3.521 \(\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=518 \[ -\frac{d \left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}+\frac{\left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}} \]

[Out]

-(d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^(3
/2)) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)) - (2*(c - 5*d)*Cos[e + f*x
])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) - ((4*c^2 - 27*c*d + 119*d^2)*Cos[e +
f*x])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(4*c^4 - 27*c^3*d + 111*c^2*d^
2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^4 - 27*
c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x
]])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d
^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^4*(c + d)
*f*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.21486, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{d \left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}+\frac{\left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

-(d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^(3
/2)) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)) - (2*(c - 5*d)*Cos[e + f*x
])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) - ((4*c^2 - 27*c*d + 119*d^2)*Cos[e +
f*x])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(4*c^4 - 27*c^3*d + 111*c^2*d^
2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^4 - 27*
c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x
]])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d
^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^4*(c + d)
*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2766

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dis
t[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*
(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d,
0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2978

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
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{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a (4 c-13 d)-\frac{7}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx}{5 a^2 (c-d)}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}+\frac{\int \frac{\frac{1}{2} a^2 \left (4 c^2-17 c d+69 d^2\right )+5 a^2 (c-5 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{15}{4} a^3 (c-33 d) d^2-\frac{3}{4} a^3 d \left (4 c^2-27 c d+119 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}+\frac{2 \int \frac{\frac{9}{8} a^3 d^2 \left (c^2-138 c d-119 d^2\right )+\frac{3}{8} a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{45 a^6 (c-d)^4 (c+d)}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 \int \frac{\frac{3}{16} a^3 d^2 \left (c^3+387 c^2 d+471 c d^2+165 d^3\right )+\frac{3}{16} a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{45 a^6 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{60 a^3 (c-d)^4 (c+d)}-\frac{\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{60 a^3 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{60 a^3 (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\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}{60 a^3 (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\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)}}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\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}}}{30 a^3 (c-d)^4 (c+d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.83506, size = 828, normalized size = 1.6 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-\frac{2 \cos (e+f x) d^4}{3 (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{4 c^4-27 d c^3+111 d^2 c^2+449 d^3 c+267 d^4}{15 (c-d)^5 (c+d)^2}+\frac{4 \left (c \sin \left (\frac{1}{2} (e+f x)\right )-8 d \sin \left (\frac{1}{2} (e+f x)\right )\right )}{15 (c-d)^4 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{4 \sin \left (\frac{1}{2} (e+f x)\right ) c^2-35 d \sin \left (\frac{1}{2} (e+f x)\right ) c+177 d^2 \sin \left (\frac{1}{2} (e+f x)\right )}{15 (c-d)^5 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{2 \left (9 \cos (e+f x) d^5+13 c \cos (e+f x) d^4\right )}{3 (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{2 (c-8 d)}{15 (c-d)^4 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{1}{5 (c-d)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}+\frac{2 \sin \left (\frac{1}{2} (e+f x)\right )}{5 (c-d)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}{f (\sin (e+f x) a+a)^3}+\frac{d \left (\frac{2 \left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sqrt{c+d \sin (e+f x)} \cos ^2(e+f x)}{d \left (1-\sin ^2(e+f x)\right )}-\frac{\left (-4 c^4+27 d c^3-111 d^2 c^2-579 d^3 c-357 d^4\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}-\frac{2 \left (-165 d^4-471 c d^3-387 c^2 d^2-c^3 d\right ) F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}{60 (c-d)^5 (c+d)^2 f (\sin (e+f x) a+a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]]*(-(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 449*c*d^
3 + 267*d^4)/(15*(c - d)^5*(c + d)^2) + (2*Sin[(e + f*x)/2])/(5*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^5) - 1/(5*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - (2*(c - 8*d))/(15*(c - d)^4*(Cos[(e + f*x)/2]
+ Sin[(e + f*x)/2])^2) + (4*(c*Sin[(e + f*x)/2] - 8*d*Sin[(e + f*x)/2]))/(15*(c - d)^4*(Cos[(e + f*x)/2] + Si
n[(e + f*x)/2])^3) + (4*c^2*Sin[(e + f*x)/2] - 35*c*d*Sin[(e + f*x)/2] + 177*d^2*Sin[(e + f*x)/2])/(15*(c - d)
^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - (2*d^4*Cos[e + f*x])/(3*(c - d)^4*(c + d)*(c + d*Sin[e + f*x])^2)
- (2*(13*c*d^4*Cos[e + f*x] + 9*d^5*Cos[e + f*x]))/(3*(c - d)^5*(c + d)^2*(c + d*Sin[e + f*x]))))/(f*(a + a*Si
n[e + f*x])^3) + (d*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*((-2*(-(c^3*d) - 387*c^2*d^2 - 471*c*d^3 - 165*d^4
)*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]] +
(2*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Si
n[e + f*x]^2)) - ((-4*c^4 + 27*c^3*d - 111*c^2*d^2 - 579*c*d^3 - 357*d^4)*((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))/(60*(c - d)^5*(c
+ d)^2*f*(a + a*Sin[e + f*x])^3)

