\(\int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx\) [744]

Optimal result

Integrand size = 27, antiderivative size = 693 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 (b c-3 d)^2 \cos (e+f x) (3+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac {8 (b c-3 d)^2 \left (9 c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac {2 (b c-3 d) \left (9 d^2 \left (71 c^2+25 d^2\right )+3 b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (432 c d^3 \left (11 c^2+13 d^2\right )-18 b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-81 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (432 c d^3 \left (11 c^2+13 d^2\right )-18 b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-81 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^3 \left (c^2-d^2\right )^4 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-3 d) \left (9 d^2 \left (71 c^2+25 d^2\right )+3 b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^3 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/7*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(7/2)+8/35*(-a*d+b*c)^2*(3*a*c*d+b
*(c^2-4*d^2))*cos(f*x+e)/d^2/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(5/2)-2/105*(-a*d+b*c)*(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(f*x+e)/d^2/(c^2-d^2)^3/f/(c+d*sin(f*x+e))^(3/2)+2/1
05*(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(f*x+e)/d^2/(c^2-d^2)^4/f/(c+d*sin(f*x+e))^(1/2)-2/105*(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))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos
(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^3/(c^2-d^2)^4/f/((c+d*sin(f*x+e))/(c+
d))^(1/2)-2/105*(-a*d+b*c)*(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))*(
sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c
+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/(c^2-d^2)^3/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2871, 3100, 2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=-\frac {2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) (b c-a d) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^3 \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^4 \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^4 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{5/2}}+\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}} \]

[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 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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 2833

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 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(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 3100

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]

Rubi steps \begin{align*} \text {integral}& = \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 c d \left (19 c^2+77 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 ) \operatorname {EllipticF}\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] (verified)

Time = 8.72 (sec) , antiderivative size = 651, normalized size of antiderivative = 0.94 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 \left (\frac {\left (-d^2 \left (27 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-1296 b \left (5 c^3 d^2+3 c d^4\right )-2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )+9 b^2 \left (51 c^4 d+298 c^2 d^3+35 d^5\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (18 b^2 \left (3 c^5 d-62 c^3 d^3-133 c d^5\right )-432 \left (11 c^3 d^3+13 c d^5\right )+81 b \left (5 c^4 d^2+102 c^2 d^4+21 d^6\right )+b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) (c+d \sin (e+f x))^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d)^4 (c+d)^4}-\frac {d \cos (e+f x) \left (15 (b c-3 d)^3 \left (c^2-d^2\right )^3-9 (b c-3 d)^2 \left (c^2-d^2\right )^2 \left (3 b c^2+12 c d-7 b d^2\right ) (c+d \sin (e+f x))+\left (c^2-d^2\right ) \left (81 b \left (5 c^3 d^2+27 c d^4\right )+b^3 \left (8 c^5-17 c^3 d^2+105 c d^4\right )+9 b^2 \left (6 c^4 d-67 c^2 d^3-35 d^5\right )-27 \left (71 c^2 d^3+25 d^5\right )\right ) (c+d \sin (e+f x))^2+\left (18 b^2 \left (3 c^5 d-62 c^3 d^3-133 c d^5\right )-432 \left (11 c^3 d^3+13 c d^5\right )+81 b \left (5 c^4 d^2+102 c^2 d^4+21 d^6\right )+b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) (c+d \sin (e+f x))^3\right )}{\left (c^2-d^2\right )^4}\right )}{105 d^3 f (c+d \sin (e+f x))^{7/2}} \]

[In]

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

[Out]

(2*(((-(d^2*(27*d*(105*c^4 + 254*c^2*d^2 + 25*d^4) - 1296*b*(5*c^3*d^2 + 3*c*d^4) - 2*b^3*(c^5 + 86*c^3*d^2 +
105*c*d^4) + 9*b^2*(51*c^4*d + 298*c^2*d^3 + 35*d^5))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (18*b
^2*(3*c^5*d - 62*c^3*d^3 - 133*c*d^5) - 432*(11*c^3*d^3 + 13*c*d^5) + 81*b*(5*c^4*d^2 + 102*c^2*d^4 + 21*d^6)
+ b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] -
c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(c + d*Sin[e + f*x])^3*Sqrt[(c + d*Sin[e + f*x])/(c + d)]
)/((c - d)^4*(c + d)^4) - (d*Cos[e + f*x]*(15*(b*c - 3*d)^3*(c^2 - d^2)^3 - 9*(b*c - 3*d)^2*(c^2 - d^2)^2*(3*b
*c^2 + 12*c*d - 7*b*d^2)*(c + d*Sin[e + f*x]) + (c^2 - d^2)*(81*b*(5*c^3*d^2 + 27*c*d^4) + b^3*(8*c^5 - 17*c^3
*d^2 + 105*c*d^4) + 9*b^2*(6*c^4*d - 67*c^2*d^3 - 35*d^5) - 27*(71*c^2*d^3 + 25*d^5))*(c + d*Sin[e + f*x])^2 +
(18*b^2*(3*c^5*d - 62*c^3*d^3 - 133*c*d^5) - 432*(11*c^3*d^3 + 13*c*d^5) + 81*b*(5*c^4*d^2 + 102*c^2*d^4 + 21
*d^6) + b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*(c + d*Sin[e + f*x])^3))/(c^2 - d^2)^4))/(105*d^3*f*
(c + d*Sin[e + f*x])^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2110\) vs. \(2(754)=1508\).

