\(\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n (A+B \sin (e+f x)+C \sin ^2(e+f x)) \, dx\) [25]

Optimal result

Integrand size = 45, antiderivative size = 383 \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=-\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\sqrt {2} (c (C+2 C m)+d (C (1-m+n)+A (2+m+n)-B (2+m+n))) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},-n,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{d f (1+2 m) (2+m+n) \sqrt {1-\sin (e+f x)}}-\frac {\sqrt {2} (c C (1+m)-d (C m+B (2+m+n))) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},-n,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a d f (3+2 m) (2+m+n) \sqrt {1-\sin (e+f x)}} \]

[Out]

-C*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1+n)/d/f/(2+m+n)+(c*(2*C*m+C)+d*(C*(1-m+n)+A*(2+m+n)-B*(2+m
+n)))*AppellF1(1/2+m,-n,1/2,3/2+m,-d*(1+sin(f*x+e))/(c-d),1/2+1/2*sin(f*x+e))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c
+d*sin(f*x+e))^n*2^(1/2)/d/f/(1+2*m)/(2+m+n)/(((c+d*sin(f*x+e))/(c-d))^n)/(1-sin(f*x+e))^(1/2)-(c*C*(1+m)-d*(C
*m+B*(2+m+n)))*AppellF1(3/2+m,-n,1/2,5/2+m,-d*(1+sin(f*x+e))/(c-d),1/2+1/2*sin(f*x+e))*cos(f*x+e)*(a+a*sin(f*x
+e))^(1+m)*(c+d*sin(f*x+e))^n*2^(1/2)/a/d/f/(3+2*m)/(2+m+n)/(((c+d*sin(f*x+e))/(c-d))^n)/(1-sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3124, 3066, 2867, 145, 144, 143} \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^m (d (A (m+n+2)-B (m+n+2)+C (-m+n+1))+c (2 C m+C)) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},-n,m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) (m+n+2) \sqrt {1-\sin (e+f x)}}+\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (B d (m+n+2)-c C (m+1)+C d m) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},-n,m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (m+n+2) \sqrt {1-\sin (e+f x)}}-\frac {C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{n+1}}{d f (m+n+2)} \]

[In]

Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]

[Out]

-((C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(2 + m + n))) + (Sqrt[2]*(c*(C + 2
*C*m) + d*(C*(1 - m + n) + A*(2 + m + n) - B*(2 + m + n)))*AppellF1[1/2 + m, 1/2, -n, 3/2 + m, (1 + Sin[e + f*
x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(d*f*(1
+ 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n) + (Sqrt[2]*(C*d*m - c*C*(1 + m) +
B*d*(2 + m + n))*AppellF1[3/2 + m, 1/2, -n, 5/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*
Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*d*f*(3 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e
+ f*x]]*((c + d*Sin[e + f*x])/(c - d))^n)

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 145

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(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*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 2867

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

Rule 3066

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A*b - a*B)/b, Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x
], x] + Dist[B/b, 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] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]

Rule 3124

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

Rubi steps \begin{align*} \text {integral}& = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n (a (A d (2+m+n)+C (d+c m+d n))+a (C d m-c C (1+m)+B d (2+m+n)) \sin (e+f x)) \, dx}{a d (2+m+n)} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {(C d m-c C (1+m)+B d (2+m+n)) \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \, dx}{a d (2+m+n)}+\frac {(c (C+2 C m)+d (C (1-m+n)+A (2+m+n)-B (2+m+n))) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx}{d (2+m+n)} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {(a (C d m-c C (1+m)+B d (2+m+n)) \cos (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (2+m+n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c (C+2 C m)+d (C (1-m+n)+A (2+m+n)-B (2+m+n))) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (2+m+n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\left (a (C d m-c C (1+m)+B d (2+m+n)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c (C+2 C m)+d (C (1-m+n)+A (2+m+n)-B (2+m+n))) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\left (a (C d m-c C (1+m)+B d (2+m+n)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} (c+d \sin (e+f x))^n \left (\frac {a (c+d \sin (e+f x))}{a c-a d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c (C+2 C m)+d (C (1-m+n)+A (2+m+n)-B (2+m+n))) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} (c+d \sin (e+f x))^n \left (\frac {a (c+d \sin (e+f x))}{a c-a d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\sqrt {2} (c (C+2 C m)+d (C (1-m+n)+A (2+m+n)-B (2+m+n))) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},-n,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{d f (1+2 m) (2+m+n) \sqrt {1-\sin (e+f x)}}+\frac {\sqrt {2} (C d m-c C (1+m)+B d (2+m+n)) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},-n,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{d f (3+2 m) (2+m+n) (a-a \sin (e+f x))} \\ \end{align*}

Mathematica [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx \]

[In]

Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]

[Out]

Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x]

Maple [F]

\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]

[In]

int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x)

[Out]

int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x)

Fricas [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

integral(-(C*cos(f*x + e)^2 - B*sin(f*x + e) - A - C)*(a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n, x)

Sympy [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**n*(A+B*sin(f*x+e)+C*sin(f*x+e)**2),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n, x)

Giac [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="giac")

[Out]

integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n, x)

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (C\,{\sin \left (e+f\,x\right )}^2+B\,\sin \left (e+f\,x\right )+A\right ) \,d x \]

[In]

int((a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n*(A + B*sin(e + f*x) + C*sin(e + f*x)^2),x)

[Out]

int((a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n*(A + B*sin(e + f*x) + C*sin(e + f*x)^2), x)