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

Optimal. Leaf size=716 \[ -\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}} \]

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(7*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(7/2)) + (8*(b*c -
a*d)^2*(3*a*c*d + b*(c^2 - 4*d^2))*Cos[e + f*x])/(35*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(5/2)) - (2*(b*
c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*Cos[e +
f*x])/(105*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d
*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c
^2*d^4 + 105*d^6))*Cos[e + f*x])/(105*d^2*(c^2 - d^2)^4*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(16*a^3*c*d^3*(11*c^2
+ 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*
c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x
]])/(105*d^3*(c^2 - d^2)^4*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) +
a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]
*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]])

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Rubi [A]  time = 1.43605, antiderivative size = 716, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2792, 3021, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(7*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(7/2)) + (8*(b*c -
a*d)^2*(3*a*c*d + b*(c^2 - 4*d^2))*Cos[e + f*x])/(35*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(5/2)) - (2*(b*
c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*Cos[e +
f*x])/(105*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d
*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c
^2*d^4 + 105*d^6))*Cos[e + f*x])/(105*d^2*(c^2 - d^2)^4*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(16*a^3*c*d^3*(11*c^2
+ 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*
c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x
]])/(105*d^3*(c^2 - d^2)^4*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) +
a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]
*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*(c^2 - d^2)^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 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(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))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 2754