Time = 72.35 (sec) , antiderivative size = 2111, normalized size of antiderivative = 3.05

method	result	size
default	\(\text {Expression too large to display}\)	\(2111\)
parts	\(\text {Expression too large to display}\)	\(5606\)

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(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)*(1/(c-d)*(-sin(f*x+
e)-1)*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)*(1/(c-d)*(-sin(f*x+
e)-1)*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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*Elliptic
F(((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)*(1/(c-d)*(-sin(f*x+e)-1)*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*(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)/(sin(f*x+e)+c/d)^2+2
/15*cos(f*x+e)^2*d/(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)*(1/(c-
d)*(-sin(f*x+e)-1)*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)*(1/(c-d)*(-sin(f*x+e)-1)*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*cos(f*x+e)^2*
d/(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)*(1/(c-d)*(-sin(f*x+e)-1)*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))+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)*(1/(c-d)*(-sin(f*x+e)-1)*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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.52 (sec) , antiderivative size = 4693, normalized size of antiderivative = 6.77 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/315*((sqrt(2)*(16*b^3*c^7*d^4 + 36*a*b^2*c^6*d^5 + 2*(45*a^2*b - 26*b^3)*c^5*d^6 - (37*a^3 + 285*a*b^2)*c^4*
d^7 - 36*(9*a^2*b - 2*b^3)*c^3*d^8 + 2*(173*a^3 + 543*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*(5*a
^3 + 21*a*b^2)*d^11)*cos(f*x + e)^4 - 2*sqrt(2)*(48*b^3*c^9*d^2 + 108*a*b^2*c^8*d^3 + 10*(27*a^2*b - 14*b^3)*c
^7*d^4 - 3*(37*a^3 + 273*a*b^2)*c^6*d^5 - 2*(441*a^2*b - 82*b^3)*c^5*d^6 + (1001*a^3 + 2973*a*b^2)*c^4*d^7 - 5
4*(57*a^2*b + 22*b^3)*c^3*d^8 + (571*a^3 + 2031*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*(5*a^3 + 2
1*a*b^2)*d^11)*cos(f*x + e)^2 - 4*(sqrt(2)*(16*b^3*c^8*d^3 + 36*a*b^2*c^7*d^4 + 2*(45*a^2*b - 26*b^3)*c^6*d^5
- (37*a^3 + 285*a*b^2)*c^5*d^6 - 36*(9*a^2*b - 2*b^3)*c^4*d^7 + 2*(173*a^3 + 543*a*b^2)*c^3*d^8 - 6*(153*a^2*b
+ 70*b^3)*c^2*d^9 + 15*(5*a^3 + 21*a*b^2)*c*d^10)*cos(f*x + e)^2 - sqrt(2)*(16*b^3*c^10*d + 36*a*b^2*c^9*d^2
