\(\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\) [300]

Optimal result

Integrand size = 37, antiderivative size = 534 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{45045 d^2 f}-\frac {4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f} \]

[Out]

-4/15015*a*(c+d)*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(f*x+e)*(a+a*sin(f
*x+e))^(3/2)/d/f-2/13*a*B*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^4/d/f-4/45045*a^3*(c+d)*(15*c^2+1
0*c*d+7*d^2)*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(f*x+e)/d^3/f/(a+a*sin
(f*x+e))^(1/2)-2/9009*a^3*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(f*x+e)*(
c+d*sin(f*x+e))^3/d^3/f/(a+a*sin(f*x+e))^(1/2)-2/1287*a^3*(-39*A*c*d+299*A*d^2+15*B*c^2-75*B*c*d+280*B*d^2)*co
s(f*x+e)*(c+d*sin(f*x+e))^4/d^3/f/(a+a*sin(f*x+e))^(1/2)-8/45045*a^2*(5*c-d)*(c+d)*(13*A*d*(3*c^2-38*c*d+355*d
^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d^2/f+2/143*a^2*(-13*A*d+5*B*c-
16*B*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4*(a+a*sin(f*x+e))^(1/2)/d^2/f

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3055, 3060, 2849, 2840, 2830, 2725} \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {2 a^3 \left (-39 A c d+299 A d^2+15 B c^2-75 B c d+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{45045 d^2 f}+\frac {2 a^2 (-13 A d+5 B c-16 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{15015 d f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^4}{13 d f} \]

[In]

Int[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]

[Out]

(-4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d
^2 - 4184*d^3))*Cos[e + f*x])/(45045*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (8*a^2*(5*c - d)*(c + d)*(13*A*d*(3*c^2
- 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d^2 - 4184*d^3))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(
45045*d^2*f) - (4*a*(c + d)*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d^2 - 4184*d^3)
)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(15015*d*f) - (2*a^3*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3
- 150*c^2*d + 799*c*d^2 - 4184*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(9009*d^3*f*Sqrt[a + a*Sin[e + f*x]
]) - (2*a^3*(15*B*c^2 - 39*A*c*d - 75*B*c*d + 299*A*d^2 + 280*B*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(128
7*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 13*A*d - 16*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c
+ d*Sin[e + f*x])^4)/(143*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^4)/(13*
d*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 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 2840

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-
d^2)*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[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], 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, -1]

Rule 2849

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] + Dist[2*n*((b*c + a*d)
/(b*(2*n + 1))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rule 3055

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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x
])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*
x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))
*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3060

