\(\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx\) [302]

Optimal result

Integrand size = 35, antiderivative size = 212 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {64 a^3 (21 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {2 (9 B c+9 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f} \]

[Out]

-2/105*a*(21*A*c+15*A*d+15*B*c+13*B*d)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f-2/63*(9*A*d+9*B*c-2*B*d)*cos(f*x+e)
*(a+a*sin(f*x+e))^(5/2)/f-2/9*B*d*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/a/f-64/315*a^3*(21*A*c+15*A*d+15*B*c+13*B*
d)*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-16/315*a^2*(21*A*c+15*A*d+15*B*c+13*B*d)*cos(f*x+e)*(a+a*sin(f*x+e))^(1
/2)/f

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3047, 3102, 2830, 2726, 2725} \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {64 a^3 (21 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{315 f \sqrt {a \sin (e+f x)+a}}-\frac {16 a^2 (21 A c+15 A d+15 B c+13 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{315 f}-\frac {2 (9 A d+9 B c-2 B d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac {2 a (21 A c+15 A d+15 B c+13 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac {2 B d \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f} \]

[In]

Int[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]

[Out]

(-64*a^3*(21*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x])/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(21*A*c
+ 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) - (2*a*(21*A*c + 15*B*c + 15*A*d +
13*B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) - (2*(9*B*c + 9*A*d - 2*B*d)*Cos[e + f*x]*(a + a*Sin[
e + f*x])^(5/2))/(63*f) - (2*B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f)

Rule 2725

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

Rule 2726

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

Rule 2830

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

Rule 3047

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

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((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 + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int (a+a \sin (e+f x))^{5/2} \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {2 \int (a+a \sin (e+f x))^{5/2} \left (\frac {1}{2} a (9 A c+7 B d)+\frac {1}{2} a (9 B c+9 A d-2 B d) \sin (e+f x)\right ) \, dx}{9 a} \\ & = -\frac {2 (9 B c+9 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {1}{21} (21 A c+15 B c+15 A d+13 B d) \int (a+a \sin (e+f x))^{5/2} \, dx \\ & = -\frac {2 a (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {2 (9 B c+9 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {1}{105} (8 a (21 A c+15 B c+15 A d+13 B d)) \int (a+a \sin (e+f x))^{3/2} \, dx \\ & = -\frac {16 a^2 (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {2 (9 B c+9 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {1}{315} \left (32 a^2 (21 A c+15 B c+15 A d+13 B d)\right ) \int \sqrt {a+a \sin (e+f x)} \, dx \\ & = -\frac {64 a^3 (21 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a (21 A c+15 B c+15 A d+13 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {2 (9 B c+9 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.73 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.95 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (7476 A c+6240 B c+6240 A d+5653 B d-4 (63 A c+180 B c+180 A d+254 B d) \cos (2 (e+f x))+35 B d \cos (4 (e+f x))+2352 A c \sin (e+f x)+3030 B c \sin (e+f x)+3030 A d \sin (e+f x)+3116 B d \sin (e+f x)-90 B c \sin (3 (e+f x))-90 A d \sin (3 (e+f x))-260 B d \sin (3 (e+f x)))}{1260 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]

[Out]

-1/1260*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(7476*A*c + 6240*B*c + 6240*A*d
+ 5653*B*d - 4*(63*A*c + 180*B*c + 180*A*d + 254*B*d)*Cos[2*(e + f*x)] + 35*B*d*Cos[4*(e + f*x)] + 2352*A*c*Si
n[e + f*x] + 3030*B*c*Sin[e + f*x] + 3030*A*d*Sin[e + f*x] + 3116*B*d*Sin[e + f*x] - 90*B*c*Sin[3*(e + f*x)] -
90*A*d*Sin[3*(e + f*x)] - 260*B*d*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 7.57 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (35 B \left (\cos ^{4}\left (f x +e \right )\right ) d +\left (-45 d A -45 B c -130 d B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-63 A c -180 d A -180 B c -289 d B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (294 A c +390 d A +390 B c +422 d B \right ) \sin \left (f x +e \right )+966 A c +870 d A +870 B c +838 d B \right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(152\)
parts	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (d A +B c \right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )+12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )+46\right )}{21 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 A c \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d B \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )+130 \left (\sin ^{3}\left (f x +e \right )\right )+219 \left (\sin ^{2}\left (f x +e \right )\right )+292 \sin \left (f x +e \right )+584\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(235\)

