3.758 \(\int \frac{(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=816 \[ \frac{\left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} (b c-a d)^3}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\cos (e+f x) (c+d \sin (e+f x))^{5/2} (b c-a d)^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}-\frac{\left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (a^2-b^2\right )^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

[Out]

(d*(36*a^3*b*c*d^2 - 35*a^4*d^3 + b^4*d*(45*c^2 - 8*d^2) - 18*a*b^3*c*(c^2 + 5*d^2) + a^2*b^2*d*(9*c^2 + 61*d^
2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(12*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(6*a*b*c + 7*a^2*d - 13*b
^2*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*C
os[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((185*a^4*b*c*d^3 - 105*a
^5*d^4 - b^5*c*d*(51*c^2 - 104*d^2) - 15*a^3*b^2*d^2*(3*c^2 - 13*d^2) - a^2*b^3*c*d*(21*c^2 + 361*d^2) + 9*a*b
^4*(2*c^4 + 17*c^2*d^2 - 8*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(12*b^
4*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((150*a^5*b*c*d^4 - 105*a^6*d^5 - 12*a^3*b^3*c*d^2*(4*
c^2 + 29*d^2) + a^4*b^2*d^3*(26*c^2 + 223*d^2) - b^6*d*(57*c^4 + 136*c^2*d^2 + 8*d^4) + 6*a*b^5*c*(3*c^4 + 38*
c^2*d^2 + 48*d^4) - a^2*b^4*d*(33*c^4 + 70*c^2*d^2 + 128*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sq
rt[(c + d*Sin[e + f*x])/(c + d)])/(12*b^5*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^3*(20*a^3*b
*c*d - 44*a*b^3*c*d + 35*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 43*d^2) + b^4*(4*c^2 + 63*d^2))*EllipticPi[(2*b)/(a + b)
, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^5*(a + b)^3*f*Sqrt[c +
d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 3.17912, antiderivative size = 816, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {2792, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{\left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} (b c-a d)^3}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\cos (e+f x) (c+d \sin (e+f x))^{5/2} (b c-a d)^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}-\frac{\left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (a^2-b^2\right )^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3,x]

[Out]

(d*(36*a^3*b*c*d^2 - 35*a^4*d^3 + b^4*d*(45*c^2 - 8*d^2) - 18*a*b^3*c*(c^2 + 5*d^2) + a^2*b^2*d*(9*c^2 + 61*d^
2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(12*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(6*a*b*c + 7*a^2*d - 13*b
^2*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*C
os[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((185*a^4*b*c*d^3 - 105*a
^5*d^4 - b^5*c*d*(51*c^2 - 104*d^2) - 15*a^3*b^2*d^2*(3*c^2 - 13*d^2) - a^2*b^3*c*d*(21*c^2 + 361*d^2) + 9*a*b
^4*(2*c^4 + 17*c^2*d^2 - 8*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(12*b^
4*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((150*a^5*b*c*d^4 - 105*a^6*d^5 - 12*a^3*b^3*c*d^2*(4*
c^2 + 29*d^2) + a^4*b^2*d^3*(26*c^2 + 223*d^2) - b^6*d*(57*c^4 + 136*c^2*d^2 + 8*d^4) + 6*a*b^5*c*(3*c^4 + 38*
c^2*d^2 + 48*d^4) - a^2*b^4*d*(33*c^4 + 70*c^2*d^2 + 128*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sq
rt[(c + d*Sin[e + f*x])/(c + d)])/(12*b^5*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^3*(20*a^3*b
*c*d - 44*a*b^3*c*d + 35*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 43*d^2) + b^4*(4*c^2 + 63*d^2))*EllipticPi[(2*b)/(a + b)
, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^5*(a + b)^3*f*Sqrt[c +
d*Sin[e + f*x]])

Rule 2792

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(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*Sin[e + f*x])^(m - 3)*(c + d*Sin[e +
f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*
d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*
n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], 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] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3047

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[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(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*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*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] && GtQ[m, 0] && LtQ[n, -1]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3059

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(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 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3002

