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

Optimal. Leaf size=605 \[ \frac{3 \left (a^2 b^2 d \left (c^2-11 d^2\right )+4 a^3 b c d^2+5 a^4 d^3-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^4 f \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (a^2 b^2 d \left (5 c^2+29 d^2\right )+8 a^3 b c d^2-15 a^4 d^3-2 a b^3 c \left (3 c^2+13 d^2\right )+b^4 d \left (13 c^2-8 d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^3 f \left (a^2-b^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (2 a^2 b^2 \left (4 c^2-19 d^2\right )+12 a^3 b c d+15 a^4 d^2-36 a b^3 c d+b^4 \left (4 c^2+35 d^2\right )\right ) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{\left (5 a^2 d+6 a b c-11 b^2 d\right ) (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))} \]

[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]]
)

________________________________________________________________________________________

Rubi [A]  time = 2.20118, antiderivative size = 605, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {2792, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{3 \left (a^2 b^2 d \left (c^2-11 d^2\right )+4 a^3 b c d^2+5 a^4 d^3-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^4 f \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (a^2 b^2 d \left (5 c^2+29 d^2\right )+8 a^3 b c d^2-15 a^4 d^3-2 a b^3 c \left (3 c^2+13 d^2\right )+b^4 d \left (13 c^2-8 d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^3 f \left (a^2-b^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (2 a^2 b^2 \left (4 c^2-19 d^2\right )+12 a^3 b c d+15 a^4 d^2-36 a b^3 c d+b^4 \left (4 c^2+35 d^2\right )\right ) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{\left (5 a^2 d+6 a b c-11 b^2 d\right ) (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))} \]

Antiderivative was successfully verified.

[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 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 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))^{7/2}}{(a+b \sin (e+f x))^3} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{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 ) 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}}}{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 ) \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^4 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 8.31852, size = 1323, normalized size = 2.19 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{-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)}{2 b^2 \left (b^2-a^2\right ) (a+b \sin (e+f x))^2}+\frac{7 d^3 \cos (e+f x) a^4-8 b c d^2 \cos (e+f x) a^3-13 b^2 d^3 \cos (e+f x) a^2-5 b^2 c^2 d \cos (e+f x) a^2+6 b^3 c^3 \cos (e+f x) a+26 b^3 c d^2 \cos (e+f x) a-13 b^4 c^2 d \cos (e+f x)}{4 b^2 \left (b^2-a^2\right )^2 (a+b \sin (e+f x))}\right )}{f}+\frac{-\frac{2 \left (8 c^4 b^4+8 d^4 b^4+57 c^2 d^2 b^4-50 a c d^3 b^3-78 a c^3 d b^3+16 a^2 c^4 b^2-7 a^2 d^4 b^2+33 a^2 c^2 d^2 b^2+8 a^3 c d^3 b+5 a^4 d^4\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 (20 c d^3 a^4+8 b d^4 a^3-8 b c^2 d^2 a^3-12 b^2 c d^3 a^2+20 b^2 c^3 d a^2-32 b^3 d^4 a-64 b^3 c^2 d^2 a+64 b^4 c d^3+4 b^4 c^3 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 (-8 d^4 b^4+13 c^2 d^2 b^4-26 a c d^3 b^3-6 a c^3 d b^3+29 a^2 d^4 b^2+5 a^2 c^2 d^2 b^2+8 a^3 c d^3 b-15 a^4 d^4\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}}}}{16 (a-b)^2 b^2 (a+b)^2 f} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c + d*Sin[e + f*x]]*((-(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])/(2*b^2*(-a^2 + b^2)*(a + b*Sin[e + f*x])^2) + (6*a*b^3*c^3*Cos[e + f*x] - 5*a^2*b^2*c^
2*d*Cos[e + f*x] - 13*b^4*c^2*d*Cos[e + f*x] - 8*a^3*b*c*d^2*Cos[e + f*x] + 26*a*b^3*c*d^2*Cos[e + f*x] + 7*a^
4*d^3*Cos[e + f*x] - 13*a^2*b^2*d^3*Cos[e + f*x])/(4*b^2*(-a^2 + b^2)^2*(a + b*Sin[e + f*x]))))/f + ((-2*(16*a
^2*b^2*c^4 + 8*b^4*c^4 - 78*a*b^3*c^3*d + 33*a^2*b^2*c^2*d^2 + 57*b^4*c^2*d^2 + 8*a^3*b*c*d^3 - 50*a*b^3*c*d^3
+ 5*a^4*d^4 - 7*a^2*b^2*d^4 + 8*b^4*d^4)*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)*(20*a^2*b^2*c^3*d + 4*b^4*c^3*d - 8*
a^3*b*c^2*d^2 - 64*a*b^3*c^2*d^2 + 20*a^4*c*d^3 - 12*a^2*b^2*c*d^3 + 64*b^4*c*d^3 + 8*a^3*b*d^4 - 32*a*b^3*d^4
)*Cos[e + f*x]*((b*c - a*d)*EllipticF[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)*(-6*a*b^3*c^3*d + 5*a^2*b^2*c^2*d^2
+ 13*b^4*c^2*d^2 + 8*a^3*b*c*d^3 - 26*a*b^3*c*d^3 - 15*a^4*d^4 + 29*a^2*b^2*d^4 - 8*b^4*d^4)*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*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^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)]))/(16*(a - b)^2*b^2*(a + b)^2*f)

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Maple [B]  time = 9.81, size = 2237, normalized size = 3.7 \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))^(7/2)/(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^4*(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)*((-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)
))-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)*((-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*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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+12/b^5*d^2*(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)*((-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))+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)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*Ellip
ticF(((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)*((-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)))-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)*((-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))-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))^(7/2)/(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))^(7/2)/(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))**(7/2)/(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{7}{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))^(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)