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt
[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac {2 \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \left (\frac {1}{2} a (3 B c+13 A d+8 B d)-\frac {1}{2} a (5 B c-13 A d-16 B d) \sin (e+f x)\right ) \, dx}{13 d} \\ & = \frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac {4 \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \left (\frac {1}{4} a^2 \left (13 A d (c+19 d)-B \left (5 c^2-9 c d-216 d^2\right )\right )+\frac {1}{4} a^2 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \sin (e+f x)\right ) \, dx}{143 d^2} \\ & = -\frac {2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac {\left (a^2 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{1287 d^3} \\ & = -\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac {\left (2 a^2 (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{3003 d^3} \\ & = -\frac {4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac {\left (4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{15015 d^3} \\ & = -\frac {8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{45045 d^2 f}-\frac {4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac {\left (2 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{45045 d^3} \\ & = -\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{45045 d^2 f}-\frac {4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.55 (sec) , antiderivative size = 1565, normalized size of antiderivative = 2.93 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {B d^3 \cos \left (\frac {13}{2} (e+f x)\right ) (a (1+\sin (e+f x)))^{5/2}}{416 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (40 A c^3+30 B c^3+90 A c^2 d+78 B c^2 d+78 A c d^2+69 B c d^2+23 A d^3+21 B d^3\right ) \left (\left (-\frac {1}{16}-\frac {i}{16}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{16}-\frac {i}{16}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{5/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (40 A c^3+30 B c^3+90 A c^2 d+78 B c^2 d+78 A c d^2+69 B c d^2+23 A d^3+21 B d^3\right ) \left (\left (-\frac {1}{16}+\frac {i}{16}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{5/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (80 A c^3+88 B c^3+264 A c^2 d+240 B c^2 d+240 A c d^2+228 B c d^2+76 A d^3+71 B d^3\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (-\frac {1}{192}+\frac {i}{192}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{192}+\frac {i}{192}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (80 A c^3+88 B c^3+264 A c^2 d+240 B c^2 d+240 A c d^2+228 B c d^2+76 A d^3+71 B d^3\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (-\frac {1}{192}-\frac {i}{192}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{192}-\frac {i}{192}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (16 A c^3+40 B c^3+120 A c^2 d+144 B c^2 d+144 A c d^2+150 B c d^2+50 A d^3+51 B d^3\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (\frac {1}{320}-\frac {i}{320}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{320}+\frac {i}{320}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (16 A c^3+40 B c^3+120 A c^2 d+144 B c^2 d+144 A c d^2+150 B c d^2+50 A d^3+51 B d^3\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (\frac {1}{320}+\frac {i}{320}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{320}-\frac {i}{320}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (4 B c^3+12 A c^2 d+30 B c^2 d+30 A c d^2+39 B c d^2+13 A d^3+15 B d^3\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (\frac {1}{224}+\frac {i}{224}\right ) \cos \left (\frac {7}{2} (e+f x)\right )+\left (\frac {1}{224}-\frac {i}{224}\right ) \sin \left (\frac {7}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (4 B c^3+12 A c^2 d+30 B c^2 d+30 A c d^2+39 B c d^2+13 A d^3+15 B d^3\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (\frac {1}{224}-\frac {i}{224}\right ) \cos \left (\frac {7}{2} (e+f x)\right )+\left (\frac {1}{224}+\frac {i}{224}\right ) \sin \left (\frac {7}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (6 B c^2+6 A c d+15 B c d+5 A d^2+7 B d^2\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (-\frac {1}{288}-\frac {i}{288}\right ) d \cos \left (\frac {9}{2} (e+f x)\right )+\left (\frac {1}{288}-\frac {i}{288}\right ) d \sin \left (\frac {9}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {\left (6 B c^2+6 A c d+15 B c d+5 A d^2+7 B d^2\right ) (a (1+\sin (e+f x)))^{5/2} \left (\left (-\frac {1}{288}+\frac {i}{288}\right ) d \cos \left (\frac {9}{2} (e+f x)\right )+\left (\frac {1}{288}+\frac {i}{288}\right ) d \sin \left (\frac {9}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {(6 B c+2 A d+5 B d) (a (1+\sin (e+f x)))^{5/2} \left (\left (-\frac {1}{704}+\frac {i}{704}\right ) d^2 \cos \left (\frac {11}{2} (e+f x)\right )-\left (\frac {1}{704}+\frac {i}{704}\right ) d^2 \sin \left (\frac {11}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {(6 B c+2 A d+5 B d) (a (1+\sin (e+f x)))^{5/2} \left (\left (-\frac {1}{704}-\frac {i}{704}\right ) d^2 \cos \left (\frac {11}{2} (e+f x)\right )-\left (\frac {1}{704}-\frac {i}{704}\right ) d^2 \sin \left (\frac {11}{2} (e+f x)\right )\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {B d^3 (a (1+\sin (e+f x)))^{5/2} \sin \left (\frac {13}{2} (e+f x)\right )}{416 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]

[In]

Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]

[Out]

