\(\int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx\) [799]

Optimal result

Integrand size = 29, antiderivative size = 655 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 b^2 \cos (e+f x)}{3 \left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}+\frac {8 b^2 \left (3 b c-18 d+b^2 d\right ) \cos (e+f x)}{3 \left (9-b^2\right )^2 (b c-3 d)^2 f \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (243 d^3-b^4 d \left (5 c^2-8 d^2\right )+27 b^2 d \left (3 c^2-5 d^2\right )-12 b^3 c \left (c^2-d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {3+b} \left (9-b^2\right ) (b c-3 d)^4 (c-d) \sqrt {c+d} f}-\frac {2 \left (27 b (2 c-3 d) d-81 d^2-9 b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {3+b} \left (9-b^2\right ) (b c-3 d)^3 (c-d) \sqrt {c+d} f} \]

[Out]

2/3*(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((c+d)^(1/2)*(a+b*s
in(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*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^2-b^2)/(c-d)/(-a*d+b*c)^4/f/(a+b)^(1/2)/(c+d)^(1/2)-2/3*(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((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*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^2-b^2)/(c-d)/(-a*d+b*c)^3/f/(a+b)^(1/2)/
(c+d)^(1/2)+2/3*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2)+8/3*b^2*(-
2*a^2*d+a*b*c+b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2881, 3134, 3077, 2897, 3075} \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{3 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f \sqrt {a+b} \left (a^2-b^2\right ) (c-d) \sqrt {c+d} (b c-a d)^4}-\frac {2 \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f \sqrt {a+b} \left (a^2-b^2\right ) (c-d) \sqrt {c+d} (b c-a d)^3} \]

[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 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-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 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 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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \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 (\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))}{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 ) \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))}{3 (a-b) (a+b)^{3/2} (c-d) \sqrt {c+d} (b c-a d)^3 f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2255\) vs. \(2(655)=1310\).

Time = 6.96 (sec) , antiderivative size = 2255, normalized size of antiderivative = 3.44 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[1/((3 + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^3*Cos[e + f*x])/(3*(-9 + b^2)*(b*c - 3*d)^2*(3 + b*S
in[e + f*x])^2) + (2*(12*b^4*c*Cos[e + f*x] - 81*b^3*d*Cos[e + f*x] + 5*b^5*d*Cos[e + f*x]))/(3*(-9 + b^2)^2*(
b*c - 3*d)^3*(3 + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - 3*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f
+ ((-4*(-(b*c) + 3*d)*(27*b^3*c^4 + b^5*c^4 - 243*b^2*c^3*d + 15*b^4*c^3*d + 729*b*c^2*d^2 - 180*b^3*c^2*d^2
+ 7*b^5*c^2*d^2 - 729*c*d^3 + 405*b^2*c*d^3 - 24*b^4*c*d^3 - 729*b*d^4 + 153*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[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[
e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + 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*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e +
Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*S
in[e + f*x]]) - 4*(-(b*c) + 3*d)*(12*b^4*c^4 - 45*b^3*c^3*d + 5*b^5*c^3*d - 243*b^2*c^2*d^2 + 3*b^4*c^2*d^2 -
243*b*c*d^3 + 99*b^3*c*d^3 - 8*b^5*c*d^3 - 729*d^4 + 405*b^2*d^4 - 24*b^4*d^4)*((Sqrt[((c + d)*Cot[(-e + Pi/2
- f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c
) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + 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*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*
(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + 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) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 -
b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + 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*(3 + b*Sin[e + f*x]))/
(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sq
rt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-12*b^4*c^3*d + 81*b^3*c^2*d^2 - 5*b^5*c^2*d^2 + 12*b^4
*c*d^3 + 243*b*d^4 - 135*b^3*d^4 + 8*b^5*d^4)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[3 + b*Sin[e + f
*x]]) + (Sqrt[(3 - b)/(3 + b)]*(3 + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(3 - b)/(3 + b)]*Sin[(-
e + Pi/2 - f*x)/2])/Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]], (2*(-(b*c) + 3*d))/((3 - b)*(c + d))]*Sqrt[c + d*Sin[
e + f*x]])/(b*d*Sqrt[((3 + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(3 + b*Sin[e + f*x])]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[
(3 + b*Sin[e + f*x])/(3 + b)]*Sqrt[((3 + b)*(c + d*Sin[e + f*x]))/((c + d)*(3 + b*Sin[e + f*x]))]) - (2*(-(b*c
) + 3*d)*((((3 + b)*c + 3*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 -
b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + 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*(3 + b*Sin[e + f*x]))/
(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c +
d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + 3*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2
)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin
[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + 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*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e +
Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]])))/(b*d)))/(3*(-3 + b)^2*(3 + b)^2*(b*c - 3*d)^3*(c - d)*(c + d)*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(373778\) vs. \(2(642)=1284\).

Time = 16.30 (sec) , antiderivative size = 373779, normalized size of antiderivative = 570.65

method	result	size
default	\(\text {Expression too large to display}\)	\(373779\)

[In]

int(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

int(1/((a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2)),x)

[Out]

int(1/((a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2)), x)