\(\int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx\) [496]

Optimal result

Integrand size = 27, antiderivative size = 445 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {12 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{77 d^2 f}-\frac {12 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{77 d^2 f}-\frac {12 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{77 d^2 f}+\frac {24 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{11 d^2 f}-\frac {2 \cos (e+f x) (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {12 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{77 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {12 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{77 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-4/693*a^3*(4*c^3-33*c^2*d+182*c*d^2+231*d^3)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f-4/693*a^3*(4*c^2-33*c*d+
189*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d^2/f+8/99*a^3*(c-6*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(7/2)/d^2/f-2/11
*cos(f*x+e)*(a^3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^(7/2)/d/f-4/693*a^3*(4*c^4-33*c^3*d+177*c^2*d^2+561*c*d^3+31
5*d^4)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-4/693*a^3*(c+3*d)*(4*c^4-45*c^3*d+309*c^2*d^2+525*c*d^3+231*d^4
)*(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)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4/693*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)*(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)/d^3/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2842, 3047, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{693 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{693 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f} \]

[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 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 2831

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 2832

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)*Sim
p[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 2842

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 - 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 3047

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 3102

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \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}}}{693 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.77 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\frac {3 \left (-32 \left (d^2 \left (c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^5-33 c^4 d+174 c^3 d^2+1452 c^2 d^3+1806 c d^4+693 d^5\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (2 \left (32 c^4-264 c^3 d-8994 c^2 d^2-13926 c d^3-5859 d^4\right ) \cos (e+f x)+d^2 \left (452 c^2+2508 c d+1701 d^2\right ) \cos (3 (e+f x))-63 d^4 \cos (5 (e+f x))-4 d \left (6 c^3+990 c^2 d+2401 c d^2+1155 d^3\right ) \sin (2 (e+f x))+14 d^3 (23 c+33 d) \sin (4 (e+f x))\right )\right )}{616 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(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 - 5859*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)])))/(616*d^3*f*Sqrt[c + d*Sin[e + f
*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1925\) vs. \(2(501)=1002\).

Time = 43.33 (sec) , antiderivative size = 1926, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1926\)
parts	\(\text {Expression too large to display}\)	\(4388\)

[In]

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

[Out]

2/693*a^3*(4*c^5*d^2-1096*c^3*d^4-1584*c^2*d^5-33*c^4*d^3-630*c*d^6-120*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(
f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(
-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x
+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-
d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e
))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^6*d-72*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(
-d*(sin(f*x+e)+1)/(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+2128*((c+
d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d
))^(1/2)*(-d*(sin(f*x+e)+1)/(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
-2016*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(
c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(
((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^7+224*c*d^6*sin(f*x+e)^6+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)+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))/d^4/cos(f
*x+e)/(c+d*sin(f*x+e))^(1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.86 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{6} - 66 \, a^{3} c^{5} d + 345 \, a^{3} c^{4} d^{2} + 330 \, a^{3} c^{3} d^{3} - 1392 \, a^{3} c^{2} d^{4} - 2376 \, a^{3} c d^{5} - 945 \, a^{3} d^{6}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (8 \, a^{3} c^{6} - 66 \, a^{3} c^{5} d + 345 \, a^{3} c^{4} d^{2} + 330 \, a^{3} c^{3} d^{3} - 1392 \, a^{3} c^{2} d^{4} - 2376 \, a^{3} c d^{5} - 945 \, a^{3} d^{6}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (4 i \, a^{3} c^{5} d - 33 i \, a^{3} c^{4} d^{2} + 174 i \, a^{3} c^{3} d^{3} + 1452 i \, a^{3} c^{2} d^{4} + 1806 i \, a^{3} c d^{5} + 693 i \, a^{3} d^{6}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-4 i \, a^{3} c^{5} d + 33 i \, a^{3} c^{4} d^{2} - 174 i \, a^{3} c^{3} d^{3} - 1452 i \, a^{3} c^{2} d^{4} - 1806 i \, a^{3} c d^{5} - 693 i \, a^{3} d^{6}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (63 \, a^{3} d^{6} \cos \left (f x + e\right )^{5} - {\left (113 \, a^{3} c^{2} d^{4} + 627 \, a^{3} c d^{5} + 504 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{3} c^{4} d^{2} - 33 \, a^{3} c^{3} d^{3} - 1209 \, a^{3} c^{2} d^{4} - 2211 \, a^{3} c d^{5} - 1071 \, a^{3} d^{6}\right )} \cos \left (f x + e\right ) - {\left (7 \, {\left (23 \, a^{3} c d^{5} + 33 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c^{3} d^{3} + 165 \, a^{3} c^{2} d^{4} + 427 \, a^{3} c d^{5} + 231 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{2079 \, d^{4} f} \]

[In]

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

[Out]

-2/2079*(sqrt(2)*(8*a^3*c^6 - 66*a^3*c^5*d + 345*a^3*c^4*d^2 + 330*a^3*c^3*d^3 - 1392*a^3*c^2*d^4 - 2376*a^3*c
*d^5 - 945*a^3*d^6)*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1
/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + sqrt(2)*(8*a^3*c^6 - 66*a^3*c^5*d + 345*a^3*c^4*d^2 +
330*a^3*c^3*d^3 - 1392*a^3*c^2*d^4 - 2376*a^3*c*d^5 - 945*a^3*d^6)*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2
- 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*sqr
t(2)*(4*I*a^3*c^5*d - 33*I*a^3*c^4*d^2 + 174*I*a^3*c^3*d^3 + 1452*I*a^3*c^2*d^4 + 1806*I*a^3*c*d^5 + 693*I*a^3
*d^6)*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse
(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c
)/d)) + 3*sqrt(2)*(-4*I*a^3*c^5*d + 33*I*a^3*c^4*d^2 - 174*I*a^3*c^3*d^3 - 1452*I*a^3*c^2*d^4 - 1806*I*a^3*c*d
^5 - 693*I*a^3*d^6)*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, wei
erstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin
(f*x + e) + 2*I*c)/d)) + 3*(63*a^3*d^6*cos(f*x + e)^5 - (113*a^3*c^2*d^4 + 627*a^3*c*d^5 + 504*a^3*d^6)*cos(f*
x + e)^3 - (4*a^3*c^4*d^2 - 33*a^3*c^3*d^3 - 1209*a^3*c^2*d^4 - 2211*a^3*c*d^5 - 1071*a^3*d^6)*cos(f*x + e) -
(7*(23*a^3*c*d^5 + 33*a^3*d^6)*cos(f*x + e)^3 - 3*(a^3*c^3*d^3 + 165*a^3*c^2*d^4 + 427*a^3*c*d^5 + 231*a^3*d^6
)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4*f)

Sympy [F]

\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=a^{3} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{5}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]

[In]

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

[Out]

a**3*(Integral(c**2*sqrt(c + d*sin(e + f*x)), x) + Integral(3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) +
Integral(3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(c**2*sqrt(c + d*sin(e + f*x))*sin(e +
f*x)**3, x) + Integral(d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(3*d**2*sqrt(c + d*sin(e +
f*x))*sin(e + f*x)**3, x) + Integral(3*d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4, x) + Integral(d**2*sqrt
(c + d*sin(e + f*x))*sin(e + f*x)**5, x) + Integral(2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral
(6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3,
x) + Integral(2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4, x))

Maxima [F]

\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[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)

Giac [F]

\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

[In]

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

[Out]

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