+ 18*(5*a^2*b - 2*b^3)*c^8*d^3 - (37*a^3 + 249*a*b^2)*c^7*d^4 - 2*(117*a^2*b - 10*b^3)*c^6*d^5 + 3*(103*a^3 +
267*a*b^2)*c^5*d^6 - 6*(207*a^2*b + 58*b^3)*c^4*d^7 + (421*a^3 + 1401*a*b^2)*c^3*d^8 - 6*(153*a^2*b + 70*b^3)*
c^2*d^9 + 15*(5*a^3 + 21*a*b^2)*c*d^10))*sin(f*x + e) + sqrt(2)*(16*b^3*c^11 + 36*a*b^2*c^10*d + 2*(45*a^2*b +
22*b^3)*c^9*d^2 - (37*a^3 + 69*a*b^2)*c^8*d^3 + 8*(27*a^2*b - 28*b^3)*c^7*d^4 + 4*(31*a^3 - 147*a*b^2)*c^6*d^
5 - 4*(693*a^2*b + 10*b^3)*c^5*d^6 + 2*(1057*a^3 + 3273*a*b^2)*c^4*d^7 - 72*(81*a^2*b + 34*b^3)*c^3*d^8 + 4*(1
99*a^3 + 744*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*(5*a^3 + 21*a*b^2)*d^11))*sqrt(I*d)*weierstra
ssPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x +
e) - 2*I*c)/d) + (sqrt(2)*(16*b^3*c^7*d^4 + 36*a*b^2*c^6*d^5 + 2*(45*a^2*b - 26*b^3)*c^5*d^6 - (37*a^3 + 285*a
*b^2)*c^4*d^7 - 36*(9*a^2*b - 2*b^3)*c^3*d^8 + 2*(173*a^3 + 543*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10
+ 15*(5*a^3 + 21*a*b^2)*d^11)*cos(f*x + e)^4 - 2*sqrt(2)*(48*b^3*c^9*d^2 + 108*a*b^2*c^8*d^3 + 10*(27*a^2*b -
14*b^3)*c^7*d^4 - 3*(37*a^3 + 273*a*b^2)*c^6*d^5 - 2*(441*a^2*b - 82*b^3)*c^5*d^6 + (1001*a^3 + 2973*a*b^2)*c
^4*d^7 - 54*(57*a^2*b + 22*b^3)*c^3*d^8 + (571*a^3 + 2031*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*
(5*a^3 + 21*a*b^2)*d^11)*cos(f*x + e)^2 - 4*(sqrt(2)*(16*b^3*c^8*d^3 + 36*a*b^2*c^7*d^4 + 2*(45*a^2*b - 26*b^3
)*c^6*d^5 - (37*a^3 + 285*a*b^2)*c^5*d^6 - 36*(9*a^2*b - 2*b^3)*c^4*d^7 + 2*(173*a^3 + 543*a*b^2)*c^3*d^8 - 6*
(153*a^2*b + 70*b^3)*c^2*d^9 + 15*(5*a^3 + 21*a*b^2)*c*d^10)*cos(f*x + e)^2 - sqrt(2)*(16*b^3*c^10*d + 36*a*b^
2*c^9*d^2 + 18*(5*a^2*b - 2*b^3)*c^8*d^3 - (37*a^3 + 249*a*b^2)*c^7*d^4 - 2*(117*a^2*b - 10*b^3)*c^6*d^5 + 3*(
103*a^3 + 267*a*b^2)*c^5*d^6 - 6*(207*a^2*b + 58*b^3)*c^4*d^7 + (421*a^3 + 1401*a*b^2)*c^3*d^8 - 6*(153*a^2*b
+ 70*b^3)*c^2*d^9 + 15*(5*a^3 + 21*a*b^2)*c*d^10))*sin(f*x + e) + sqrt(2)*(16*b^3*c^11 + 36*a*b^2*c^10*d + 2*(
45*a^2*b + 22*b^3)*c^9*d^2 - (37*a^3 + 69*a*b^2)*c^8*d^3 + 8*(27*a^2*b - 28*b^3)*c^7*d^4 + 4*(31*a^3 - 147*a*b
^2)*c^6*d^5 - 4*(693*a^2*b + 10*b^3)*c^5*d^6 + 2*(1057*a^3 + 3273*a*b^2)*c^4*d^7 - 72*(81*a^2*b + 34*b^3)*c^3*
d^8 + 4*(199*a^3 + 744*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*(5*a^3 + 21*a*b^2)*d^11))*sqrt(-I*d
)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*
d*sin(f*x + e) + 2*I*c)/d) + 3*(sqrt(2)*(8*I*b^3*c^6*d^5 + 18*I*a*b^2*c^5*d^6 + I*(45*a^2*b - 23*b^3)*c^4*d^7