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2807

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \frac{(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{(c+d \sin (e+f x))^{3/2} \left (\frac{1}{2} \left (5 d (b c-a d)^2+4 b c \left (2 b c d-a \left (c^2+d^2\right )\right )\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)+\frac{1}{2} d \left (6 a b c d-7 a^2 d^2-b^2 \left (3 c^2-4 d^2\right )\right ) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\int \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{1}{4} \left (28 a^3 b c d^3-21 a^4 d^4-2 a b^3 c d \left (27 c^2+47 d^2\right )+b^4 c^2 \left (4 c^2+63 d^2\right )+a^2 b^2 \left (8 c^4+27 c^2 d^2+39 d^4\right )\right )+\frac{1}{2} d \left (7 a^4 c d^2-b^4 c \left (c^2-16 d^2\right )+a^2 b^2 c \left (7 c^2-5 d^2\right )-2 a^3 b d \left (3 c^2-d^2\right )-4 a b^3 d \left (3 c^2+2 d^2\right )\right ) \sin (e+f x)-\frac{1}{4} d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac{(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\int \frac{\frac{1}{8} \left (-99 a^4 b c d^4+35 a^5 d^5+a^3 b^2 d^3 \left (75 c^2-61 d^2\right )+3 b^5 c^3 \left (4 c^2+63 d^2\right )-a b^4 d \left (162 c^4+327 c^2 d^2-8 d^4\right )+3 a^2 b^3 c \left (8 c^4+33 c^2 d^2+69 d^4\right )\right )-\frac{1}{4} d \left (35 a^5 c d^3+a^3 b^2 c d \left (9 c^2-91 d^2\right )+a b^4 c d \left (63 c^2+128 d^2\right )-a^4 b \left (57 c^2 d^2-14 d^4\right )-b^5 \left (3 c^4+120 c^2 d^2+4 d^4\right )-a^2 b^3 \left (15 c^4-69 c^2 d^2+28 d^4\right )\right ) \sin (e+f x)+\frac{1}{8} d \left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac{(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{-\frac{1}{8} d \left (105 a^6 c d^4+3 a^4 b^2 c d^2 \left (15 c^2-98 d^2\right )-5 a^5 b d^3 \left (37 c^2-7 d^2\right )+3 b^6 c^3 \left (4 c^2+63 d^2\right )+a^3 b^3 d \left (21 c^4+436 c^2 d^2-61 d^4\right )-a b^5 d \left (111 c^4+431 c^2 d^2-8 d^4\right )+3 a^2 b^4 c \left (2 c^4-18 c^2 d^2+93 d^4\right )\right )+\frac{1}{8} d \left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 b^4 \left (a^2-b^2\right )^2 d}+\frac{\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{24 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac{(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left ((b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{8 b^5 \left (a^2-b^2\right )^2}-\frac{\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{24 b^5 \left (a^2-b^2\right )^2}+\frac{\left (\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{24 b^4 \left (a^2-b^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac{(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{8 b^5 \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{24 b^5 \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac{(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (a^2-b^2\right )^2 f \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 9.05907, size = 1526, normalized size = 1.87 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-\frac{2 \cos (e+f x) d^4}{3 b^3}+\frac{-11 d^4 \cos (e+f x) a^5+27 b c d^3 \cos (e+f x) a^4+17 b^2 d^4 \cos (e+f x) a^3-15 b^2 c^2 d^2 \cos (e+f x) a^3-51 b^3 c d^3 \cos (e+f x) a^2-7 b^3 c^3 d \cos (e+f x) a^2+6 b^4 c^4 \cos (e+f x) a+51 b^4 c^2 d^2 \cos (e+f x) a-17 b^5 c^3 d \cos (e+f x)}{4 b^3 \left (b^2-a^2\right )^2 (a+b \sin (e+f x))}+\frac{-b^4 \cos (e+f x) c^4+4 a b^3 d \cos (e+f x) c^3-6 a^2 b^2 d^2 \cos (e+f x) c^2+4 a^3 b d^3 \cos (e+f x) c-a^4 d^4 \cos (e+f x)}{2 b^3 \left (b^2-a^2\right ) (a+b \sin (e+f x))^2}\right )}{f}-\frac{-\frac{2 \left (-24 c^5 b^5-104 c d^4 b^5-327 c^3 d^2 b^5+56 a d^5 b^4+501 a c^2 d^3 b^4+306 a c^4 d b^4-48 a^2 c^5 b^3-53 a^2 c d^4 b^3-177 a^2 c^3 d^2 b^3-73 a^3 d^5 b^2-105 a^3 c^2 d^3 b^2+13 a^4 c d^4 b+35 a^5 d^5\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left (140 c d^4 a^5+56 b d^5 a^4-228 b c^2 d^3 a^4-364 b^2 c d^4 a^3+36 b^2 c^3 d^2 a^3-112 b^3 d^5 a^2+276 b^3 c^2 d^3 a^2-60 b^3 c^4 d a^2+512 b^4 c d^4 a+252 b^4 c^3 d^2 a-16 b^5 d^5-480 b^5 c^2 d^3-12 b^5 c^4 d\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left (104 c d^4 b^5-51 c^3 d^2 b^5-72 a d^5 b^4+153 a c^2 d^3 b^4+18 a c^4 d b^4-361 a^2 c d^4 b^3-21 a^2 c^3 d^2 b^3+195 a^3 d^5 b^2-45 a^3 c^2 d^3 b^2+185 a^4 c d^4 b-105 a^5 d^5\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{48 (a-b)^2 b^3 (a+b)^2 f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3,x]

[Out]

