\(\int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx\) [759]

Optimal result

Integrand size = 27, antiderivative size = 561 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-3 d)^2 \left (18 b c+45 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b^2 \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2}-\frac {\left (216 b c d^2-1215 d^3+b^4 d \left (13 c^2-8 d^2\right )-6 b^3 c \left (3 c^2+13 d^2\right )+9 b^2 d \left (5 c^2+29 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b^3 \left (9-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {3 (b c-3 d) \left (108 b c d^2+405 d^3+9 b^2 d \left (c^2-11 d^2\right )-6 b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b^4 \left (9-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d)^2 \left (324 b c d-108 b^3 c d+1215 d^2+18 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 b^4 (3+b)^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

1/2*(-a*d+b*c)^2*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2+1/4*(-a*d+b*c)^2*(5*a^2*d+
6*a*b*c-11*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b^2/(a^2-b^2)^2/f/(a+b*sin(f*x+e))+1/4*(8*a^3*b*c*d^2-15*a
^4*d^3+b^4*d*(13*c^2-8*d^2)-2*a*b^3*c*(3*c^2+13*d^2)+a^2*b^2*d*(5*c^2+29*d^2))*(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)/b^3/(a^2-b^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-3/4*(-a*d+b*c)*(4*a^3*b*c*d^2+5*a^4*d^3+a^2*b^2*d*(c^2-1
1*d^2)-2*a*b^3*c*(c^2+5*d^2)+b^4*d*(5*c^2+8*d^2))*(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)/b^4/(a^2-b^2)^2/
f/(c+d*sin(f*x+e))^(1/2)-1/4*(-a*d+b*c)^2*(12*a^3*b*c*d-36*a*b^3*c*d+15*a^4*d^2+2*a^2*b^2*(4*c^2-19*d^2)+b^4*(
4*c^2+35*d^2))*(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)*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f
*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)^2/b^4/(a+b)^3/f/(c+d*sin(f*x+e))^(
1/2)

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2871, 3126, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {\left (5 a^2 d+6 a b c-11 b^2 d\right ) (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {\left (15 a^4 d^2+12 a^3 b c d+2 a^2 b^2 \left (4 c^2-19 d^2\right )-36 a b^3 c d+b^4 \left (4 c^2+35 d^2\right )\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt {c+d \sin (e+f x)}}+\frac {3 \left (5 a^4 d^3+4 a^3 b c d^2+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 b^4 f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (-15 a^4 d^3+8 a^3 b c d^2+a^2 b^2 d \left (5 c^2+29 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+b^4 d \left (13 c^2-8 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^3 f \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]

[In]

Int[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^3,x]

[Out]

((b*c - a*d)^2*(6*a*b*c + 5*a^2*d - 11*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b^2*(a^2 - b^2)^2*f*(a
+ b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e
+ f*x])^2) - ((8*a^3*b*c*d^2 - 15*a^4*d^3 + b^4*d*(13*c^2 - 8*d^2) - 2*a*b^3*c*(3*c^2 + 13*d^2) + a^2*b^2*d*(5
*c^2 + 29*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b^3*(a^2 - b^2)^2*f*
Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (3*(b*c - a*d)*(4*a^3*b*c*d^2 + 5*a^4*d^3 + a^2*b^2*d*(c^2 - 11*d^2) - 2
*a*b^3*c*(c^2 + 5*d^2) + b^4*d*(5*c^2 + 8*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e
+ f*x])/(c + d)])/(4*b^4*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^2*(12*a^3*b*c*d - 36*a*b^3*
c*d + 15*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 19*d^2) + b^4*(4*c^2 + 35*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^4*(a + b)^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 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 2884

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

Rule 2886

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/
(c + d))*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] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3081