(B*d^3*Cos[(13*(e + f*x))/2]*(a*(1 + Sin[e + f*x]))^(5/2))/(416*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + (
(40*A*c^3 + 30*B*c^3 + 90*A*c^2*d + 78*B*c^2*d + 78*A*c*d^2 + 69*B*c*d^2 + 23*A*d^3 + 21*B*d^3)*((-1/16 - I/16
)*Cos[(e + f*x)/2] + (1/16 - I/16)*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[
(e + f*x)/2])^5) + ((40*A*c^3 + 30*B*c^3 + 90*A*c^2*d + 78*B*c^2*d + 78*A*c*d^2 + 69*B*c*d^2 + 23*A*d^3 + 21*B
*d^3)*((-1/16 + I/16)*Cos[(e + f*x)/2] + (1/16 + I/16)*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((80*A*c^3 + 88*B*c^3 + 264*A*c^2*d + 240*B*c^2*d + 240*A*c*d^2 + 228*B
*c*d^2 + 76*A*d^3 + 71*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/192 + I/192)*Cos[(3*(e + f*x))/2] - (1/192 + I
/192)*Sin[(3*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((80*A*c^3 + 88*B*c^3 + 264*A*c^2*d
+ 240*B*c^2*d + 240*A*c*d^2 + 228*B*c*d^2 + 76*A*d^3 + 71*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/192 - I/19
2)*Cos[(3*(e + f*x))/2] - (1/192 - I/192)*Sin[(3*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) +
((16*A*c^3 + 40*B*c^3 + 120*A*c^2*d + 144*B*c^2*d + 144*A*c*d^2 + 150*B*c*d^2 + 50*A*d^3 + 51*B*d^3)*(a*(1 +
Sin[e + f*x]))^(5/2)*((1/320 - I/320)*Cos[(5*(e + f*x))/2] - (1/320 + I/320)*Sin[(5*(e + f*x))/2]))/(f*(Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2])^5) + ((16*A*c^3 + 40*B*c^3 + 120*A*c^2*d + 144*B*c^2*d + 144*A*c*d^2 + 150*B*c*
d^2 + 50*A*d^3 + 51*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/320 + I/320)*Cos[(5*(e + f*x))/2] - (1/320 - I/320
)*Sin[(5*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((4*B*c^3 + 12*A*c^2*d + 30*B*c^2*d + 3
0*A*c*d^2 + 39*B*c*d^2 + 13*A*d^3 + 15*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/224 + I/224)*Cos[(7*(e + f*x))/
2] + (1/224 - I/224)*Sin[(7*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((4*B*c^3 + 12*A*c^2
*d + 30*B*c^2*d + 30*A*c*d^2 + 39*B*c*d^2 + 13*A*d^3 + 15*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/224 - I/224)
*Cos[(7*(e + f*x))/2] + (1/224 + I/224)*Sin[(7*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + (
(6*B*c^2 + 6*A*c*d + 15*B*c*d + 5*A*d^2 + 7*B*d^2)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/288 - I/288)*d*Cos[(9*(e
+ f*x))/2] + (1/288 - I/288)*d*Sin[(9*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c^2
+ 6*A*c*d + 15*B*c*d + 5*A*d^2 + 7*B*d^2)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/288 + I/288)*d*Cos[(9*(e + f*x))/2
] + (1/288 + I/288)*d*Sin[(9*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c + 2*A*d + 5
*B*d)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/704 + I/704)*d^2*Cos[(11*(e + f*x))/2] - (1/704 + I/704)*d^2*Sin[(11*(
e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c + 2*A*d + 5*B*d)*(a*(1 + Sin[e + f*x]))^(
5/2)*((-1/704 - I/704)*d^2*Cos[(11*(e + f*x))/2] - (1/704 - I/704)*d^2*Sin[(11*(e + f*x))/2]))/(f*(Cos[(e + f*
x)/2] + Sin[(e + f*x)/2])^5) - (B*d^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sin[(13*(e + f*x))/2])/(416*f*(Cos[(e + f*x
)/2] + Sin[(e + f*x)/2])^5)

Maple [A] (verified)

