3.799 \(\int \frac{1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=688 \[ -\frac{2 \left (3 a^2 b d (2 c-3 d)-3 a^3 d^2-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f \sqrt{a+b} \left (a^2-b^2\right ) (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 \left (3 a^2 b^2 d \left (3 c^2-5 d^2\right )+3 a^4 d^3-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f \sqrt{a+b} \left (a^2-b^2\right ) (c-d) \sqrt{c+d} (b c-a d)^4}+\frac{8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{3 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \]
[Out]
(2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) + (8*b^
2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f*x])/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d
*Sin[e + f*x]]) + (2*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3*a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - d^2))
*EllipticE[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]))]*Sq
rt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a
^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^4*f) - (2*(3*a^2*b*(2*c - 3*d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2)
+ b^3*(c^2 - 6*c*d + 8*d^2))*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]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f)
________________________________________________________________________________________
Rubi [A] time = 1.84657, antiderivative size = 688, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used =
{2802, 3055, 2998, 2818, 2996} \[ -\frac{2 \left (3 a^2 b d (2 c-3 d)-3 a^3 d^2-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f \sqrt{a+b} \left (a^2-b^2\right ) (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 \left (3 a^2 b^2 d \left (3 c^2-5 d^2\right )+3 a^4 d^3-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f \sqrt{a+b} \left (a^2-b^2\right ) (c-d) \sqrt{c+d} (b c-a d)^4}+\frac{8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{3 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]
[Out]
(2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) + (8*b^
2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f*x])/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d
*Sin[e + f*x]]) + (2*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3*a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - d^2))
*EllipticE[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]))]*Sq
rt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a
^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^4*f) - (2*(3*a^2*b*(2*c - 3*d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2)
+ b^3*(c^2 - 6*c*d + 8*d^2))*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]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f)
Rule 2802
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Rule 3055
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))
Rule 2998
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 2818
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])*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))])/(f*(b*c - a*d)*Rt[(c + d)/(a
+ b), 2]*Cos[e + f*x]), 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 2996
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])*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))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*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] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx &=\frac{2 b^2 \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}-\frac{2 \int \frac{\frac{1}{2} \left (-4 b^2 d-3 a (b c-a d)\right )+\frac{1}{2} b (b c-3 a d) \sin (e+f x)+b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac{2 b^2 \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}+\frac{8 b^2 \left (a b c-2 a^2 d+b^2 d\right ) \cos (e+f x)}{3 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{4 \int \frac{\frac{1}{4} \left (-6 a^3 b c d+6 a b^3 c d+3 a^4 d^2+3 a^2 b^2 \left (c^2-5 d^2\right )+b^4 \left (c^2+8 d^2\right )\right )-\frac{1}{2} b \left (3 a^2 b c d-3 b^3 c d+3 a^3 d^2-a b^2 \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=\frac{2 b^2 \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}+\frac{8 b^2 \left (a b c-2 a^2 d+b^2 d\right ) \cos (e+f