[In]

int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

2/315*(1+sin(f*x+e))*a^3*(sin(f*x+e)-1)*(35*B*cos(f*x+e)^4*d+(-45*A*d-45*B*c-130*B*d)*cos(f*x+e)^2*sin(f*x+e)+
(-63*A*c-180*A*d-180*B*c-289*B*d)*cos(f*x+e)^2+(294*A*c+390*A*d+390*B*c+422*B*d)*sin(f*x+e)+966*A*c+870*d*A+87
0*B*c+838*d*B)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.70 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {2 \, {\left (35 \, B a^{2} d \cos \left (f x + e\right )^{5} - 5 \, {\left (9 \, B a^{2} c + {\left (9 \, A + 19 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )^{4} + 96 \, {\left (7 \, A + 5 \, B\right )} a^{2} c + 32 \, {\left (15 \, A + 13 \, B\right )} a^{2} d - {\left (9 \, {\left (7 \, A + 20 \, B\right )} a^{2} c + {\left (180 \, A + 289 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )^{3} + {\left (3 \, {\left (77 \, A + 85 \, B\right )} a^{2} c + {\left (255 \, A + 263 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (161 \, A + 145 \, B\right )} a^{2} c + {\left (435 \, A + 419 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right ) - {\left (35 \, B a^{2} d \cos \left (f x + e\right )^{4} + 96 \, {\left (7 \, A + 5 \, B\right )} a^{2} c + 32 \, {\left (15 \, A + 13 \, B\right )} a^{2} d + 5 \, {\left (9 \, B a^{2} c + {\left (9 \, A + 26 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (3 \, {\left (7 \, A + 15 \, B\right )} a^{2} c + {\left (45 \, A + 53 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (3 \, {\left (49 \, A + 65 \, B\right )} a^{2} c + {\left (195 \, A + 211 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

-2/315*(35*B*a^2*d*cos(f*x + e)^5 - 5*(9*B*a^2*c + (9*A + 19*B)*a^2*d)*cos(f*x + e)^4 + 96*(7*A + 5*B)*a^2*c +
32*(15*A + 13*B)*a^2*d - (9*(7*A + 20*B)*a^2*c + (180*A + 289*B)*a^2*d)*cos(f*x + e)^3 + (3*(77*A + 85*B)*a^2
*c + (255*A + 263*B)*a^2*d)*cos(f*x + e)^2 + 2*(3*(161*A + 145*B)*a^2*c + (435*A + 419*B)*a^2*d)*cos(f*x + e)
- (35*B*a^2*d*cos(f*x + e)^4 + 96*(7*A + 5*B)*a^2*c + 32*(15*A + 13*B)*a^2*d + 5*(9*B*a^2*c + (9*A + 26*B)*a^2
*d)*cos(f*x + e)^3 - 3*(3*(7*A + 15*B)*a^2*c + (45*A + 53*B)*a^2*d)*cos(f*x + e)^2 - 2*(3*(49*A + 65*B)*a^2*c
+ (195*A + 211*B)*a^2*d)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e)
+ f)

Sympy [F]

\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]

[In]

integrate((a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x)

[Out]

Integral((a*(sin(e + f*x) + 1))**(5/2)*(A + B*sin(e + f*x))*(c + d*sin(e + f*x)), x)

Maxima [F]

\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (192) = 384\).

Time = 0.36 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.91 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {\sqrt {2} {\left (35 \, B a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) + 630 \, {\left (20 \, A a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15 \, B a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15 \, A a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 13 \, B a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 210 \, {\left (10 \, A a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 11 \, B a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 11 \, A a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 10 \, B a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 126 \, {\left (2 \, A a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, A a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6 \, B a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 45 \, {\left (2 \, B a^{2} c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, A a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right )\right )} \sqrt {a}}{2520 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

1/2520*sqrt(2)*(35*B*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-9/4*pi + 9/2*f*x + 9/2*e) + 630*(20*A*a^2*
c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 15*B*a^2*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 15*A*a^2*d*sgn(cos(-1
/4*pi + 1/2*f*x + 1/2*e)) + 13*B*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 2
10*(10*A*a^2*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 11*B*a^2*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 11*A*a^2
*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*B*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x
+ 3/2*e) + 126*(2*A*a^2*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*B*a^2*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))
+ 5*A*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*B*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-5/4*pi
+ 5/2*f*x + 5/2*e) + 45*(2*B*a^2*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*A*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x +
1/2*e)) + 5*B*a^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-7/4*pi + 7/2*f*x + 7/2*e))*sqrt(a)/f

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x)),x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x)), x)