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Maple [B]  time = 9.667, size = 2311, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(-d^3/(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))))+1/(c-d)^2*(-1/5/(c-d)*(
-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^3-2/15*(c-3*d)/(c-d)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)
^(1/2)/(1+sin(f*x+e))^2-1/30*(-sin(f*x+e)^2*d-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^3*(4*c^2-15*c*d+27*d^2)/((-d*
sin(f*x+e)-c)*(-1+sin(f*x+e))*(1+sin(f*x+e)))^(1/2)+2*(-c*d^2-15*d^3)/(60*c^3-180*c^2*d+180*c*d^2-60*d^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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-1/30*d*(4*c^2-15*c*d
+27*d^2)/(c-d)^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))))-2/(c-d)^3*d*(-1/3/(c-d)*(-(-d*sin(f
*x+e)-c)*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^2-1/3*(-sin(f*x+e)^2*d-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d
)/((-d*sin(f*x+e)-c)*(-1+sin(f*x+e))*(1+sin(f*x+e)))^(1/2)+2*d^2/(3*c^2-6*c*d+3*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))-1/3*d*(c-3*d)/(c-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)*((-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*d^3/(c-d)^4*(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/(c-d)^4*d^2*(-(-sin(f*x+e)^
2*d-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)/((-d*sin(f*x+e)-c)*(-1+sin(f*x+e))*(1+sin(f*x+e)))^(1/2)-2*d/(2*c-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*si
n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-d/(c-d)*(c/d-1)*
((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)
-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+
d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/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(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{d \sin \left (f x + e\right ) + c}}{a^{3} d^{3} \cos \left (f x + e\right )^{6} - 4 \, a^{3} c^{3} - 12 \, a^{3} c^{2} d - 12 \, a^{3} c d^{2} - 4 \, a^{3} d^{3} - 3 \,{\left (a^{3} c^{2} d + 3 \, a^{3} c d^{2} + 2 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (a^{3} c^{3} + 5 \, a^{3} c^{2} d + 7 \, a^{3} c d^{2} + 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{2} -{\left (4 \, a^{3} c^{3} + 12 \, a^{3} c^{2} d + 12 \, a^{3} c d^{2} + 4 \, a^{3} d^{3} + 3 \,{\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} -{\left (a^{3} c^{3} + 9 \, a^{3} c^{2} d + 15 \, a^{3} c d^{2} + 7 \, a^{3} d^{3}\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(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(d*sin(f*x + e) + c)/(a^3*d^3*cos(f*x + e)^6 - 4*a^3*c^3 - 12*a^3*c^2*d - 12*a^3*c*d^2 - 4*a^3*d
^3 - 3*(a^3*c^2*d + 3*a^3*c*d^2 + 2*a^3*d^3)*cos(f*x + e)^4 + 3*(a^3*c^3 + 5*a^3*c^2*d + 7*a^3*c*d^2 + 3*a^3*d
^3)*cos(f*x + e)^2 - (4*a^3*c^3 + 12*a^3*c^2*d + 12*a^3*c*d^2 + 4*a^3*d^3 + 3*(a^3*c*d^2 + a^3*d^3)*cos(f*x +
e)^4 - (a^3*c^3 + 9*a^3*c^2*d + 15*a^3*c*d^2 + 7*a^3*d^3)*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(1/(a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((a*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(5/2)), x)