\(\int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx\) [115]

Optimal result

Integrand size = 38, antiderivative size = 242 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {2048 (7 A-13 B) c^4 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{105 a^2 f}-\frac {512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f} \]

[Out]

-512/105*(7*A-13*B)*c^3*sec(f*x+e)*(c-c*sin(f*x+e))^(3/2)/a^2/f-64/105*(7*A-13*B)*c^2*sec(f*x+e)*(c-c*sin(f*x+
e))^(5/2)/a^2/f-16/105*(7*A-13*B)*c*sec(f*x+e)*(c-c*sin(f*x+e))^(7/2)/a^2/f-1/21*(7*A-13*B)*sec(f*x+e)*(c-c*si
n(f*x+e))^(9/2)/a^2/f-1/3*(A-B)*sec(f*x+e)^3*(c-c*sin(f*x+e))^(13/2)/a^2/c^2/f+2048/105*(7*A-13*B)*c^4*sec(f*x
+e)*(c-c*sin(f*x+e))^(1/2)/a^2/f

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2934, 2753, 2752} \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {2048 c^4 (7 A-13 B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{105 a^2 f}-\frac {512 c^3 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {64 c^2 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {16 c (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f} \]

[In]

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

[Out]

(2048*(7*A - 13*B)*c^4*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(105*a^2*f) - (512*(7*A - 13*B)*c^3*Sec[e + f*x]
*(c - c*Sin[e + f*x])^(3/2))/(105*a^2*f) - (64*(7*A - 13*B)*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(105*
a^2*f) - (16*(7*A - 13*B)*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(105*a^2*f) - ((7*A - 13*B)*Sec[e + f*x]*
(c - c*Sin[e + f*x])^(9/2))/(21*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(13/2))/(3*a^2*c^2*f)

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2753

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(
g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[a*((2*m + p - 1)/(m + p)), Int
[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2,
0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rule 2934

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

Rule 3046

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^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] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{13/2} \, dx}{a^2 c^2} \\ & = -\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {(7 A-13 B) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{11/2} \, dx}{6 a^2 c} \\ & = -\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {(8 (7 A-13 B)) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{21 a^2} \\ & = -\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {(32 (7 A-13 B) c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{35 a^2} \\ & = -\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {\left (256 (7 A-13 B) c^2\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{105 a^2} \\ & = -\frac {512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {\left (1024 (7 A-13 B) c^3\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{105 a^2} \\ & = \frac {2048 (7 A-13 B) c^4 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{105 a^2 f}-\frac {512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.73 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {c^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (95550 A-179340 B-72 (203 A-402 B) \cos (2 (e+f x))+6 (7 A-38 B) \cos (4 (e+f x))+119952 A \sin (e+f x)-219618 B \sin (e+f x)+784 A \sin (3 (e+f x))-2131 B \sin (3 (e+f x))+15 B \sin (5 (e+f x)))}{840 a^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2} \]

[In]

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

[Out]

(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(95550*A - 179340*B - 72*(203*A - 402*B)*C
os[2*(e + f*x)] + 6*(7*A - 38*B)*Cos[4*(e + f*x)] + 119952*A*Sin[e + f*x] - 219618*B*Sin[e + f*x] + 784*A*Sin[
3*(e + f*x)] - 2131*B*Sin[3*(e + f*x)] + 15*B*Sin[5*(e + f*x)]))/(840*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/
2])*(1 + Sin[e + f*x])^2)

Maple [A] (verified)

Time = 20.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {2 c^{5} \left (\sin \left (f x +e \right )-1\right ) \left (15 B \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (21 A -114 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (196 A -544 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-1848 A +3732 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (7448 A -13592 B \right ) \sin \left (f x +e \right )+6888 A -13032 B \right )}{105 a^{2} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(143\)

[In]

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

[Out]

