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

Optimal. Leaf size=467 \[ -\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{4 a^3 \left (-33 c^2 d+4 c^3+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left (c^2-d^2\right ) \left (177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{693 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 (c+3 d) \left (309 c^2 d^2-45 c^3 d+4 c^4+525 c d^3+231 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{693 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f} \]

[Out]

(-4*a^3*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(693*d^2
*f) - (4*a^3*(4*c^3 - 33*c^2*d + 182*c*d^2 + 231*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(693*d^2*f) - (
4*a^3*(4*c^2 - 33*c*d + 189*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8*a^3*(c - 6*d)*Cos[e
+ f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])
^(7/2))/(11*d*f) + (4*a^3*(c + 3*d)*(4*c^4 - 45*c^3*d + 309*c^2*d^2 + 525*c*d^3 + 231*d^4)*EllipticE[(e - Pi/2
+ f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(693*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c
^2 - d^2)*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*
Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(693*d^3*f*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.03167, antiderivative size = 467, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2763, 2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{4 a^3 \left (-33 c^2 d+4 c^3+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left (c^2-d^2\right ) \left (177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{693 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 (c+3 d) \left (309 c^2 d^2-45 c^3 d+4 c^4+525 c d^3+231 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{693 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^3*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(693*d^2
*f) - (4*a^3*(4*c^3 - 33*c^2*d + 182*c*d^2 + 231*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(693*d^2*f) - (
4*a^3*(4*c^2 - 33*c*d + 189*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8*a^3*(c - 6*d)*Cos[e
+ f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])
^(7/2))/(11*d*f) + (4*a^3*(c + 3*d)*(4*c^4 - 45*c^3*d + 309*c^2*d^2 + 525*c*d^3 + 231*d^4)*EllipticE[(e - Pi/2
+ f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(693*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c
^2 - d^2)*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*
Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(693*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2763

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

Rule 2968

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

Rule 3023

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

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*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] && GtQ[m, 0] && 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 (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx &=-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{2 \int (a+a \sin (e+f x)) \left (a^2 (c+9 d)-2 a^2 (c-6 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2} \, dx}{11 d}\\ &=-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{2 \int (c+d \sin (e+f x))^{5/2} \left (a^3 (c+9 d)+\left (-2 a^3 (c-6 d)+a^3 (c+9 d)\right ) \sin (e+f x)-2 a^3 (c-6 d) \sin ^2(e+f x)\right ) \, dx}{11 d}\\ &=\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{4 \int (c+d \sin (e+f x))^{5/2} \left (-\frac{5}{2} a^3 (c-33 d) d+\frac{1}{2} a^3 \left (4 c^2-33 c d+189 d^2\right ) \sin (e+f x)\right ) \, dx}{99 d^2}\\ &=-\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{8 \int (c+d \sin (e+f x))^{3/2} \left (-\frac{15}{4} a^3 d \left (c^2-66 c d-63 d^2\right )+\frac{5}{4} a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \sin (e+f x)\right ) \, dx}{693 d^2}\\ &=-\frac{4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{16 \int \sqrt{c+d \sin (e+f x)} \left (-\frac{15}{8} a^3 d \left (c^3-297 c^2 d-497 c d^2-231 d^3\right )+\frac{15}{8} a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sin (e+f x)\right ) \, dx}{3465 d^2}\\ &=-\frac{4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{32 \int \frac{\frac{15}{16} a^3 d \left (c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right )+\frac{15}{16} a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{10395 d^2}\\ &=-\frac{4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{\left (2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{693 d^3}-\frac{\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{693 d^3}\\ &=-\frac{4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{\left (2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 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}{693 d^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\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}{693 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac{4 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 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)}}{693 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\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}}}{693 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 1.89943, size = 377, normalized size = 0.81 \[ \frac{a^3 (\sin (e+f x)+1)^3 \left (d (c+d \sin (e+f x)) \left (-4 d \left (990 c^2 d+6 c^3+2401 c d^2+1155 d^3\right ) \sin (2 (e+f x))+d^2 \left (452 c^2+2508 c d+1701 d^2\right ) \cos (3 (e+f x))+2 \left (-8994 c^2 d^2-264 c^3 d+32 c^4-13926 c d^3-5859 d^4\right ) \cos (e+f x)+14 d^3 (23 c+33 d) \sin (4 (e+f x))-63 d^4 \cos (5 (e+f x))\right )-32 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (1668 c^2 d^2+858 c^3 d+c^4+1254 c d^3+315 d^4\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (174 c^3 d^2+1452 c^2 d^3-33 c^4 d+4 c^5+1806 c d^4+693 d^5\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )\right )}{5544 d^3 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[e + f*x])^3*(-32*(d^2*(c^4 + 858*c^3*d + 1668*c^2*d^2 + 1254*c*d^3 + 315*d^4)*EllipticF[(-2*e +
Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^5 - 33*c^4*d + 174*c^3*d^2 + 1452*c^2*d^3 + 1806*c*d^4 + 693*d^5)*((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)]))*Sqrt[(
c + d*Sin[e + f*x])/(c + d)] + d*(c + d*Sin[e + f*x])*(2*(32*c^4 - 264*c^3*d - 8994*c^2*d^2 - 13926*c*d^3 - 58
59*d^4)*Cos[e + f*x] + d^2*(452*c^2 + 2508*c*d + 1701*d^2)*Cos[3*(e + f*x)] - 63*d^4*Cos[5*(e + f*x)] - 4*d*(6
*c^3 + 990*c^2*d + 2401*c*d^2 + 1155*d^3)*Sin[2*(e + f*x)] + 14*d^3*(23*c + 33*d)*Sin[4*(e + f*x)])))/(5544*d^
3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])

