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

Optimal. Leaf size=532 \[ -\frac{2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (-21 c^2 d^2+8 c^4+45 d^4\right )\right ) (b c-a d) \cos (e+f x)}{15 d^2 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (-3 a^2 b d^2 \left (3 c^2+5 d^2\right )+8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )+b^3 \left (-\left (-15 c^2 d^2+8 c^4+15 d^4\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \left (c^2-d^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (-21 c^2 d^2+8 c^4+45 d^4\right )\right ) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \left (c^2-d^2\right )^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{15 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}} \]

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*(b*c -
a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(b*
c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[e + f
*x])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7
*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*
Sin[e + f*x]])/(15*d^3*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^
2 - 5*d^2) - 3*a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2
*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.10912, antiderivative size = 532, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2792, 3021, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (-21 c^2 d^2+8 c^4+45 d^4\right )\right ) (b c-a d) \cos (e+f x)}{15 d^2 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (-3 a^2 b d^2 \left (3 c^2+5 d^2\right )+8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )+b^3 \left (-\left (-15 c^2 d^2+8 c^4+15 d^4\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \left (c^2-d^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (-21 c^2 d^2+8 c^4+45 d^4\right )\right ) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \left (c^2-d^2\right )^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{15 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*(b*c -
a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(b*
c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[e + f
*x])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7
*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*
Sin[e + f*x]])/(15*d^3*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^
2 - 5*d^2) - 3*a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2
*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2792

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(
d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e +
f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*
d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*
n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3021

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

Rule 2754

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

Rule 2752

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

Rule 2663

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

Rule 2661

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

Rule 2655

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

Rule 2653

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

Rubi steps

\begin{align*} \int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \int \frac{\frac{1}{2} \left (2 b (b c-a d)^2-5 a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac{1}{2} \left (3 a (b c-a d)^2-5 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 (2 a b c d-a^2 d^2+b^2 \left (4 c^2-5 d^2\right )\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d \left (c^2-d^2\right )}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac{8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac{4 \int \frac{-\frac{3}{4} d \left (24 a^2 b c d^2-a^3 d \left (5 c^2+3 d^2\right )-3 a b^2 d \left (3 c^2+5 d^2\right )-2 b^3 \left (c^3-5 c d^2\right )\right )-\frac{1}{4} \left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 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))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac{8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{8 \int \frac{\frac{1}{8} d \left (3 a^2 b d^2 \left (27 c^2+5 d^2\right )-a^3 c d \left (15 c^2+17 d^2\right )-3 a b^2 d \left (7 c^3+25 c d^2\right )+b^3 \left (2 c^4+15 c^2 d^2+15 d^4\right )\right )+\frac{1}{8} (b c-a d) \left (8 b^2 c^4+14 a b c^3 d+23 a^2 c^2 d^2-21 b^2 c^2 d^2-78 a b c d^3+9 a^2 d^4+45 b^2 d^4\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 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))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac{8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d^3 \left (c^2-d^2\right )^2}-\frac{\left ((b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{15 d^3 \left (c^2-d^2\right )^3}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac{8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left ((b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\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}{15 d^3 \left (c^2-d^2\right )^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (\left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\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}{15 d^3 \left (c^2-d^2\right )^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac{8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\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)}}{15 d^3 \left (c^2-d^2\right )^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\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}}}{15 d^3 \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 5.30722, size = 584, normalized size = 1.1 \[ \frac{2 \left (\frac{d (b c-a d) \cos (e+f x) \left (2 d \left (2 a^2 c d^2 \left (27 c^2+5 d^2\right )+a b d \left (-170 c^2 d^2+27 c^4+15 d^4\right )+b^2 \left (-20 c^3 d^2+9 c^5+75 c d^4\right )\right ) \sin (e+f x)-d^2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (-21 c^2 d^2+8 c^4+45 d^4\right )\right ) \cos (2 (e+f x))+68 a^2 c^4 d^2+13 a^2 c^2 d^4+15 a^2 d^6-146 a b c^3 d^3+14 a b c^5 d-60 a b c d^5-2 b^2 c^4 d^2+45 b^2 c^2 d^4+8 b^2 c^6+45 b^2 d^6\right )}{2 \left (d^2-c^2\right )^3}+\frac{\left (\frac{c+d \sin (e+f x)}{c+d}\right )^{5/2} \left (d^2 \left (3 a^2 b d^2 \left (27 c^2+5 d^2\right )+a^3 (-c) d \left (15 c^2+17 d^2\right )-3 a b^2 d \left (7 c^3+25 c d^2\right )+b^3 \left (15 c^2 d^2+2 c^4+15 d^4\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (3 a^2 b c d^2 \left (3 c^2+29 d^2\right )-a^3 d^3 \left (23 c^2+9 d^2\right )-3 a b^2 d \left (19 c^2 d^2-2 c^4+15 d^4\right )+b^3 \left (-21 c^3 d^2+8 c^5+45 c d^4\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )}{(c-d)^3 (c+d)}\right )}{15 d^3 f (c+d \sin (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((d^2*(3*a^2*b*d^2*(27*c^2 + 5*d^2) - a^3*c*d*(15*c^2 + 17*d^2) - 3*a*b^2*d*(7*c^3 + 25*c*d^2) + b^3*(2*c^
4 + 15*c^2*d^2 + 15*d^4))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (-(a^3*d^3*(23*c^2 + 9*d^2)) + 3*a
^2*b*c*d^2*(3*c^2 + 29*d^2) - 3*a*b^2*d*(-2*c^4 + 19*c^2*d^2 + 15*d^4) + b^3*(8*c^5 - 21*c^3*d^2 + 45*c*d^4))*
((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])/(c + d))^(5/2))/((c - d)^3*(c + d)) + (d*(b*c - a*d)*Cos[e + f*x]*(8*b^2*c^6 + 14*a*b*c
^5*d + 68*a^2*c^4*d^2 - 2*b^2*c^4*d^2 - 146*a*b*c^3*d^3 + 13*a^2*c^2*d^4 + 45*b^2*c^2*d^4 - 60*a*b*c*d^5 + 15*
a^2*d^6 + 45*b^2*d^6 - d^2*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 +
45*d^4))*Cos[2*(e + f*x)] + 2*d*(2*a^2*c*d^2*(27*c^2 + 5*d^2) + a*b*d*(27*c^4 - 170*c^2*d^2 + 15*d^4) + b^2*(9
*c^5 - 20*c^3*d^2 + 75*c*d^4))*Sin[e + f*x]))/(2*(-c^2 + d^2)^3)))/(15*d^3*f*(c + d*Sin[e + f*x])^(5/2))

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Maple [B]  time = 8.145, size = 1621, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*b^3/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/d^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^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*d*cos(f*x+e)^2/(c^2-d^2)^3*(23*c^2+9*d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/
2)+2*(15*c^3+17*c*d^2)/(15*c^6-45*c^4*d^2+45*c^2*d^4-15*d^6)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(
f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*s
in(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/15*d*(23*c^2+9*d^2)/(c^2-d^2)^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d)
)^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)
*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1
/2),((c-d)/(c+d))^(1/2))))+3*b^2/d^3*(a*d-b*c)*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(
1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c
-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2
))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d
))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+
d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+3*b/d^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)
*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))
^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*
EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c
-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1
/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))
^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{d^{4} \cos \left (f x + e\right )^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \,{\left (3 \, c^{2} d^{2} + d^{4}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left (c d^{3} \cos \left (f x + e\right )^{2} - c^{3} d - c d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*
sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*
cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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