*(c - c*Sin[e +
f*x])^m)

Maple [F]

\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{-4-m}d x\]

[In]

int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x)

[Out]

int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.77 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\frac {{\left (4 \, {\left (2 \, B m^{2} - {\left (3 \, A - 8 \, B\right )} m - 6 \, A + 8 \, B\right )} \cos \left (f x + e\right )^{3} + {\left (8 \, A m^{3} + 12 \, {\left (4 \, A - B\right )} m^{2} + 2 \, {\left (47 \, A - 24 \, B\right )} m + 60 \, A - 45 \, B\right )} \cos \left (f x + e\right ) - {\left (2 \, {\left (2 \, B m - 3 \, A + 4 \, B\right )} \cos \left (f x + e\right )^{3} - {\left (8 \, B m^{3} - 12 \, {\left (A - 4 \, B\right )} m^{2} - 2 \, {\left (24 \, A - 47 \, B\right )} m - 45 \, A + 60 \, B\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 4}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x, algorithm="fricas")

[Out]

(4*(2*B*m^2 - (3*A - 8*B)*m - 6*A + 8*B)*cos(f*x + e)^3 + (8*A*m^3 + 12*(4*A - B)*m^2 + 2*(47*A - 24*B)*m + 60
*A - 45*B)*cos(f*x + e) - (2*(2*B*m - 3*A + 4*B)*cos(f*x + e)^3 - (8*B*m^3 - 12*(A - 4*B)*m^2 - 2*(24*A - 47*B
)*m - 45*A + 60*B)*cos(f*x + e))*sin(f*x + e))*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 4)/(16*f*m^4
+ 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)

Sympy [F]

\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{- m - 4} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]

[In]

integrate((a+a*sin(f*x+e))**m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(-4-m),x)

[Out]

Integral((a*(sin(e + f*x) + 1))**m*(-c*(sin(e + f*x) - 1))**(-m - 4)*(A + B*sin(e + f*x)), x)

Maxima [F]

\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 4} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 4), x)

Giac [F]

\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 4} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 4), x)

Mupad [B] (verification not implemented)

Time = 20.86 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.38 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx=-\frac {\sin \left (4\,e+4\,f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,B-3\,A+2\,B\,m\right )\,1{}\mathrm {i}}{4\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+4}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}+\frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,168{}\mathrm {i}-B\,84{}\mathrm {i}+A\,m\,340{}\mathrm {i}-B\,m\,96{}\mathrm {i}+A\,m^2\,192{}\mathrm {i}+A\,m^3\,32{}\mathrm {i}-B\,m^2\,24{}\mathrm {i}\right )}{4\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+4}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}+\frac {\sin \left (2\,e+2\,f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (2\,m^2+8\,m+7\right )\,\left (4\,B-3\,A+2\,B\,m\right )\,1{}\mathrm {i}}{f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+4}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}+\frac {\cos \left (3\,e+3\,f\,x\right )\,\left (m+2\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (-A\,3{}\mathrm {i}+B\,4{}\mathrm {i}+B\,m\,2{}\mathrm {i}\right )}{f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+4}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(m + 4),x)

[Out]

(cos(e + f*x)*(a + a*sin(e + f*x))^m*(A*168i - B*84i + A*m*340i - B*m*96i + A*m^2*192i + A*m^3*32i - B*m^2*24i
))/(4*f*(c - c*sin(e + f*x))^(m + 4)*(m*352i + m^2*344i + m^3*128i + m^4*16i + 105i)) - (sin(4*e + 4*f*x)*(a +
a*sin(e + f*x))^m*(4*B - 3*A + 2*B*m)*1i)/(4*f*(c - c*sin(e + f*x))^(m + 4)*(m*352i + m^2*344i + m^3*128i + m
^4*16i + 105i)) + (sin(2*e + 2*f*x)*(a + a*sin(e + f*x))^m*(8*m + 2*m^2 + 7)*(4*B - 3*A + 2*B*m)*1i)/(f*(c - c
*sin(e + f*x))^(m + 4)*(m*352i + m^2*344i + m^3*128i + m^4*16i + 105i)) + (cos(3*e + 3*f*x)*(m + 2)*(a + a*sin
(e + f*x))^m*(B*4i - A*3i + B*m*2i))/(f*(c - c*sin(e + f*x))^(m + 4)*(m*352i + m^2*344i + m^3*128i + m^4*16i +
105i))