\(\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx\) [160]

Optimal result

Integrand size = 40, antiderivative size = 250 \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=-\frac {a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 (9 A-B) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac {a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac {a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f} \]

[Out]

-1/84*a^2*(9*A-B)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/f-1/72*a*(9*A-B)*cos(f*x+e)*(a+a*si
n(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(9/2)/f-1/9*B*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2)/f-1/31
5*a^4*(9*A-B)*cos(f*x+e)*(c-c*sin(f*x+e))^(9/2)/f/(a+a*sin(f*x+e))^(1/2)-1/126*a^3*(9*A-B)*cos(f*x+e)*(c-c*sin
(f*x+e))^(9/2)*(a+a*sin(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {3052, 2819, 2817} \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=-\frac {a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt {a \sin (e+f x)+a}}-\frac {a^3 (9 A-B) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac {a^2 (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac {a (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f} \]

[In]

Int[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2),x]

[Out]

-1/315*(a^4*(9*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(9*A - B)*C
os[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(126*f) - (a^2*(9*A - B)*Cos[e + f*x]*(a + a*
Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(84*f) - (a*(9*A - B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*
(c - c*Sin[e + f*x])^(9/2))/(72*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9
*f)

Rule 2817

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

Rule 2819

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 - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Dist[a*((2*m - 1)/(
m + n)), 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, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m
]) &&  !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])

Rule 3052

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((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/(f*
(m + n + 1))), x] - Dist[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Si
n[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)] && NeQ[m + n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac {1}{9} (9 A-B) \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx \\ & = -\frac {a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac {1}{12} (a (9 A-B)) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2} \, dx \\ & = -\frac {a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac {a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac {1}{21} \left (a^2 (9 A-B)\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2} \, dx \\ & = -\frac {a^3 (9 A-B) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac {a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac {a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac {1}{63} \left (a^3 (9 A-B)\right ) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2} \, dx \\ & = -\frac {a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 (9 A-B) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac {a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac {a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.51 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.08 \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=\frac {a^3 c^4 (-1+\sin (e+f x))^4 (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (17640 (A-B) \cos (2 (e+f x))+8820 (A-B) \cos (4 (e+f x))+2520 A \cos (6 (e+f x))-2520 B \cos (6 (e+f x))+315 A \cos (8 (e+f x))-315 B \cos (8 (e+f x))+176400 A \sin (e+f x)-17640 B \sin (e+f x)+35280 A \sin (3 (e+f x))+7056 A \sin (5 (e+f x))+2016 B \sin (5 (e+f x))+720 A \sin (7 (e+f x))+900 B \sin (7 (e+f x))+140 B \sin (9 (e+f x)))}{322560 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]

[In]

Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(a^3*c^4*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(17640
*(A - B)*Cos[2*(e + f*x)] + 8820*(A - B)*Cos[4*(e + f*x)] + 2520*A*Cos[6*(e + f*x)] - 2520*B*Cos[6*(e + f*x)]
+ 315*A*Cos[8*(e + f*x)] - 315*B*Cos[8*(e + f*x)] + 176400*A*Sin[e + f*x] - 17640*B*Sin[e + f*x] + 35280*A*Sin
[3*(e + f*x)] + 7056*A*Sin[5*(e + f*x)] + 2016*B*Sin[5*(e + f*x)] + 720*A*Sin[7*(e + f*x)] + 900*B*Sin[7*(e +
f*x)] + 140*B*Sin[9*(e + f*x)]))/(322560*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^7)

Maple [A] (verified)

Time = 74.96 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.01

method	result	size
parts	\(\frac {A \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a^{3} c^{4} \left (35 \left (\cos ^{7}\left (f x +e \right )\right )+40 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+64 \cos \left (f x +e \right ) \sin \left (f x +e \right )+128 \tan \left (f x +e \right )-35 \sec \left (f x +e \right )\right )}{280 f}+\frac {B \sec \left (f x +e \right ) \left (280 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )-315 \left (\cos ^{6}\left (f x +e \right )\right )+240 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-315 \left (\cos ^{4}\left (f x +e \right )\right )+192 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-315 \left (\cos ^{2}\left (f x +e \right )\right )+128 \sin \left (f x +e \right )-315\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, c^{4} a^{3} \left (\cos ^{2}\left (f x +e \right )-1\right )}{2520 f}\)	\(253\)
default	\(-\frac {a^{3} c^{4} \tan \left (f x +e \right ) \left (280 B \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{6}\left (f x +e \right )\right )+315 A \sin \left (f x +e \right ) \left (\cos ^{6}\left (f x +e \right )\right )+315 B \left (\sin ^{3}\left (f x +e \right )\right ) \left (\cos ^{4}\left (f x +e \right )\right )-360 A \left (\cos ^{6}\left (f x +e \right )\right )+240 B \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right )+315 A \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+630 B \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{3}\left (f x +e \right )\right )-432 A \left (\cos ^{4}\left (f x +e \right )\right )+192 B \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right )+315 A \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+945 B \left (\sin ^{3}\left (f x +e \right )\right )-576 A \left (\cos ^{2}\left (f x +e \right )\right )+128 B \left (\sin ^{2}\left (f x +e \right )\right )+315 A \sin \left (f x +e \right )-1260 B \sin \left (f x +e \right )-1152 A \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{2520 f}\)	\(266\)