Time = 168.51 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (-3465 B \left (\cos ^{6}\left (f x +e \right )\right ) d^{3}+\left (4095 A \,d^{3}+12285 d^{2} c B +11970 d^{3} B \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (15015 d^{2} c A +14560 A \,d^{3}+15015 c^{2} d B +43680 d^{2} c B +28700 d^{3} B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-19305 c^{2} d A -55770 d^{2} c A -31265 A \,d^{3}-6435 B \,c^{3}-55770 c^{2} d B -93795 d^{2} c B -44860 d^{3} B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-9009 A \,c^{3}-77220 c^{2} d A -123981 d^{2} c A -56810 A \,d^{3}-25740 B \,c^{3}-123981 c^{2} d B -170430 d^{2} c B -72109 d^{3} B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (42042 A \,c^{3}+167310 c^{2} d A +181038 d^{2} c A +64090 A \,d^{3}+55770 B \,c^{3}+181038 c^{2} d B +192270 d^{2} c B +66362 d^{3} B \right ) \sin \left (f x +e \right )+138138 A \,c^{3}+373230 c^{2} d A +359502 d^{2} c A +116090 A \,d^{3}+124410 B \,c^{3}+359502 c^{2} d B +348270 d^{2} c B +113818 d^{3} B \right )}{45045 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(374\)
parts	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) d^{2} \left (d A +3 B c \right ) \left (63 \left (\sin ^{5}\left (f x +e \right )\right )+224 \left (\sin ^{4}\left (f x +e \right )\right )+355 \left (\sin ^{3}\left (f x +e \right )\right )+426 \left (\sin ^{2}\left (f x +e \right )\right )+568 \sin \left (f x +e \right )+1136\right )}{693 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (3 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 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) c d \left (d A +B c \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 )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 A \,c^{3} \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^{3} B \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3465 \left (\sin ^{6}\left (f x +e \right )\right )+11970 \left (\sin ^{5}\left (f x +e \right )\right )+18305 \left (\sin ^{4}\left (f x +e \right )\right )+20920 \left (\sin ^{3}\left (f x +e \right )\right )+25104 \left (\sin ^{2}\left (f x +e \right )\right )+33472 \sin \left (f x +e \right )+66944\right )}{45045 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(461\)

[In]

int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/45045*(1+sin(f*x+e))*a^3*(sin(f*x+e)-1)*(-3465*B*cos(f*x+e)^6*d^3+(4095*A*d^3+12285*B*c*d^2+11970*B*d^3)*cos
(f*x+e)^4*sin(f*x+e)+(15015*A*c*d^2+14560*A*d^3+15015*B*c^2*d+43680*B*c*d^2+28700*B*d^3)*cos(f*x+e)^4+(-19305*
A*c^2*d-55770*A*c*d^2-31265*A*d^3-6435*B*c^3-55770*B*c^2*d-93795*B*c*d^2-44860*B*d^3)*cos(f*x+e)^2*sin(f*x+e)+
(-9009*A*c^3-77220*A*c^2*d-123981*A*c*d^2-56810*A*d^3-25740*B*c^3-123981*B*c^2*d-170430*B*c*d^2-72109*B*d^3)*c
os(f*x+e)^2+(42042*A*c^3+167310*A*c^2*d+181038*A*c*d^2+64090*A*d^3+55770*B*c^3+181038*B*c^2*d+192270*B*c*d^2+6
6362*B*d^3)*sin(f*x+e)+138138*A*c^3+373230*c^2*d*A+359502*d^2*c*A+116090*A*d^3+124410*B*c^3+359502*c^2*d*B+348
270*d^2*c*B+113818*d^3*B)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.62 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

