\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx\) [353]

Optimal result

Integrand size = 39, antiderivative size = 840 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\frac {(c-d) \sqrt {c+d} \left (2 A b^2 c-2 a b B c-2 a A b d+3 a^2 B d-b^2 B d\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt {a+b} (b c-a d) f}+\frac {\sqrt {c+d} (3 b B c+2 A b d-3 a B d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt {a+b} f}+\frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} \left (2 A b (b (c-2 d)+a d)-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt {c+d} f} \]

[Out]

(c-d)*(-2*A*a*b*d+2*A*b^2*c+3*B*a^2*d-2*B*a*b*c-B*b^2*d)*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1
/2)/(a+b*sin(f*x+e))^(1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*
c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/(a-b)
/b^2/(-a*d+b*c)/f/(a+b)^(1/2)+(2*A*b*d-3*B*a*d+3*B*b*c)*EllipticPi((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1
/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^
(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+
e)))^(1/2)/b^3/f/(a+b)^(1/2)+(2*A*b*(b*(c-2*d)+a*d)-B*(3*a^2*d-6*a*b*d+b^2*(2*c+d)))*EllipticF((c+d)^(1/2)*(a+
b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*
x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(
c+d*sin(f*x+e)))^(1/2)/(a-b)/b^3/f/(c+d)^(1/2)+2*(A*b-B*a)*(-a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2
-b^2)/f/(a+b*sin(f*x+e))^(1/2)-(2*A*b*(-a*d+b*c)-B*(-3*a^2*d+2*a*b*c+b^2*d))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)
/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 840, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3068, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\frac {2 (A b-a B) (b c-a d) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (-3 d a^2+2 b c a+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}+\frac {(c-d) \sqrt {c+d} \left (3 B d a^2-2 b B c a-2 A b d a+2 A b^2 c-b^2 B d\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt {a+b} (b c-a d) f}+\frac {\sqrt {c+d} (3 b B c+2 A b d-3 a B d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt {a+b} f}+\frac {\sqrt {a+b} \left (2 A b (b (c-2 d)+a d)-B \left (3 d a^2-6 b d a+b^2 (2 c+d)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt {c+d} f} \]

[In]

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

[Out]

((c - d)*Sqrt[c + d]*(2*A*b^2*c - 2*a*b*B*c - 2*a*A*b*d + 3*a^2*B*d - b^2*B*d)*EllipticE[ArcSin[(Sqrt[a + b]*S
qrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e +
f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e +
f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*b^2*Sqrt[a + b]*(b*c - a*d)*f) + (Sqrt[c
+ d]*(3*b*B*c + 2*A*b*d - 3*a*B*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e +
f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*
c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(
a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^3*Sqrt[a + b]*f) + (2*(A*b - a*B)*(b*c - a*d)*Cos[e + f*x]*Sqrt
[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) - ((2*A*b*(b*c - a*d) - B*(2*a*b*c - 3*a^2*d
+ b^2*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[a + b]*(2*
A*b*(b*(c - 2*d) + a*d) - B*(3*a^2*d - 6*a*b*d + b^2*(2*c + d)))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[
e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b
*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)
*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*b^3*Sqrt[c + d]*f)

Rule 2890

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[
2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c -
d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi
[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])],
(a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]

Rule 2897

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e +
f*x])/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))
)]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c -
d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 3068

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

Rule 3075

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

Rule 3077

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

Rule 3132

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

Rule 3140

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} \left (a^2 B d^2+b^2 c (B c+2 A d)-a b \left (2 B c d+A \left (c^2+d^2\right )\right )\right )-\frac {1}{2} \left (A b^2 \left (c^2-d^2\right )+B \left (2 a^2 c d-2 b^2 c d-a b \left (c^2-d^2\right )\right )\right ) \sin (e+f x)-\frac {1}{2} d \left (\left (3 a^2-b^2\right ) B d+2 b (A b c-a B c-a A d)\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{2} \left (a^2-b^2\right ) d \left (2 A b c^2-B d (b c-a d)\right )-\left (a^2-b^2\right ) c d (b B c+2 A b d-a B d) \sin (e+f x)-\frac {1}{2} \left (a^2-b^2\right ) d^2 (3 b B c+2 A b d-3 a B d) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right ) d} \\ & = \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {\frac {1}{2} a^2 \left (a^2-b^2\right ) d^2 (3 b B c+2 A b d-3 a B d)-\frac {1}{2} b^2 \left (a^2-b^2\right ) d \left (2 A b c^2-B d (b c-a d)\right )+b \left (a \left (a^2-b^2\right ) d^2 (3 b B c+2 A b d-3 a B d)-b \left (a^2-b^2\right ) c d (b B c+2 A b d-a B d)\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{b^3 \left (a^2-b^2\right ) d}+\frac {(d (3 b B c+2 A b d-3 a B d)) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 b^3} \\ & = \frac {\sqrt {c+d} (3 b B c+2 A b d-3 a B d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt {a+b} f}+\frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left ((b c-a d) \left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right )\right ) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{2 (a-b) b^2}+\frac {\left ((b c-a d) \left (2 A b (b (c-2 d)+a d)-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{2 (a-b) b^3} \\ & = \frac {(c-d) \sqrt {c+d} \left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt {a+b} (b c-a d) f}+\frac {\sqrt {c+d} (3 b B c+2 A b d-3 a B d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt {a+b} f}+\frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} \left (2 A b (b (c-2 d)+a d)-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt {c+d} f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2042\) vs. \(2(840)=1680\).

Time = 8.06 (sec) , antiderivative size = 2042, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-2*(A*b^2*c*Cos[e + f*x] - a*b*B*c*Cos[e + f*x] - a*A*b*d*Cos[e + f*x] + a^2*B*d*Cos[e + f*x])*Sqrt[c + d*Sin
[e + f*x]])/(b*(-a^2 + b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((-4*(-(b*c) + a*d)*(2*a*A*b*c^2 - 2*b^2*B*c^2 - 2*A
*b^2*c*d + 2*a*b*B*c*d + a^2*B*d^2 - b^2*B*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[
ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) +
a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a
+ b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a
*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(2*A*b^2*c^2 - 2*
a*b*B*c^2 + 4*a^2*B*c*d - 4*b^2*B*c*d - 2*A*b^2*d^2 + 2*a*b*B*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)
/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sq
rt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e +
Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e
+ f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d
)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e +
Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e
+ f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*
d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin
[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-2*A*b^2*c*d + 2*a*b*B*c*d + 2*a*A*b*d^2 - 3*a^2*B*d^2 + b^2*B*d^2)
*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-
e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])
/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2
- f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(
c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Co
t[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e +
f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^
4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2
- f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b
)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b
*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]
^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*
c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(2*(a - b)*b*(a + b)*f)

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 114.64 (sec) , antiderivative size = 1442707, normalized size of antiderivative = 1717.51

method	result	size
parts	\(\text {Expression too large to display}\)	\(1442707\)
default	\(\text {Expression too large to display}\)	\(1467567\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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