x)}{3 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 a^4 d^3-b^4 d \left (5 c^2-8 d^2\right )+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )\right ) \int \frac{1+\sin (e+f x)}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )^2 (c-d) (b c-a d)^2}-\frac{\left (3 a^2 b (2 c-3 d) d-3 a^3 d^2-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{3 (a-b) (a+b)^2 (c-d) (b c-a d)^2}\\ &=\frac{2 b^2 \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}+\frac{8 b^2 \left (a b c-2 a^2 d+b^2 d\right ) \cos (e+f x)}{3 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (3 a^4 d^3-b^4 d \left (5 c^2-8 d^2\right )+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )\right ) E\left (\sin ^{-1}\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))}{3 (a-b) (a+b)^{3/2} (c-d) \sqrt{c+d} (b c-a d)^4 f}-\frac{2 \left (3 a^2 b (2 c-3 d) d-3 a^3 d^2-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) F\left (\sin ^{-1}\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))}{3 (a-b) (a+b)^{3/2} (c-d) \sqrt{c+d} (b c-a d)^3 f}\\ \end{align*}
Mathematica [B] time = 7.37752, size = 2322, normalized size = 3.38 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]
[Out]
(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((2*b^3*Cos[e + f*x])/(3*(a^2 - b^2)*(-(b*c) + a*d)^2*(a +
b*Sin[e + f*x])^2) - (2*(4*a*b^4*c*Cos[e + f*x] - 9*a^2*b^3*d*Cos[e + f*x] + 5*b^5*d*Cos[e + f*x]))/(3*(a^2 -
b^2)^2*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - a*d)^3*(c^2 - d^2)*(c + d*Sin[e +
f*x]))))/f + ((-4*(-(b*c) + a*d)*(-3*a^2*b^3*c^4 - b^5*c^4 + 9*a^3*b^2*c^3*d - 5*a*b^4*c^3*d - 9*a^4*b*c^2*d^
2 + 20*a^2*b^3*c^2*d^2 - 7*b^5*c^2*d^2 + 3*a^5*c*d^3 - 15*a^3*b^2*c*d^3 + 8*a*b^4*c*d^3 + 9*a^4*b*d^4 - 17*a^2
*b^3*d^4 + 8*b^5*d^4)*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)*(-4*a*b^4*c^4 + 5*a^2*b^3*c^3*d - 5*b^5
*c^3*d + 9*a^3*b^2*c^2*d^2 - a*b^4*c^2*d^2 + 3*a^4*b*c*d^3 - 11*a^2*b^3*c*d^3 + 8*b^5*c*d^3 + 3*a^5*d^4 - 15*a
^3*b^2*d^4 + 8*a*b^4*d^4)*((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]]) - (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*(4*a*b^4*c^3*d - 9*a^2*b^3*c^2*d^2 + 5*b^5*c^2*d^2 - 4*a*b^4*c*d^3 - 3*a^4*b*d^4 + 15*a^2*b^3*d^4 - 8
*b^5*d^4)*((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)*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)*Cs
c[(-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]]*Sqr
t[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))]/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)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(3*(a - b
)^2*(a + b)^2*(c - d)*(c + d)*(-(b*c) + a*d)^3*f)
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Maple [B] time = 8.531, size = 439275, normalized size = 638.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x)
[Out]
result too large to display
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
integrate(1/((b*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^(3/2)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right ) + c}}{{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} +{\left (a^{3} + 3 \, a b^{2}\right )} c^{2} + 2 \,{\left (3 \, a^{2} b + b^{3}\right )} c d +{\left (a^{3} + 3 \, a b^{2}\right )} d^{2} -{\left (3 \, a b^{2} c^{2} + 2 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d +{\left (a^{3} + 6 \, a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (b^{3} d^{2} \cos \left (f x + e\right )^{4} +{\left (3 \, a^{2} b + b^{3}\right )} c^{2} + 2 \,{\left (a^{3} + 3 \, a b^{2}\right )} c d +{\left (3 \, a^{2} b + b^{3}\right )} d^{2} -{\left (b^{3} c^{2} + 6 \, a b^{2} c d +{\left (3 \, a^{2} b + 2 \, b^{3}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((2*b^3*c*d + 3*a*b^2*d^2)*cos(f*x + e)^4 + (a^3 +
3*a*b^2)*c^2 + 2*(3*a^2*b + b^3)*c*d + (a^3 + 3*a*b^2)*d^2 - (3*a*b^2*c^2 + 2*(3*a^2*b + 2*b^3)*c*d + (a^3 + 6
*a*b^2)*d^2)*cos(f*x + e)^2 + (b^3*d^2*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^2 + 2*(a^3 + 3*a*b^2)*c*d + (3*a^2*b
+ b^3)*d^2 - (b^3*c^2 + 6*a*b^2*c*d + (3*a^2*b + 2*b^3)*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(3/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")
[Out]
integrate(1/((b*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^(3/2)), x)