- 4*I*(44*a^3 + 93*a*b^2)*c^3*d^8 + 6*I*(153*a^2*b + 49*b^3)*c^2*d^9 - 2*I*(104*a^3 + 399*a*b^2)*c*d^10 + 21*I
*(9*a^2*b + 5*b^3)*d^11)*cos(f*x + e)^4 + 2*sqrt(2)*(-24*I*b^3*c^8*d^3 - 54*I*a*b^2*c^7*d^4 - I*(135*a^2*b - 6
1*b^3)*c^6*d^5 + 6*I*(88*a^3 + 183*a*b^2)*c^5*d^6 - I*(2799*a^2*b + 859*b^3)*c^4*d^7 + 2*I*(400*a^3 + 1383*a*b
^2)*c^3*d^8 - 3*I*(495*a^2*b + 203*b^3)*c^2*d^9 + 2*I*(104*a^3 + 399*a*b^2)*c*d^10 - 21*I*(9*a^2*b + 5*b^3)*d^
11)*cos(f*x + e)^2 + 4*(sqrt(2)*(-8*I*b^3*c^7*d^4 - 18*I*a*b^2*c^6*d^5 - I*(45*a^2*b - 23*b^3)*c^5*d^6 + 4*I*(
44*a^3 + 93*a*b^2)*c^4*d^7 - 6*I*(153*a^2*b + 49*b^3)*c^3*d^8 + 2*I*(104*a^3 + 399*a*b^2)*c^2*d^9 - 21*I*(9*a^
2*b + 5*b^3)*c*d^10)*cos(f*x + e)^2 + sqrt(2)*(8*I*b^3*c^9*d^2 + 18*I*a*b^2*c^8*d^3 + 15*I*(3*a^2*b - b^3)*c^7
*d^4 - 2*I*(88*a^3 + 177*a*b^2)*c^6*d^5 + I*(963*a^2*b + 271*b^3)*c^5*d^6 - 6*I*(64*a^3 + 195*a*b^2)*c^4*d^7 +
3*I*(369*a^2*b + 133*b^3)*c^3*d^8 - 2*I*(104*a^3 + 399*a*b^2)*c^2*d^9 + 21*I*(9*a^2*b + 5*b^3)*c*d^10))*sin(f
*x + e) + sqrt(2)*(8*I*b^3*c^10*d + 18*I*a*b^2*c^9*d^2 + 5*I*(9*a^2*b + 5*b^3)*c^8*d^3 - 88*I*(2*a^3 + 3*a*b^2
)*c^7*d^4 + 4*I*(297*a^2*b + 41*b^3)*c^6*d^5 - 4*I*(316*a^3 + 753*a*b^2)*c^5*d^6 + 2*I*(2871*a^2*b + 923*b^3)*
c^4*d^7 - 8*I*(178*a^3 + 645*a*b^2)*c^3*d^8 + 12*I*(171*a^2*b + 77*b^3)*c^2*d^9 - 2*I*(104*a^3 + 399*a*b^2)*c*
d^10 + 21*I*(9*a^2*b + 5*b^3)*d^11))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*
c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x +
e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(sqrt(2)*(-8*I*b^3*c^6*d^5 - 18*I*a*b^2*c^5*d^6 - I*(45*a^2*b - 23*b^
3)*c^4*d^7 + 4*I*(44*a^3 + 93*a*b^2)*c^3*d^8 - 6*I*(153*a^2*b + 49*b^3)*c^2*d^9 + 2*I*(104*a^3 + 399*a*b^2)*c*
d^10 - 21*I*(9*a^2*b + 5*b^3)*d^11)*cos(f*x + e)^4 + 2*sqrt(2)*(24*I*b^3*c^8*d^3 + 54*I*a*b^2*c^7*d^4 + I*(135
*a^2*b - 61*b^3)*c^6*d^5 - 6*I*(88*a^3 + 183*a*b^2)*c^5*d^6 + I*(2799*a^2*b + 859*b^3)*c^4*d^7 - 2*I*(400*a^3
+ 1383*a*b^2)*c^3*d^8 + 3*I*(495*a^2*b + 203*b^3)*c^2*d^9 - 2*I*(104*a^3 + 399*a*b^2)*c*d^10 + 21*I*(9*a^2*b +
5*b^3)*d^11)*cos(f*x + e)^2 + 4*(sqrt(2)*(8*I*b^3*c^7*d^4 + 18*I*a*b^2*c^6*d^5 + I*(45*a^2*b - 23*b^3)*c^5*d^
6 - 4*I*(44*a^3 + 93*a*b^2)*c^4*d^7 + 6*I*(153*a^2*b + 49*b^3)*c^3*d^8 - 2*I*(104*a^3 + 399*a*b^2)*c^2*d^9 + 2
1*I*(9*a^2*b + 5*b^3)*c*d^10)*cos(f*x + e)^2 + sqrt(2)*(-8*I*b^3*c^9*d^2 - 18*I*a*b^2*c^8*d^3 - 15*I*(3*a^2*b