(Sqrt[c + d*Sin[e + f*x]]*((-2*d^4*Cos[e + f*x])/(3*b^3) + (-(b^4*c^4*Cos[e + f*x]) + 4*a*b^3*c^3*d*Cos[e + f*
x] - 6*a^2*b^2*c^2*d^2*Cos[e + f*x] + 4*a^3*b*c*d^3*Cos[e + f*x] - a^4*d^4*Cos[e + f*x])/(2*b^3*(-a^2 + b^2)*(
a + b*Sin[e + f*x])^2) + (6*a*b^4*c^4*Cos[e + f*x] - 7*a^2*b^3*c^3*d*Cos[e + f*x] - 17*b^5*c^3*d*Cos[e + f*x]
- 15*a^3*b^2*c^2*d^2*Cos[e + f*x] + 51*a*b^4*c^2*d^2*Cos[e + f*x] + 27*a^4*b*c*d^3*Cos[e + f*x] - 51*a^2*b^3*c
*d^3*Cos[e + f*x] - 11*a^5*d^4*Cos[e + f*x] + 17*a^3*b^2*d^4*Cos[e + f*x])/(4*b^3*(-a^2 + b^2)^2*(a + b*Sin[e
+ f*x]))))/f - ((-2*(-48*a^2*b^3*c^5 - 24*b^5*c^5 + 306*a*b^4*c^4*d - 177*a^2*b^3*c^3*d^2 - 327*b^5*c^3*d^2 -
105*a^3*b^2*c^2*d^3 + 501*a*b^4*c^2*d^3 + 13*a^4*b*c*d^4 - 53*a^2*b^3*c*d^4 - 104*b^5*c*d^4 + 35*a^5*d^5 - 73*
a^3*b^2*d^5 + 56*a*b^4*d^5)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e +
f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-60*a^2*b^3*c^4*d - 12*b^5*c^4*d + 36*a^3*b^2*c^3
*d^2 + 252*a*b^4*c^3*d^2 - 228*a^4*b*c^2*d^3 + 276*a^2*b^3*c^2*d^3 - 480*b^5*c^2*d^3 + 140*a^5*c*d^4 - 364*a^3
*b^2*c*d^4 + 512*a*b^4*c*d^4 + 56*a^4*b*d^5 - 112*a^2*b^3*d^5 - 16*b^5*d^5)*Cos[e + f*x]*((b*c - a*d)*Elliptic
F[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c
- a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(
c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-
1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (
c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(18*a*b^4*c^4*d - 21*a^2*b^3*c^3*d^2 - 51*b^5*c^3*d^2 - 45*a^3*b^2*c^2*
d^3 + 153*a*b^4*c^2*d^3 + 185*a^4*b*c*d^4 - 361*a^2*b^3*c*d^4 + 104*b^5*c*d^4 - 105*a^5*d^5 + 195*a^3*b^2*d^5
- 72*a*b^4*d^5)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]
*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^
(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcS
inh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[
-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*
d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f
*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(48*(a - b)^2*b^3*(a +
b)^2*f)

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Maple [B]  time = 11.506, size = 2775, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(d^3/b^5*(b^2*d^2*(-2/3/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/3*
(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*si
n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-4/3*c/d*(c/d-1)*
((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)
-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+
d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+2*(-3*a*b*d^2+5*b^2*c*d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1
/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-
c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),
((c-d)/(c+d))^(1/2)))+12*a^2*d^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(
f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-
d)/(c+d))^(1/2))-30*a*b*c*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e
)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c
+d))^(1/2))+20*c^2*b^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*
d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^
(1/2)))-20/b^6*d^2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-
sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*El
lipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+1/b^5*(-a^5*d^5+5*a^4*b*c*d^4
-10*a^3*b^2*c^2*d^3+10*a^2*b^3*c^3*d^2-5*a*b^4*c^4*d+b^5*c^5)*(-1/2*b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*si
n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))^2-3/4*b^2*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*
c)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-1/4*d*(7*a^3*d-4*a^2*b*c-a*b^2*d-2*b^3*c)/(a^3*d
-a^2*b*c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-
1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d
))^(1/2))-3/4*b*d*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/
2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c
/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),(
(c-d)/(c+d))^(1/2)))+1/4*(15*a^4*d^2-20*a^3*b*c*d+8*a^2*b^2*c^2-6*a^2*b^2*d^2-4*a*b^3*c*d+4*b^4*c^2+3*b^4*d^2)
/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-si
n(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-
d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2)))+5/b^5*d*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c
^3*d+b^4*c^4)*(-b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-a*d
/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*
x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)
/(c+d))^(1/2))-b*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d
))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin
(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a
^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(
c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+
d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [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((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [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((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

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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((c+d*sin(f*x+e))**(9/2)/(a+b*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^(9/2)/(b*sin(f*x + e) + a)^3, x)