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 3126

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 3138

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{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}{2} \left (3 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 (2 a b c d-5 a^2 d^2-b^2 \left (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+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {\frac {1}{4} \left (8 a^3 b c d^3-5 a^4 d^4-2 a b^3 c d \left (21 c^2+19 d^2\right )+b^4 c^2 \left (4 c^2+35 d^2\right )+a^2 b^2 \left (8 c^4+19 c^2 d^2+11 d^4\right )\right )+\frac {1}{2} d \left (5 a^4 c d^2+a^2 b^2 c \left (5 c^2-3 d^2\right )-2 a^3 b d \left (c^2-d^2\right )-8 a b^3 d \left (2 c^2+d^2\right )+b^4 c \left (c^2+16 d^2\right )\right ) \sin (e+f x)-\frac {1}{4} d \left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-\frac {1}{4} d (b c-a d) \left (2 a^2 b^2 c^3+4 b^4 c^3+7 a^3 b c^2 d-25 a b^3 c^2 d+15 a^4 c d^2-32 a^2 b^2 c d^2+35 b^4 c d^2+5 a^3 b d^3-11 a b^3 d^3\right )+\frac {3}{4} d (b c-a d) \left (2 a b^3 c^3-a^2 b^2 c^2 d-5 b^4 c^2 d-4 a^3 b c d^2+10 a b^3 c d^2-5 a^4 d^3+11 a^2 b^2 d^3-8 b^4 d^3\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{8 b^3 \left (a^2-b^2\right )^2} \\ & = \frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 (b c-a d) \left (4 a^3 b c d^2+5 a^4 d^3+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}+\frac {\left ((b c-a d)^2 \left (12 a^3 b c d-36 a b^3 c d+15 a^4 d^2+2 a^2 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 d^2\right )\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}-\frac {\left (\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\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}{8 b^3 \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\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)}}{4 b^3 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (3 (b c-a d) \left (4 a^3 b c d^2+5 a^4 d^3+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\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}{8 b^4 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left ((b c-a d)^2 \left (12 a^3 b c d-36 a b^3 c d+15 a^4 d^2+2 a^2 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 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^4 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\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)}}{4 b^3 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {3 (b c-a d) \left (4 a^3 b c d^2+5 a^4 d^3+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\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}}}{4 b^4 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^2 \left (12 a^3 b c d-36 a b^3 c d+15 a^4 d^2+2 a^2 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 d^2\right )\right ) \operatorname {EllipticPi}\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^4 (a+b)^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.62 (sec) , antiderivative size = 1233, normalized size of antiderivative = 2.20 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (\frac {-b^3 c^3 \cos (e+f x)+9 b^2 c^2 d \cos (e+f x)-27 b c d^2 \cos (e+f x)+27 d^3 \cos (e+f x)}{2 b^2 \left (-9+b^2\right ) (3+b \sin (e+f x))^2}+\frac {18 b^3 c^3 \cos (e+f x)-45 b^2 c^2 d \cos (e+f x)-13 b^4 c^2 d \cos (e+f x)-216 b c d^2 \cos (e+f x)+78 b^3 c d^2 \cos (e+f x)+567 d^3 \cos (e+f x)-117 b^2 d^3 \cos (e+f x)}{4 b^2 \left (-9+b^2\right )^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (144 b^2 c^4+8 b^4 c^4-234 b^3 c^3 d+297 b^2 c^2 d^2+57 b^4 c^2 d^2+216 b c d^3-150 b^3 c d^3+405 d^4-63 b^2 d^4+8 b^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+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}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (180 b^2 c^3 d+4 b^4 c^3 d-216 b c^2 d^2-192 b^3 c^2 d^2+1620 c d^3-108 b^2 c d^3+64 b^4 c d^3+216 b d^4-96 b^3 d^4\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\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 {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-18 b^3 c^3 d+45 b^2 c^2 d^2+13 b^4 c^2 d^2+216 b c d^3-78 b^3 c d^3-1215 d^4+261 b^2 d^4-8 b^4 d^4\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\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 {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{16 (-3+b)^2 b^2 (3+b)^2 f} \]