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

Rule 2752

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

Rubi steps

\begin{align*} \int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}-\frac{2 \int \frac{\frac{1}{2} \left (2 b (b c-a d)^2-7 a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac{1}{2} \left (5 a (b c-a d)^2-7 b \left (a b c^2+\left (a^2+b^2\right ) c d-3 a b d^2\right )\right ) \sin (e+f x)-\frac{1}{2} b \left (6 a b c d-3 a^2 d^2+b^2 \left (4 c^2-7 d^2\right )\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d \left (c^2-d^2\right )}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}+\frac{4 \int \frac{-\frac{5}{4} d \left (36 a^2 b c d^2-a^3 d \left (7 c^2+5 d^2\right )-3 a b^2 d \left (5 c^2+7 d^2\right )-2 b^3 \left (c^3-7 c d^2\right )\right )-\frac{1}{4} \left (36 a^3 c d^3-18 a b^2 c d \left (c^2-7 d^2\right )-9 a^2 b d^2 \left (5 c^2+7 d^2\right )-b^3 \left (8 c^4-7 c^2 d^2+35 d^4\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 d^2 \left (c^2-d^2\right )^2}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}-\frac{8 \int \frac{\frac{3}{8} d \left (9 a^2 b d^2 \left (25 c^2+7 d^2\right )-a^3 c d \left (35 c^2+61 d^2\right )-3 a b^2 d \left (19 c^3+77 c d^2\right )-b^3 \left (2 c^4-63 c^2 d^2-35 d^4\right )\right )-\frac{1}{8} (b c-a d) \left (8 b^2 c^4+26 a b c^3 d+71 a^2 c^2 d^2-17 b^2 c^2 d^2-218 a b c d^3+25 a^2 d^4+105 b^2 d^4\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 d^2 \left (c^2-d^2\right )^3}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{16 \int \frac{-\frac{1}{16} d \left (144 a^2 b c d^2 \left (5 c^2+3 d^2\right )-a^3 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-3 a b^2 d \left (51 c^4+298 c^2 d^2+35 d^4\right )+2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )\right )+\frac{1}{16} \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^2 \left (c^2-d^2\right )^4}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{105 d^3 \left (c^2-d^2\right )^4}--\frac{\left (16 \left (-\frac{1}{16} d^2 \left (144 a^2 b c d^2 \left (5 c^2+3 d^2\right )-a^3 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-3 a b^2 d \left (51 c^4+298 c^2 d^2+35 d^4\right )+2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )\right )-\frac{1}{16} c \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^3 \left (c^2-d^2\right )^4}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\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}{105 d^3 \left (c^2-d^2\right )^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}--\frac{\left (16 \left (-\frac{1}{16} d^2 \left (144 a^2 b c d^2 \left (5 c^2+3 d^2\right )-a^3 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-3 a b^2 d \left (51 c^4+298 c^2 d^2+35 d^4\right )+2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )\right )-\frac{1}{16} c \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\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}{105 d^3 \left (c^2-d^2\right )^4 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\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)}}{105 d^3 \left (c^2-d^2\right )^4 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d) \left (8 b^2 c^4+26 a b c^3 d+71 a^2 c^2 d^2-17 b^2 c^2 d^2-218 a b c d^3+25 a^2 d^4+105 b^2 d^4\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}}}{105 d^3 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.94125, size = 1127, normalized size = 1.57 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-\frac{2 \left (-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)\right )}{7 d^2 \left (d^2-c^2\right ) (c+d \sin (e+f x))^4}-\frac{2 \left (8 b^3 \cos (e+f x) c^6+18 a b^2 d \cos (e+f x) c^5-23 b^3 d^2 \cos (e+f x) c^4+45 a^2 b d^2 \cos (e+f x) c^4-176 a^3 d^3 \cos (e+f x) c^3-372 a b^2 d^3 \cos (e+f x) c^3+294 b^3 d^4 \cos (e+f x) c^2+918 a^2 b d^4 \cos (e+f x) c^2-208 a^3 d^5 \cos (e+f x) c-798 a b^2 d^5 \cos (e+f x) c+105 b^3 d^6 \cos (e+f x)+189 a^2 b d^6 \cos (e+f x)\right )}{105 d^2 \left (d^2-c^2\right )^4 (c+d \sin (e+f x))}-\frac{2 \left (-8 b^3 \cos (e+f x) c^5-18 a b^2 d \cos (e+f x) c^4+17 b^3 d^2 \cos (e+f x) c^3-45 a^2 b d^2 \cos (e+f x) c^3+71 a^3 d^3 \cos (e+f x) c^2+201 a b^2 d^3 \cos (e+f x) c^2-105 b^3 d^4 \cos (e+f x) c-243 a^2 b d^4 \cos (e+f x) c+25 a^3 d^5 \cos (e+f x)+105 a b^2 d^5 \cos (e+f x)\right )}{105 d^2 \left (d^2-c^2\right )^3 (c+d \sin (e+f x))^2}-\frac{6 \left (-3 b^3 \cos (e+f x) c^4+2 a b^2 d \cos (e+f x) c^3+7 b^3 d^2 \cos (e+f x) c^2+5 a^2 b d^2 \cos (e+f x) c^2-4 a^3 d^3 \cos (e+f x) c-14 a b^2 d^3 \cos (e+f x) c+7 a^2 b d^4 \cos (e+f x)\right )}{35 d^2 \left (d^2-c^2\right )^2 (c+d \sin (e+f x))^3}\right )}{f}-\frac{-\frac{2 \left (-25 a^3 d^6-105 a b^2 d^6+210 b^3 c d^5+432 a^2 b c d^5-254 a^3 c^2 d^4-894 a b^2 c^2 d^4+172 b^3 c^3 d^3+720 a^2 b c^3 d^3-105 a^3 c^4 d^2-153 a b^2 c^4 d^2+2 b^3 c^5 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\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)}}-\frac{\left (8 b^3 c^6+18 a b^2 d c^5-23 b^3 d^2 c^4+45 a^2 b d^2 c^4-176 a^3 d^3 c^3-372 a b^2 d^3 c^3+294 b^3 d^4 c^2+918 a^2 b d^4 c^2-208 a^3 d^5 c-798 a b^2 d^5 c+105 b^3 d^6+189 a^2 b d^6\right ) \left (\frac{2 (c+d) E\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}}}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c 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}}}{\sqrt{c+d \sin (e+f x)}}\right )}{d}}{105 (c-d)^4 d^2 (c+d)^4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*Sin[e + f*x]]*((-2*(-(b^3*c^3*Cos[e + f*x]) + 3*a*b^2*c^2*d*Cos[e + f*x] - 3*a^2*b*c*d^2*Cos[e + f
*x] + a^3*d^3*Cos[e + f*x]))/(7*d^2*(-c^2 + d^2)*(c + d*Sin[e + f*x])^4) - (6*(-3*b^3*c^4*Cos[e + f*x] + 2*a*b
^2*c^3*d*Cos[e + f*x] + 5*a^2*b*c^2*d^2*Cos[e + f*x] + 7*b^3*c^2*d^2*Cos[e + f*x] - 4*a^3*c*d^3*Cos[e + f*x] -
14*a*b^2*c*d^3*Cos[e + f*x] + 7*a^2*b*d^4*Cos[e + f*x]))/(35*d^2*(-c^2 + d^2)^2*(c + d*Sin[e + f*x])^3) - (2*
(-8*b^3*c^5*Cos[e + f*x] - 18*a*b^2*c^4*d*Cos[e + f*x] - 45*a^2*b*c^3*d^2*Cos[e + f*x] + 17*b^3*c^3*d^2*Cos[e
+ f*x] + 71*a^3*c^2*d^3*Cos[e + f*x] + 201*a*b^2*c^2*d^3*Cos[e + f*x] - 243*a^2*b*c*d^4*Cos[e + f*x] - 105*b^3
*c*d^4*Cos[e + f*x] + 25*a^3*d^5*Cos[e + f*x] + 105*a*b^2*d^5*Cos[e + f*x]))/(105*d^2*(-c^2 + d^2)^3*(c + d*Si
n[e + f*x])^2) - (2*(8*b^3*c^6*Cos[e + f*x] + 18*a*b^2*c^5*d*Cos[e + f*x] + 45*a^2*b*c^4*d^2*Cos[e + f*x] - 23
*b^3*c^4*d^2*Cos[e + f*x] - 176*a^3*c^3*d^3*Cos[e + f*x] - 372*a*b^2*c^3*d^3*Cos[e + f*x] + 918*a^2*b*c^2*d^4*
Cos[e + f*x] + 294*b^3*c^2*d^4*Cos[e + f*x] - 208*a^3*c*d^5*Cos[e + f*x] - 798*a*b^2*c*d^5*Cos[e + f*x] + 189*
a^2*b*d^6*Cos[e + f*x] + 105*b^3*d^6*Cos[e + f*x]))/(105*d^2*(-c^2 + d^2)^4*(c + d*Sin[e + f*x]))))/f - ((-2*(
2*b^3*c^5*d - 105*a^3*c^4*d^2 - 153*a*b^2*c^4*d^2 + 720*a^2*b*c^3*d^3 + 172*b^3*c^3*d^3 - 254*a^3*c^2*d^4 - 89
4*a*b^2*c^2*d^4 + 432*a^2*b*c*d^5 + 210*b^3*c*d^5 - 25*a^3*d^6 - 105*a*b^2*d^6)*EllipticF[(-e + Pi/2 - f*x)/2,
(2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - ((8*b^3*c^6 + 18*a*b^2*c^5*d +
45*a^2*b*c^4*d^2 - 23*b^3*c^4*d^2 - 176*a^3*c^3*d^3 - 372*a*b^2*c^3*d^3 + 918*a^2*b*c^2*d^4 + 294*b^3*c^2*d^4
- 208*a^3*c*d^5 - 798*a*b^2*c*d^5 + 189*a^2*b*d^6 + 105*b^3*d^6)*((2*(c + d)*EllipticE[(-e + Pi/2 - f*x)/2, (2
*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e + Pi/2 - f*x)/2
, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]]))/d)/(105*(c - d)^4*d^2*(c + d)^
4*f)

