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

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

[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]  time = 1.20, antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2976, 2981, 2770, 2761, 2751, 2646} \[ -\frac {2 a^3 \left (-39 A c d+299 A d^2+15 B c^2-75 B c d+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{45045 d^2 f}-\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (-13 A d+5 B c-16 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac {4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{15015 d f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^4}{13 d f} \]

Antiderivative was successfully verified.

[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 2646

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 2751

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*Sin
[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 2761

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 2770

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 2976

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)))*S
in[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 2981

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)*Sqr
t[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*} \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 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]  time = 6.87, size = 1565, normalized size = 2.93 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 863, normalized size = 1.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)sqrt(2*a)*(-288*f*(2*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*p
i))+4*B*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+6*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi)))*cos(1/4*(
18*f*x+18*exp(1)+pi))/(288*f)^2-352*f*(2*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+4*B*a^2*d^3*sign(cos(1/2
*(f*x+exp(1))-1/4*pi))+6*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi)))*cos(1/4*(22*f*x+22*exp(1)-pi))/(352*f
)^2+224*f*(8*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+10*B*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+12*A
*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+24*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+12*B*a^2*c^2*d
*sign(cos(1/2*(f*x+exp(1))-1/4*pi)))*sin(1/4*(14*f*x+14*exp(1)+pi))/(224*f)^2+288*f*(8*A*a^2*d^3*sign(cos(1/2*
(f*x+exp(1))-1/4*pi))+10*B*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+12*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))
-1/4*pi))+24*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+12*B*a^2*c^2*d*sign(cos(1/2*(f*x+exp(1))-1/4*pi)))
*sin(1/4*(18*f*x+18*exp(1)-pi))/(288*f)^2-160*f*(-18*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-8*B*a^2*c^3*
sign(cos(1/2*(f*x+exp(1))-1/4*pi))-20*B*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-48*A*a^2*c*d^2*sign(cos(1/2
*(f*x+exp(1))-1/4*pi))-24*A*a^2*c^2*d*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-54*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(
1))-1/4*pi))-48*B*a^2*c^2*d*sign(cos(1/2*(f*x+exp(1))-1/4*pi)))*cos(1/4*(10*f*x+10*exp(1)+pi))/(160*f)^2-224*f
*(-18*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-8*B*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-20*B*a^2*d^3
*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-48*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-24*A*a^2*c^2*d*sign(cos(
1/2*(f*x+exp(1))-1/4*pi))-54*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-48*B*a^2*c^2*d*sign(cos(1/2*(f*x+e
xp(1))-1/4*pi)))*cos(1/4*(14*f*x+14*exp(1)-pi))/(224*f)^2-16*f*(32*A*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi)
)+22*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+28*B*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+20*B*a^2*d^3
*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+72*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+84*A*a^2*c^2*d*sign(cos(
1/2*(f*x+exp(1))-1/4*pi))+66*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+72*B*a^2*c^2*d*sign(cos(1/2*(f*x+e
xp(1))-1/4*pi)))*cos(1/4*(2*f*x+2*exp(1)+pi))/(16*f)^2-48*f*(32*A*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+2
2*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+28*B*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+20*B*a^2*d^3*si
gn(cos(1/2*(f*x+exp(1))-1/4*pi))+72*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+84*A*a^2*c^2*d*sign(cos(1/2
*(f*x+exp(1))-1/4*pi))+66*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+72*B*a^2*c^2*d*sign(cos(1/2*(f*x+exp(
1))-1/4*pi)))*cos(1/4*(6*f*x+6*exp(1)-pi))/(48*f)^2+16*f*(48*A*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+24*A
*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+32*B*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+22*B*a^2*d^3*sign(
cos(1/2*(f*x+exp(1))-1/4*pi))+84*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+96*A*a^2*c^2*d*sign(cos(1/2*(f
*x+exp(1))-1/4*pi))+72*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+84*B*a^2*c^2*d*sign(cos(1/2*(f*x+exp(1))
-1/4*pi)))*sin(1/4*(2*f*x-pi)+1/2*exp(1))/(16*f)^2+192*f*(-32*A*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-64*
A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-64*B*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-62*B*a^2*d^3*sign
(cos(1/2*(f*x+exp(1))-1/4*pi))-192*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-192*A*a^2*c^2*d*sign(cos(1/2
*(f*x+exp(1))-1/4*pi))-192*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-192*B*a^2*c^2*d*sign(cos(1/2*(f*x+ex
p(1))-1/4*pi)))*sin(1/4*(6*f*x+6*exp(1)+pi))/(192*f)^2+320*f*(-32*A*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))
-64*A*a^2*d^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-64*B*a^2*c^3*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-62*B*a^2*d^3*
sign(cos(1/2*(f*x+exp(1))-1/4*pi))-192*A*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-192*A*a^2*c^2*d*sign(cos
(1/2*(f*x+exp(1))-1/4*pi))-192*B*a^2*c*d^2*sign(cos(1/2*(f*x+exp(1))-1/4*pi))-192*B*a^2*c^2*d*sign(cos(1/2*(f*
x+exp(1))-1/4*pi)))*sin(1/4*(10*f*x+10*exp(1)-pi))/(320*f)^2-1408*B*a^2*d^3*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi
))*sin(1/4*(22*f*x+22*exp(1)+pi))/(704*f)^2-1664*B*a^2*d^3*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(1/4*(26*f*
x+26*exp(1)-pi))/(832*f)^2)

________________________________________________________________________________________

maple [A]  time = 1.40, size = 374, normalized size = 0.70 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (\left (4095 A \,d^{3}+12285 B c \,d^{2}+11970 B \,d^{3}\right ) \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-19305 A \,c^{2} d -55770 A c \,d^{2}-31265 A \,d^{3}-6435 B \,c^{3}-55770 B \,c^{2} d -93795 B c \,d^{2}-44860 B \,d^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (42042 A \,c^{3}+167310 A \,c^{2} d +181038 A c \,d^{2}+64090 A \,d^{3}+55770 B \,c^{3}+181038 B \,c^{2} d +192270 B c \,d^{2}+66362 B \,d^{3}\right ) \sin \left (f x +e \right )-3465 B \,d^{3} \left (\cos ^{6}\left (f x +e \right )\right )+\left (15015 A c \,d^{2}+14560 A \,d^{3}+15015 B \,c^{2} d +43680 B c \,d^{2}+28700 B \,d^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-9009 A \,c^{3}-77220 A \,c^{2} d -123981 A c \,d^{2}-56810 A \,d^{3}-25740 B \,c^{3}-123981 B \,c^{2} d -170430 B c \,d^{2}-72109 B \,d^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+138138 A \,c^{3}+373230 A \,c^{2} d +359502 A c \,d^{2}+116090 A \,d^{3}+124410 B \,c^{3}+359502 B \,c^{2} d +348270 B c \,d^{2}+113818 B \,d^{3}\right )}{45045 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(1+sin(f*x+e))*a^3*(sin(f*x+e)-1)*((4095*A*d^3+12285*B*c*d^2+11970*B*d^3)*sin(f*x+e)*cos(f*x+e)^4+(-19
305*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)+(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+66362*B*d
^3)*sin(f*x+e)-3465*B*d^3*cos(f*x+e)^6+(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+(-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^2-7210
9*B*d^3)*cos(f*x+e)^2+138138*A*c^3+373230*A*c^2*d+359502*A*c*d^2+116090*A*d^3+124410*B*c^3+359502*B*c^2*d+3482
70*B*c*d^2+113818*B*d^3)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________