\(\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\) [293]

Optimal result

Integrand size = 37, antiderivative size = 374 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {4 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x)}{3465 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {8 a (5 c-d) (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d f}+\frac {4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac {2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f} \]

[Out]

4/1155*(c+d)*(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f+4/3465*a^2*(c+d)*(15
*c^2+10*c*d+7*d^2)*(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(f*x+e)/d^2/f/(a+a*sin(f*x+e))^(1/2)+2/693*a^2*
(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^2/f/(a+a*sin(f*x+e))^(1/2)+2/99*a^2*(
3*B*(c-4*d)-11*A*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/d^2/f/(a+a*sin(f*x+e))^(1/2)+8/3465*a*(5*c-d)*(c+d)*(11*A*(c
-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d/f-2/11*a*B*cos(f*x+e)*(c+d*sin(f*x+e))^4*
(a+a*sin(f*x+e))^(1/2)/d/f

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3055, 3060, 2849, 2840, 2830, 2725} \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {2 a^2 \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {4 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x)}{3465 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {4 (c+d) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}+\frac {8 a (5 c-d) (c+d) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3465 d f}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f} \]

[In]

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

[Out]

(4*a^2*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x])/(3465*
d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(5*c - d)*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[
e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d*f) + (4*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Co
s[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) + (2*a^2*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[
e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(3*B*(c - 4*d) - 11*A*d)*Cos[e
+ f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*
x]]*(c + d*Sin[e + f*x])^4)/(11*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) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac {2 \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \left (\frac {1}{2} a (11 A d+B (c+8 d))-\frac {1}{2} a (3 B (c-4 d)-11 A d) \sin (e+f x)\right ) \, dx}{11 d} \\ & = \frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac {\left (a \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{99 d^2} \\ & = \frac {2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac {\left (2 a (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^2} \\ & = \frac {4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac {2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac {\left (4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\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}{1155 d^2} \\ & = \frac {8 a (5 c-d) (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d f}+\frac {4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac {2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac {\left (2 a (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{3465 d^2} \\ & = \frac {4 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x)}{3465 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {8 a (5 c-d) (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d f}+\frac {4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac {2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.34 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.04 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (92400 A c^3+72072 B c^3+216216 A c^2 d+195624 B c^2 d+195624 A c d^2+177474 B c d^2+59158 A d^3+55482 B d^3-8 \left (11 A d \left (189 c^2+351 c d+137 d^2\right )+3 B \left (231 c^3+1287 c^2 d+1507 c d^2+581 d^3\right )\right ) \cos (2 (e+f x))+70 d^2 (33 B c+11 A d+21 B d) \cos (4 (e+f x))+18480 A c^3 \sin (e+f x)+33264 B c^3 \sin (e+f x)+99792 A c^2 d \sin (e+f x)+100188 B c^2 d \sin (e+f x)+100188 A c d^2 \sin (e+f x)+105468 B c d^2 \sin (e+f x)+35156 A d^3 \sin (e+f x)+34734 B d^3 \sin (e+f x)-5940 B c^2 d \sin (3 (e+f x))-5940 A c d^2 \sin (3 (e+f x))-11220 B c d^2 \sin (3 (e+f x))-3740 A d^3 \sin (3 (e+f x))-4935 B d^3 \sin (3 (e+f x))+315 B d^3 \sin (5 (e+f x))\right )}{27720 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