-2/105*c^5/a^2*(sin(f*x+e)-1)/(1+sin(f*x+e))*(15*B*cos(f*x+e)^4*sin(f*x+e)+(21*A-114*B)*cos(f*x+e)^4+(196*A-54
4*B)*cos(f*x+e)^2*sin(f*x+e)+(-1848*A+3732*B)*cos(f*x+e)^2+(7448*A-13592*B)*sin(f*x+e)+6888*A-13032*B)/cos(f*x
+e)/(c-c*sin(f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.63 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {2 \, {\left (3 \, {\left (7 \, A - 38 \, B\right )} c^{4} \cos \left (f x + e\right )^{4} - 12 \, {\left (154 \, A - 311 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 24 \, {\left (287 \, A - 543 \, B\right )} c^{4} + {\left (15 \, B c^{4} \cos \left (f x + e\right )^{4} + 4 \, {\left (49 \, A - 136 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 8 \, {\left (931 \, A - 1699 \, B\right )} c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, {\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )\right )}} \]

[In]

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

[Out]

2/105*(3*(7*A - 38*B)*c^4*cos(f*x + e)^4 - 12*(154*A - 311*B)*c^4*cos(f*x + e)^2 + 24*(287*A - 543*B)*c^4 + (1
5*B*c^4*cos(f*x + e)^4 + 4*(49*A - 136*B)*c^4*cos(f*x + e)^2 + 8*(931*A - 1699*B)*c^4)*sin(f*x + e))*sqrt(-c*s
in(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (218) = 436\).

Time = 0.33 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.15 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2/105*(7*(723*c^(9/2) + 2184*c^(9/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 5370*c^(9/2)*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 10696*c^(9/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15021*c^(9/2)*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 + 21168*c^(9/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20748*c^(9/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 +
21168*c^(9/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 15021*c^(9/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 10696
*c^(9/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5370*c^(9/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 2184*c^(9/
2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 723*c^(9/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)*A/((a^2 + 3*a^2*
sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(9/2)) - 2*(4707*c^(9/2) + 14121*c^(9/2)*sin(f*x + e)/(cos(
f*x + e) + 1) + 35250*c^(9/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 68549*c^(9/2)*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 + 99549*c^(9/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 134802*c^(9/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 + 138012*c^(9/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 134802*c^(9/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 +
99549*c^(9/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 68549*c^(9/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 35250
*c^(9/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 14121*c^(9/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 4707*c^
(9/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)*B/((a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1
)^(9/2)))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (218) = 436\).

Time = 0.84 (sec) , antiderivative size = 978, normalized size of antiderivative = 4.04 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-16/105*sqrt(2)*sqrt(c)*(35*(11*A*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 17*B*c^4*sgn(sin(-1/4*pi + 1/2*f*x
+ 1/2*e)) + 24*A*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi +
1/2*f*x + 1/2*e) + 1) - 36*B*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos
(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 9*A*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x +
1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 15*B*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*
pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2)/(a^2*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos
(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 1)^3) - (511*A*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 1069*B*c^4*sgn(sin
(-1/4*pi + 1/2*f*x + 1/2*e)) - 3262*A*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2
*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 6958*B*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi +
1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 8421*A*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2*sgn
(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 18459*B*c^4*(cos(-1/4*pi + 1/2*f*x +
1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 10780*A*c^4*(cos(-
1/4*pi + 1/2*f*x + 1/2*e) - 1)^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 +
24220*B*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x
+ 1/2*e) + 1)^3 + 7105*A*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-
1/4*pi + 1/2*f*x + 1/2*e) + 1)^4 - 13195*B*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*
x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4 - 2310*A*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5*sgn(sin
(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5 + 3990*B*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*
e) - 1)^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5 + 315*A*c^4*(cos(-1/4*pi
+ 1/2*f*x + 1/2*e) - 1)^6*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^6 - 525*B*c
^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^6*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e)
+ 1)^6)/(a^2*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 1)^7))/f

Mupad [F(-1)]

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

[In]

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

[Out]

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