________________________________________________________________________________________

Maple [B]  time = 12.731, size = 2111, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(3*b*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^3*(2/5/(c^2-d^2)/d^2*(-(-d*sin(f*x+
e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^3+16/15*c/(c^2-d^2)^2/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(si
n(f*x+e)+c/d)^2+2/15*d*cos(f*x+e)^2/(c^2-d^2)^3*(23*c^2+9*d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(15*c
^3+17*c*d^2)/(15*c^6-45*c^4*d^2+45*c^2*d^4-15*d^6)*(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))+2/15*d*(23*c^2+9*d^2)/(c^2-d^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*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))))+(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^3*(2/7/(c^2-d^2)/d^3*(-(-d*sin(f*x+e)-c)*cos(
f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^4+24/35/(c^2-d^2)^2/d^2*c*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)
+c/d)^3+2/105*(71*c^2+25*d^2)/d/(c^2-d^2)^3*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+32/105*
d*cos(f*x+e)^2/(c^2-d^2)^4*c*(11*c^2+13*d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(105*c^4+254*c^2*d^2+25
*d^4)/(105*c^8-420*c^6*d^2+630*c^4*d^4-420*c^2*d^6+105*d^8)*(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*si
n(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+32/105*c*d*(11*c^2+13*d^2)/(c^2-d^2)^4*(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))))+b^3/d^3*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*c
/(c^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))+2/(c^
2-d^2)*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))))+3*b^2*(a*d-b*c)/d^3*(2/3/(c^2-d^2)/d*(-(-d*
sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(
f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(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))+8/3*c*d/(c^2-d^2)^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)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{5 \, c d^{4} \cos \left (f x + e\right )^{4} + c^{5} + 10 \, c^{3} d^{2} + 5 \, c d^{4} - 10 \,{\left (c^{3} d^{2} + c d^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (d^{5} \cos \left (f x + e\right )^{4} + 5 \, c^{4} d + 10 \, c^{2} d^{3} + d^{5} - 2 \,{\left (5 \, c^{2} d^{3} + d^{5}\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((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*
sin(f*x + e) + c)/(5*c*d^4*cos(f*x + e)^4 + c^5 + 10*c^3*d^2 + 5*c*d^4 - 10*(c^3*d^2 + c*d^4)*cos(f*x + e)^2 +
(d^5*cos(f*x + e)^4 + 5*c^4*d + 10*c^2*d^3 + d^5 - 2*(5*c^2*d^3 + d^5)*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((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(9/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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