-1/27720*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(92400*A*c^3 + 72072*B*c^3 + 2162
16*A*c^2*d + 195624*B*c^2*d + 195624*A*c*d^2 + 177474*B*c*d^2 + 59158*A*d^3 + 55482*B*d^3 - 8*(11*A*d*(189*c^2
+ 351*c*d + 137*d^2) + 3*B*(231*c^3 + 1287*c^2*d + 1507*c*d^2 + 581*d^3))*Cos[2*(e + f*x)] + 70*d^2*(33*B*c +
11*A*d + 21*B*d)*Cos[4*(e + f*x)] + 18480*A*c^3*Sin[e + f*x] + 33264*B*c^3*Sin[e + f*x] + 99792*A*c^2*d*Sin[e
+ f*x] + 100188*B*c^2*d*Sin[e + f*x] + 100188*A*c*d^2*Sin[e + f*x] + 105468*B*c*d^2*Sin[e + f*x] + 35156*A*d^
3*Sin[e + f*x] + 34734*B*d^3*Sin[e + f*x] - 5940*B*c^2*d*Sin[3*(e + f*x)] - 5940*A*c*d^2*Sin[3*(e + f*x)] - 11
220*B*c*d^2*Sin[3*(e + f*x)] - 3740*A*d^3*Sin[3*(e + f*x)] - 4935*B*d^3*Sin[3*(e + f*x)] + 315*B*d^3*Sin[5*(e
+ f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (315 B \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right ) d^{3}+\left (385 A \,d^{3}+1155 d^{2} c B +735 d^{3} B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-1485 d^{2} c A -935 A \,d^{3}-1485 c^{2} d B -2805 d^{2} c B -1470 d^{3} B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-2079 c^{2} d A -3861 d^{2} c A -1892 A \,d^{3}-693 B \,c^{3}-3861 c^{2} d B -5676 d^{2} c B -2478 d^{3} B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (1155 A \,c^{3}+6237 c^{2} d A +6633 d^{2} c A +2431 A \,d^{3}+2079 B \,c^{3}+6633 c^{2} d B +7293 d^{2} c B +2499 d^{3} B \right ) \sin \left (f x +e \right )+5775 A \,c^{3}+14553 c^{2} d A +14157 d^{2} c A +4499 A \,d^{3}+4851 B \,c^{3}+14157 c^{2} d B +13497 d^{2} c B +4431 d^{3} B \right )}{3465 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(312\)
parts	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) d^{2} \left (d A +3 B c \right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )+85 \left (\sin ^{3}\left (f x +e \right )\right )+102 \left (\sin ^{2}\left (f x +e \right )\right )+136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (3 d A +B c \right ) \left (\sin ^{2}\left (f x +e \right )+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) c d \left (d A +B c \right ) \left (15 \left (\sin ^{3}\left (f x +e \right )\right )+39 \left (\sin ^{2}\left (f x +e \right )\right )+52 \sin \left (f x +e \right )+104\right )}{35 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 A \,c^{3} \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d^{3} B \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (15 \left (\sin ^{5}\left (f x +e \right )\right )+35 \left (\sin ^{4}\left (f x +e \right )\right )+40 \left (\sin ^{3}\left (f x +e \right )\right )+48 \left (\sin ^{2}\left (f x +e \right )\right )+64 \sin \left (f x +e \right )+128\right )}{165 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(407\)

[In]

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

[Out]

2/3465*(1+sin(f*x+e))*a^2*(sin(f*x+e)-1)*(315*B*cos(f*x+e)^4*sin(f*x+e)*d^3+(385*A*d^3+1155*B*c*d^2+735*B*d^3)
*cos(f*x+e)^4+(-1485*A*c*d^2-935*A*d^3-1485*B*c^2*d-2805*B*c*d^2-1470*B*d^3)*cos(f*x+e)^2*sin(f*x+e)+(-2079*A*
c^2*d-3861*A*c*d^2-1892*A*d^3-693*B*c^3-3861*B*c^2*d-5676*B*c*d^2-2478*B*d^3)*cos(f*x+e)^2+(1155*A*c^3+6237*A*
c^2*d+6633*A*c*d^2+2431*A*d^3+2079*B*c^3+6633*B*c^2*d+7293*B*c*d^2+2499*B*d^3)*sin(f*x+e)+5775*A*c^3+14553*c^2
*d*A+14157*d^2*c*A+4499*A*d^3+4851*B*c^3+14157*c^2*d*B+13497*d^2*c*B+4431*d^3*B)/cos(f*x+e)/(a+a*sin(f*x+e))^(
1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.70 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {2 \, {\left (315 \, B a d^{3} \cos \left (f x + e\right )^{6} + 35 \, {\left (33 \, B a c d^{2} + {\left (11 \, A + 21 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{5} + 924 \, {\left (5 \, A + 3 \, B\right )} a c^{3} + 396 \, {\left (21 \, A + 19 \, B\right )} a c^{2} d + 132 \, {\left (57 \, A + 47 \, B\right )} a c d^{2} + 4 \, {\left (517 \, A + 483 \, B\right )} a d^{3} - 5 \, {\left (297 \, B a c^{2} d + 33 \, {\left (9 \, A + 10 \, B\right )} a c d^{2} + 10 \, {\left (11 \, A + 21 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (693 \, B a c^{3} + 297 \, {\left (7 \, A + 13 \, B\right )} a c^{2} d + 33 \, {\left (117 \, A + 172 \, B\right )} a c d^{2} + 2 \, {\left (946 \, A + 1239 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (231 \, {\left (5 \, A + 6 \, B\right )} a c^{3} + 99 \, {\left (42 \, A + 43 \, B\right )} a c^{2} d + 33 \, {\left (129 \, A + 134 \, B\right )} a c d^{2} + {\left (1474 \, A + 1491 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (231 \, {\left (25 \, A + 21 \, B\right )} a c^{3} + 99 \, {\left (147 \, A + 143 \, B\right )} a c^{2} d + 33 \, {\left (429 \, A + 409 \, B\right )} a c d^{2} + {\left (4499 \, A + 4431 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right ) + {\left (315 \, B a d^{3} \cos \left (f x + e\right )^{5} - 924 \, {\left (5 \, A + 3 \, B\right )} a c^{3} - 396 \, {\left (21 \, A + 19 \, B\right )} a c^{2} d - 132 \, {\left (57 \, A + 47 \, B\right )} a c d^{2} - 4 \, {\left (517 \, A + 483 \, B\right )} a d^{3} - 35 \, {\left (33 \, B a c d^{2} + {\left (11 \, A + 12 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (297 \, B a c^{2} d + 33 \, {\left (9 \, A + 17 \, B\right )} a c d^{2} + {\left (187 \, A + 294 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (231 \, B a c^{3} + 99 \, {\left (7 \, A + 8 \, B\right )} a c^{2} d + 33 \, {\left (24 \, A + 29 \, B\right )} a c d^{2} + {\left (319 \, A + 336 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (231 \, {\left (5 \, A + 9 \, B\right )} a c^{3} + 99 \, {\left (63 \, A + 67 \, B\right )} a c^{2} d + 33 \, {\left (201 \, A + 221 \, B\right )} a c d^{2} + 17 \, {\left (143 \, A + 147 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3465 \, {\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))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