[In]

Integrate[(c + d*Sin[e + f*x])^(7/2)/(3 + b*Sin[e + f*x])^3,x]

[Out]

(Sqrt[c + d*Sin[e + f*x]]*((-(b^3*c^3*Cos[e + f*x]) + 9*b^2*c^2*d*Cos[e + f*x] - 27*b*c*d^2*Cos[e + f*x] + 27*
d^3*Cos[e + f*x])/(2*b^2*(-9 + b^2)*(3 + b*Sin[e + f*x])^2) + (18*b^3*c^3*Cos[e + f*x] - 45*b^2*c^2*d*Cos[e +
f*x] - 13*b^4*c^2*d*Cos[e + f*x] - 216*b*c*d^2*Cos[e + f*x] + 78*b^3*c*d^2*Cos[e + f*x] + 567*d^3*Cos[e + f*x]
- 117*b^2*d^3*Cos[e + f*x])/(4*b^2*(-9 + b^2)^2*(3 + b*Sin[e + f*x]))))/f + ((-2*(144*b^2*c^4 + 8*b^4*c^4 - 2
34*b^3*c^3*d + 297*b^2*c^2*d^2 + 57*b^4*c^2*d^2 + 216*b*c*d^3 - 150*b^3*c*d^3 + 405*d^4 - 63*b^2*d^4 + 8*b^4*d
^4)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)
*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(180*b^2*c^3*d + 4*b^4*c^3*d - 216*b*c^2*d^2 - 192*b^3*c^2*d^2 + 1620*c*d^
3 - 108*b^2*c*d^3 + 64*b^4*c*d^3 + 216*b*d^4 - 96*b^3*d^4)*Cos[e + f*x]*((b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[
-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + 3*d*EllipticPi[(b*(c + d))/(b*c - 3*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) + 3*d + b*(c + d*Sin[e + f*x])))/(b*(b*c - 3*d)*d^2*Sqrt[-(c + d)^(-1)]*(
3 + 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*b^3*c^3*d + 45*b^2*c^2*d^2 + 13*b^4*c^2*d^2 + 216*b*c*d^3 - 78*b^3*c*d^3 - 1215*d^4
+ 261*b^2*d^4 - 8*b^4*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)*EllipticE[I*ArcSinh[Sqrt[-(
c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(2*(3 + b)*(b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-
(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - (-18 + b^2)*d*EllipticPi[(b*(c + d))/(b*c - 3*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) + 3*d + b*(c + d*Sin[e + f*x])))/(b^2*(b*c - 3*d)*d*Sqrt[-(c +
d)^(-1)]*(3 + 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)]))/(16*(-3 + b)^2*b^
2*(3 + b)^2*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2236\) vs. \(2(678)=1356\).

Time = 86.06 (sec) , antiderivative size = 2237, normalized size of antiderivative = 3.99

method	result	size
default	\(\text {Expression too large to display}\)	\(2237\)

[In]

int((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(d^3/b^4*(-6*d*a*(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))+8*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)*(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*b*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))))+12*d^2/b^
5*(a^2*d^2-2*a*b*c*d+b^2*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)*(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+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))-4/b^4*d*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(-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)*(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))-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)*(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*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)*(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+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)))+1/b^4*(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)*(-1/2*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))^
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)*(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))-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)*(1/(c-d)*(-si
n(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)))+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)*(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+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

Fricas [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((c+d*sin(f*x+e))**(7/2)/(a+b*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^(7/2)/(b*sin(f*x + e) + a)^3, x)

Giac [F]

\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^(7/2)/(b*sin(f*x + e) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]

[In]

int((c + d*sin(e + f*x))^(7/2)/(a + b*sin(e + f*x))^3,x)

[Out]

int((c + d*sin(e + f*x))^(7/2)/(a + b*sin(e + f*x))^3, x)