- b^3)*c^7*d^4 + 2*I*(88*a^3 + 177*a*b^2)*c^6*d^5 - I*(963*a^2*b + 271*b^3)*c^5*d^6 + 6*I*(64*a^3 + 195*a*b^2)
*c^4*d^7 - 3*I*(369*a^2*b + 133*b^3)*c^3*d^8 + 2*I*(104*a^3 + 399*a*b^2)*c^2*d^9 - 21*I*(9*a^2*b + 5*b^3)*c*d^
10))*sin(f*x + e) + sqrt(2)*(-8*I*b^3*c^10*d - 18*I*a*b^2*c^9*d^2 - 5*I*(9*a^2*b + 5*b^3)*c^8*d^3 + 88*I*(2*a^
3 + 3*a*b^2)*c^7*d^4 - 4*I*(297*a^2*b + 41*b^3)*c^6*d^5 + 4*I*(316*a^3 + 753*a*b^2)*c^5*d^6 - 2*I*(2871*a^2*b
+ 923*b^3)*c^4*d^7 + 8*I*(178*a^3 + 645*a*b^2)*c^3*d^8 - 12*I*(171*a^2*b + 77*b^3)*c^2*d^9 + 2*I*(104*a^3 + 39
9*a*b^2)*c*d^10 - 21*I*(9*a^2*b + 5*b^3)*d^11))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8
*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(
3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) + 6*((32*b^3*c^7*d^4 + 72*a*b^2*c^6*d^5 + 2*(90*a^2*b - 47*
b^3)*c^5*d^6 - (599*a^3 + 1335*a*b^2)*c^4*d^7 + 4*(738*a^2*b + 251*b^3)*c^3*d^8 - 2*(289*a^3 + 1149*a*b^2)*c^2
*d^9 + 6*(54*a^2*b + 35*b^3)*c*d^10 + 5*(5*a^3 + 21*a*b^2)*d^11)*cos(f*x + e)^3 - (4*b^3*c^9*d^2 + 9*a*b^2*c^8
*d^3 + 4*(45*a^2*b + 14*b^3)*c^7*d^4 - 2*(149*a^3 + 273*a*b^2)*c^6*d^5 + 2*(567*a^2*b + 107*b^3)*c^5*d^6 - 2*(
322*a^3 + 951*a*b^2)*c^4*d^7 + 4*(738*a^2*b + 263*b^3)*c^3*d^8 - 2*(317*a^3 + 1137*a*b^2)*c^2*d^9 + 6*(57*a^2*
b + 35*b^3)*c*d^10 + 5*(8*a^3 + 21*a*b^2)*d^11)*cos(f*x + e) + ((8*b^3*c^6*d^5 + 18*a*b^2*c^5*d^6 + (45*a^2*b
- 23*b^3)*c^4*d^7 - 4*(44*a^3 + 93*a*b^2)*c^3*d^8 + 6*(153*a^2*b + 49*b^3)*c^2*d^9 - 2*(104*a^3 + 399*a*b^2)*c
*d^10 + 21*(9*a^2*b + 5*b^3)*d^11)*cos(f*x + e)^3 - (13*b^3*c^8*d^3 + 108*a*b^2*c^7*d^4 + 6*(45*a^2*b + b^3)*c
^6*d^5 - 2*(353*a^3 + 849*a*b^2)*c^5*d^6 + 2*(1584*a^2*b + 475*b^3)*c^4*d^7 - 12*(53*a^3 + 192*a*b^2)*c^3*d^8
+ 6*(153*a^2*b + 77*b^3)*c^2*d^9 - 2*(97*a^3 + 357*a*b^2)*c*d^10 + 21*(12*a^2*b + 5*b^3)*d^11)*cos(f*x + e))*s
in(f*x + e))*sqrt(d*sin(f*x + e) + c))/((c^8*d^8 - 4*c^6*d^10 + 6*c^4*d^12 - 4*c^2*d^14 + d^16)*f*cos(f*x + e)
^4 - 2*(3*c^10*d^6 - 11*c^8*d^8 + 14*c^6*d^10 - 6*c^4*d^12 - c^2*d^14 + d^16)*f*cos(f*x + e)^2 + (c^12*d^4 + 2
*c^10*d^6 - 17*c^8*d^8 + 28*c^6*d^10 - 17*c^4*d^12 + 2*c^2*d^14 + d^16)*f - 4*((c^9*d^7 - 4*c^7*d^9 + 6*c^5*d^
11 - 4*c^3*d^13 + c*d^15)*f*cos(f*x + e)^2 - (c^11*d^5 - 3*c^9*d^7 + 2*c^7*d^9 + 2*c^5*d^11 - 3*c^3*d^13 + c*d
^15)*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]

[In]

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

[Out]

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