[In]

int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/280*A/f*(a*(1+sin(f*x+e)))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*a^3*c^4*(35*cos(f*x+e)^7+40*cos(f*x+e)^5*sin(f*x+
e)+48*cos(f*x+e)^3*sin(f*x+e)+64*cos(f*x+e)*sin(f*x+e)+128*tan(f*x+e)-35*sec(f*x+e))+1/2520*B/f*sec(f*x+e)*(28
0*cos(f*x+e)^6*sin(f*x+e)-315*cos(f*x+e)^6+240*cos(f*x+e)^4*sin(f*x+e)-315*cos(f*x+e)^4+192*cos(f*x+e)^2*sin(f
*x+e)-315*cos(f*x+e)^2+128*sin(f*x+e)-315)*(-c*(sin(f*x+e)-1))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*c^4*a^3*(cos(f*x
+e)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.73 \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=\frac {{\left (315 \, {\left (A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{8} - 315 \, {\left (A - B\right )} a^{3} c^{4} + 8 \, {\left (35 \, B a^{3} c^{4} \cos \left (f x + e\right )^{8} + 5 \, {\left (9 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{6} + 6 \, {\left (9 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{4} + 8 \, {\left (9 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{2} + 16 \, {\left (9 \, A - B\right )} a^{3} c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2520 \, f \cos \left (f x + e\right )} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

1/2520*(315*(A - B)*a^3*c^4*cos(f*x + e)^8 - 315*(A - B)*a^3*c^4 + 8*(35*B*a^3*c^4*cos(f*x + e)^8 + 5*(9*A - B
)*a^3*c^4*cos(f*x + e)^6 + 6*(9*A - B)*a^3*c^4*cos(f*x + e)^4 + 8*(9*A - B)*a^3*c^4*cos(f*x + e)^2 + 16*(9*A -
B)*a^3*c^4)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(9/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (220) = 440\).

Time = 0.55 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.81 \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=\frac {32 \, {\left (560 \, B a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{18} - 315 \, A a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16} - 2205 \, B a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16} + 1080 \, A a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} + 3240 \, B a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 1260 \, A a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 2100 \, B a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 504 \, A a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} + 504 \, B a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}\right )} \sqrt {a} \sqrt {c}}{315 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

32/315*(560*B*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/
2*f*x + 1/2*e)^18 - 315*A*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(
-1/4*pi + 1/2*f*x + 1/2*e)^16 - 2205*B*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x +
1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^16 + 1080*A*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*p
i + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^14 + 3240*B*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*s
gn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^14 - 1260*A*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*
x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^12 - 2100*B*a^3*c^4*sgn(cos(-1/
4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^12 + 504*A*a^3*c^4
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^10 + 5
04*B*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1
/2*e)^10)*sqrt(a)*sqrt(c)/f

Mupad [B] (verification not implemented)

Time = 19.64 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.93 \[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx=\frac {{\mathrm {e}}^{-e\,9{}\mathrm {i}-f\,x\,9{}\mathrm {i}}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (-\frac {a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (A\,1{}\mathrm {i}-B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,7{}\mathrm {i}}{64\,f}-\frac {a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\left (A\,1{}\mathrm {i}-B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,7{}\mathrm {i}}{128\,f}-\frac {a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\left (A\,1{}\mathrm {i}-B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,1{}\mathrm {i}}{64\,f}-\frac {a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\cos \left (8\,e+8\,f\,x\right )\,\left (A\,1{}\mathrm {i}-B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,1{}\mathrm {i}}{512\,f}+\frac {a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\left (7\,A+2\,B\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{160\,f}+\frac {a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\left (4\,A+5\,B\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{896\,f}+\frac {7\,A\,a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}+\frac {7\,a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (10\,A-B\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}+\frac {B\,a^3\,c^4\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (9\,e+9\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{1152\,f}\right )}{2\,\cos \left (e+f\,x\right )} \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(9/2),x)

[Out]

(exp(- e*9i - f*x*9i)*(c - c*sin(e + f*x))^(1/2)*((a^3*c^4*exp(e*9i + f*x*9i)*sin(5*e + 5*f*x)*(7*A + 2*B)*(a
+ a*sin(e + f*x))^(1/2))/(160*f) - (a^3*c^4*exp(e*9i + f*x*9i)*cos(4*e + 4*f*x)*(A*1i - B*1i)*(a + a*sin(e + f
*x))^(1/2)*7i)/(128*f) - (a^3*c^4*exp(e*9i + f*x*9i)*cos(6*e + 6*f*x)*(A*1i - B*1i)*(a + a*sin(e + f*x))^(1/2)
*1i)/(64*f) - (a^3*c^4*exp(e*9i + f*x*9i)*cos(8*e + 8*f*x)*(A*1i - B*1i)*(a + a*sin(e + f*x))^(1/2)*1i)/(512*f
) - (a^3*c^4*exp(e*9i + f*x*9i)*cos(2*e + 2*f*x)*(A*1i - B*1i)*(a + a*sin(e + f*x))^(1/2)*7i)/(64*f) + (a^3*c^
4*exp(e*9i + f*x*9i)*sin(7*e + 7*f*x)*(4*A + 5*B)*(a + a*sin(e + f*x))^(1/2))/(896*f) + (7*A*a^3*c^4*exp(e*9i
+ f*x*9i)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2))/(32*f) + (7*a^3*c^4*exp(e*9i + f*x*9i)*sin(e + f*x)*(10
*A - B)*(a + a*sin(e + f*x))^(1/2))/(64*f) + (B*a^3*c^4*exp(e*9i + f*x*9i)*sin(9*e + 9*f*x)*(a + a*sin(e + f*x
))^(1/2))/(1152*f)))/(2*cos(e + f*x))