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Maple [B]  time = 1.256, size = 1926, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*a^3*(224*c*d^6*sin(f*x+e)^6-72*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+si
n(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^5*d^2+274*c^2*d^5*sin(f
*x+e)^5+858*c*d^6*sin(f*x+e)^5+116*c^3*d^4*sin(f*x+e)^4+1122*c^2*d^5*sin(f*x+e)^4+1274*c*d^6*sin(f*x+e)^4-c^4*
d^3*sin(f*x+e)^3+528*c^3*d^4*sin(f*x+e)^3+1942*c^2*d^5*sin(f*x+e)^3+1188*c*d^6*sin(f*x+e)^3-4*c^5*d^2*sin(f*x+
e)^2+33*c^4*d^3*sin(f*x+e)^2+980*c^3*d^4*sin(f*x+e)^2+462*c^2*d^5*sin(f*x+e)^2-868*c*d^6*sin(f*x+e)^2+c^4*d^3*
sin(f*x+e)-528*c^3*d^4*sin(f*x+e)-2216*c^2*d^5*sin(f*x+e)-2046*c*d^6*sin(f*x+e)-2016*((c+d*sin(f*x+e))/(c-d))^
(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2
),((c-d)/(c+d))^(1/2))*d^7-8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e)
)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^7+1386*((c+d*sin(f*x+e))/(c-d))
^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/
2),((c-d)/(c+d))^(1/2))*d^7+63*d^7*sin(f*x+e)^7+231*d^7*sin(f*x+e)^6+315*d^7*sin(f*x+e)^5+231*d^7*sin(f*x+e)^4
+252*d^7*sin(f*x+e)^3-462*d^7*sin(f*x+e)^2-630*d^7*sin(f*x+e)-1584*c^2*d^5-630*c*d^6-33*c^4*d^3+4*c^5*d^2-1096
*c^3*d^4+2128*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*
EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d^3+4176*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-
1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(
c+d))^(1/2))*c^3*d^4-120*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c
-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^5-4104*((c+d*sin(f*x+e))/(c-d))
^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/
2),((c-d)/(c+d))^(1/2))*c*d^6+66*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*
x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^6*d-340*((c+d*sin(f*x+e))/(
c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d)
)^(1/2),((c-d)/(c+d))^(1/2))*c^5*d^2-2970*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*
(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d^3-3264*((c+d*s
in(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f
*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^4+1518*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d)
)^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^5+
3612*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^6+8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*
d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c
^6*d)/d^4/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{\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((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} +{\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{4} -{\left (3 \, a^{3} c^{2} + 10 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (a^{3} d^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} -{\left (a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((4*a^3*c^2 + 8*a^3*c*d + 4*a^3*d^2 + (2*a^3*c*d + 3*a^3*d^2)*cos(f*x + e)^4 - (3*a^3*c^2 + 10*a^3*c*d
+ 7*a^3*d^2)*cos(f*x + e)^2 + (a^3*d^2*cos(f*x + e)^4 + 4*a^3*c^2 + 8*a^3*c*d + 4*a^3*d^2 - (a^3*c^2 + 6*a^3*
c*d + 5*a^3*d^2)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), 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+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{\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((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

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