-2/3465*(315*B*a*d^3*cos(f*x + e)^6 + 35*(33*B*a*c*d^2 + (11*A + 21*B)*a*d^3)*cos(f*x + e)^5 + 924*(5*A + 3*B)
*a*c^3 + 396*(21*A + 19*B)*a*c^2*d + 132*(57*A + 47*B)*a*c*d^2 + 4*(517*A + 483*B)*a*d^3 - 5*(297*B*a*c^2*d +
33*(9*A + 10*B)*a*c*d^2 + 10*(11*A + 21*B)*a*d^3)*cos(f*x + e)^4 - (693*B*a*c^3 + 297*(7*A + 13*B)*a*c^2*d + 3
3*(117*A + 172*B)*a*c*d^2 + 2*(946*A + 1239*B)*a*d^3)*cos(f*x + e)^3 + (231*(5*A + 6*B)*a*c^3 + 99*(42*A + 43*
B)*a*c^2*d + 33*(129*A + 134*B)*a*c*d^2 + (1474*A + 1491*B)*a*d^3)*cos(f*x + e)^2 + (231*(25*A + 21*B)*a*c^3 +
99*(147*A + 143*B)*a*c^2*d + 33*(429*A + 409*B)*a*c*d^2 + (4499*A + 4431*B)*a*d^3)*cos(f*x + e) + (315*B*a*d^
3*cos(f*x + e)^5 - 924*(5*A + 3*B)*a*c^3 - 396*(21*A + 19*B)*a*c^2*d - 132*(57*A + 47*B)*a*c*d^2 - 4*(517*A +
483*B)*a*d^3 - 35*(33*B*a*c*d^2 + (11*A + 12*B)*a*d^3)*cos(f*x + e)^4 - 5*(297*B*a*c^2*d + 33*(9*A + 17*B)*a*c
*d^2 + (187*A + 294*B)*a*d^3)*cos(f*x + e)^3 + 3*(231*B*a*c^3 + 99*(7*A + 8*B)*a*c^2*d + 33*(24*A + 29*B)*a*c*
d^2 + (319*A + 336*B)*a*d^3)*cos(f*x + e)^2 + (231*(5*A + 9*B)*a*c^3 + 99*(63*A + 67*B)*a*c^2*d + 33*(201*A +
221*B)*a*c*d^2 + 17*(143*A + 147*B)*a*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]

\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \]

[In]

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

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)*(A + B*sin(e + f*x))*(c + d*sin(e + f*x))**3, x)

Maxima [F]

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

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(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)^(3/2)*(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. 755 vs. \(2 (350) = 700\).

Time = 0.45 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.02 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/55440*sqrt(2)*(315*B*a*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) + 6930*(24*
A*a*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 16*B*a*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 48*A*a*c^2*d*sg
n(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 42*B*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 42*A*a*c*d^2*sgn(cos(-1
/4*pi + 1/2*f*x + 1/2*e)) + 36*B*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*A*a*d^3*sgn(cos(-1/4*pi + 1/
2*f*x + 1/2*e)) + 11*B*a*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) + 2310*(8*A*a
*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*B*a*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*A*a*c^2*d*sgn(c
os(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*B*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*A*a*c*d^2*sgn(cos(-1/4*
pi + 1/2*f*x + 1/2*e)) + 30*B*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*A*a*d^3*sgn(cos(-1/4*pi + 1/2*f
*x + 1/2*e)) + 9*B*a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 693*(8*B*a*c^3*
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 24*A*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*B*a*c^2*d*sgn(cos(
-1/4*pi + 1/2*f*x + 1/2*e)) + 36*A*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*B*a*c*d^2*sgn(cos(-1/4*pi
+ 1/2*f*x + 1/2*e)) + 12*A*a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 13*B*a*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) + 495*(12*B*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*A*a*c*d^2
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 18*B*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*A*a*d^3*sgn(cos(-1
/4*pi + 1/2*f*x + 1/2*e)) + 7*B*a*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) + 38
5*(6*B*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*A*a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a*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))*sqrt(a)/f

Mupad [F(-1)]

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

[In]

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

[Out]

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