2/45045*(3465*B*a^2*d^3*cos(f*x + e)^7 - 315*(39*B*a^2*c*d^2 + (13*A + 27*B)*a^2*d^3)*cos(f*x + e)^6 - 13728*(
7*A + 5*B)*a^2*c^3 - 13728*(15*A + 13*B)*a^2*c^2*d - 1248*(143*A + 125*B)*a^2*c*d^2 - 32*(1625*A + 1483*B)*a^2
*d^3 - 35*(429*B*a^2*c^2*d + 39*(11*A + 32*B)*a^2*c*d^2 + 4*(104*A + 205*B)*a^2*d^3)*cos(f*x + e)^5 + 5*(1287*
B*a^2*c^3 + 429*(9*A + 19*B)*a^2*c^2*d + 39*(209*A + 320*B)*a^2*c*d^2 + 2*(2080*A + 2813*B)*a^2*d^3)*cos(f*x +
e)^4 + (1287*(7*A + 20*B)*a^2*c^3 + 429*(180*A + 289*B)*a^2*c^2*d + 39*(3179*A + 4370*B)*a^2*c*d^2 + (56810*A
+ 72109*B)*a^2*d^3)*cos(f*x + e)^3 - (429*(77*A + 85*B)*a^2*c^3 + 429*(255*A + 263*B)*a^2*c^2*d + 39*(2893*A
+ 2965*B)*a^2*c*d^2 + (38545*A + 39113*B)*a^2*d^3)*cos(f*x + e)^2 - 2*(429*(161*A + 145*B)*a^2*c^3 + 429*(435*
A + 419*B)*a^2*c^2*d + 39*(4609*A + 4465*B)*a^2*c*d^2 + (58045*A + 56909*B)*a^2*d^3)*cos(f*x + e) - (3465*B*a^
2*d^3*cos(f*x + e)^6 - 13728*(7*A + 5*B)*a^2*c^3 - 13728*(15*A + 13*B)*a^2*c^2*d - 1248*(143*A + 125*B)*a^2*c*
d^2 - 32*(1625*A + 1483*B)*a^2*d^3 + 315*(39*B*a^2*c*d^2 + (13*A + 38*B)*a^2*d^3)*cos(f*x + e)^5 - 35*(429*B*a
^2*c^2*d + 39*(11*A + 23*B)*a^2*c*d^2 + (299*A + 478*B)*a^2*d^3)*cos(f*x + e)^4 - 5*(1287*B*a^2*c^3 + 429*(9*A
+ 26*B)*a^2*c^2*d + 507*(22*A + 37*B)*a^2*c*d^2 + (6253*A + 8972*B)*a^2*d^3)*cos(f*x + e)^3 + 3*(429*(7*A + 1
5*B)*a^2*c^3 + 429*(45*A + 53*B)*a^2*c^2*d + 39*(583*A + 655*B)*a^2*c*d^2 + (8515*A + 9083*B)*a^2*d^3)*cos(f*x
+ e)^2 + 2*(429*(49*A + 65*B)*a^2*c^3 + 429*(195*A + 211*B)*a^2*c^2*d + 39*(2321*A + 2465*B)*a^2*c*d^2 + (320
45*A + 33181*B)*a^2*d^3)*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(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,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)^3, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (506) = 1012\).

Time = 0.55 (sec) , antiderivative size = 1014, normalized size of antiderivative = 1.90 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/1441440*sqrt(2)*(3465*B*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-13/4*pi + 13/2*f*x + 13/2*e) + 1801
80*(40*A*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*B*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 90*A
*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 78*B*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 78*A*a^2
*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 69*B*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 23*A*a^2*d^3
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 21*B*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x
+ 1/2*e) + 15015*(80*A*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 88*B*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x +
1/2*e)) + 264*A*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 240*B*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1
/2*e)) + 240*A*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 228*B*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2
*e)) + 76*A*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 71*B*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*si
n(-3/4*pi + 3/2*f*x + 3/2*e) + 9009*(16*A*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 40*B*a^2*c^3*sgn(cos(-
1/4*pi + 1/2*f*x + 1/2*e)) + 120*A*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 144*B*a^2*c^2*d*sgn(cos(-1/
4*pi + 1/2*f*x + 1/2*e)) + 144*A*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 150*B*a^2*c*d^2*sgn(cos(-1/4*
pi + 1/2*f*x + 1/2*e)) + 50*A*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 51*B*a^2*d^3*sgn(cos(-1/4*pi + 1/2
*f*x + 1/2*e)))*sin(-5/4*pi + 5/2*f*x + 5/2*e) + 12870*(4*B*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*A
*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*B*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*A*a^2
*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 39*B*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 13*A*a^2*d^3
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 15*B*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-7/4*pi + 7/2*f*x
+ 7/2*e) + 10010*(6*B*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*A*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x
+ 1/2*e)) + 15*B*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*A*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*
e)) + 7*B*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-9/4*pi + 9/2*f*x + 9/2*e) + 4095*(6*B*a^2*c*d^2*sg
n(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*A*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*B*a^2*d^3*sgn(cos(-1/4
*pi + 1/2*f*x + 1/2*e)))*sin(-11/4*pi + 11/2*f*x + 11/2*e))*sqrt(a)/f

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3,x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3, x)