3.6.21 \(\int \frac {1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx\) [521]

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

[Out]

-1/30*d*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*cos(f*x+e)/a^3/(c-d)^4/(c+d)/f/(c+d*sin(f*x+e))^(3/2)-1/5*cos(f*x+e
)/(c-d)/f/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2)-2/15*(c-5*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2/(c+
d*sin(f*x+e))^(3/2)-1/30*(4*c^2-27*c*d+119*d^2)*cos(f*x+e)/(c-d)^3/f/(a^3+a^3*sin(f*x+e))/(c+d*sin(f*x+e))^(3/
2)-1/30*d*(4*c^4-27*c^3*d+111*c^2*d^2+579*c*d^3+357*d^4)*cos(f*x+e)/a^3/(c-d)^5/(c+d)^2/f/(c+d*sin(f*x+e))^(1/
2)+1/30*(4*c^4-27*c^3*d+111*c^2*d^2+579*c*d^3+357*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)/a^3/(c-d)^5/(c+d)^2
/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/30*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*(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)/a^3/(c-d)^4/(c+d)/f/(c+d*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.83, antiderivative size = 518, normalized size of antiderivative = 1.00, 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 = {2845, 3057, 2833, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\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 {d \left (4 c^3-27 c^2 d+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^3-27 c^2 d+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 {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 f (c-d)^5 (c+d)^2 \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 ) \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x])/(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 - 2
7*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 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 2833

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 2845

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 3057

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])

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*}

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Mathematica [A]
time = 6.73, size = 828, normalized size = 1.60 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)} \left (-\frac {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}+\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}-\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 (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 {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 c^2 \sin \left (\frac {1}{2} (e+f x)\right )-35 c d \sin \left (\frac {1}{2} (e+f x)\right )+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 d^4 \cos (e+f x)}{3 (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac {2 \left (13 c d^4 \cos (e+f x)+9 d^5 \cos (e+f x)\right )}{3 (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}\right )}{f (a+a \sin (e+f x))^3}+\frac {d \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\frac {2 \left (-c^3 d-387 c^2 d^2-471 c d^3-165 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}}}{\sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos ^2(e+f x) \sqrt {c+d \sin (e+f x)}}{d \left (1-\sin ^2(e+f x)\right )}-\frac {\left (-4 c^4+27 c^3 d-111 c^2 d^2-579 c d^3-357 d^4\right ) \left (\frac {2 (c+d) E\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}}}{\sqrt {c+d \sin (e+f x)}}-\frac {2 c 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}}}{\sqrt {c+d \sin (e+f x)}}\right )}{d}\right )}{60 (c-d)^5 (c+d)^2 f (a+a \sin (e+f x))^3} \end {gather*}

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]]*(-1/15*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 449
*c*d^3 + 267*d^4)/((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] +
Sin[(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*
Sin[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 -
Sin[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] Leaf count of result is larger than twice the leaf count of optimal. \(2310\) vs. \(2(552)=1104\).
time = 45.30, size = 2311, normalized size = 4.46

method	result	size
default	\(\text {Expression too large to display}\)	\(2311\)

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,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(-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)*(sin(f*x
+e)-1)*(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)*((-1-sin(f*x+e))*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)*((-1-sin(f*x+e))*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*Ellipti
cE(((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)*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)+2*(-c*d^2-15*d^3)/(60*c^3-1
80*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)*((-1-sin(f*x+
e))*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)*((-1-sin(f*x+e))*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))))-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)*((-1-sin(f*x+e))*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)*((-1-sin(f*x+e))*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)*(sin(f*x+e)-1)*(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)*((-1-sin(f*x+e))*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))-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)*((-1-sin(f*x+e))*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)*((-1-sin(f*x+e))*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)*((-1-sin(f*x+e))*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*s
in(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.47, size = 4857, normalized size = 9.38 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/180*((sqrt(2)*(8*c^5*d^2 - 54*c^4*d^3 + 219*c^3*d^4 - 3*c^2*d^5 - 699*c*d^6 - 495*d^7)*cos(f*x + e)^5 + sqrt
(2)*(16*c^6*d - 84*c^5*d^2 + 276*c^4*d^3 + 651*c^3*d^4 - 1407*c^2*d^5 - 3087*c*d^6 - 1485*d^7)*cos(f*x + e)^4
- sqrt(2)*(8*c^7 - 22*c^6*d + 27*c^5*d^2 + 711*c^4*d^3 - 54*c^3*d^4 - 3300*c^2*d^5 - 4077*c*d^6 - 1485*d^7)*co
s(f*x + e)^3 - sqrt(2)*(24*c^7 - 82*c^6*d + 173*c^5*d^2 + 1803*c^4*d^3 - 594*c^3*d^4 - 8496*c^2*d^5 - 9843*c*d
^6 - 3465*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4*d^3 - 486*c^3*d^4 - 1896*c
^2*d^5 - 1689*c*d^6 - 495*d^7)*cos(f*x + e) + (sqrt(2)*(8*c^5*d^2 - 54*c^4*d^3 + 219*c^3*d^4 - 3*c^2*d^5 - 699
*c*d^6 - 495*d^7)*cos(f*x + e)^4 - 2*sqrt(2)*(8*c^6*d - 46*c^5*d^2 + 165*c^4*d^3 + 216*c^3*d^4 - 702*c^2*d^5 -
1194*c*d^6 - 495*d^7)*cos(f*x + e)^3 - sqrt(2)*(8*c^7 - 6*c^6*d - 65*c^5*d^2 + 1041*c^4*d^3 + 378*c^3*d^4 - 4
704*c^2*d^5 - 6465*c*d^6 - 2475*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4*d^3
- 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7)*cos(f*x + e) + 4*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2
+ 381*c^4*d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7))*sin(f*x + e) + 4*sqrt(2)*(8*c^7 - 38*c^6*d
+ 119*c^5*d^2 + 381*c^4*d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7))*sqrt(I*d)*weierstrassPInver
se(-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*c^5*d^2 - 54*c^4*d^3 + 219*c^3*d^4 - 3*c^2*d^5 - 699*c*d^6 - 495*d^7)*cos(f*x + e)^5 + sq
rt(2)*(16*c^6*d - 84*c^5*d^2 + 276*c^4*d^3 + 651*c^3*d^4 - 1407*c^2*d^5 - 3087*c*d^6 - 1485*d^7)*cos(f*x + e)^
4 - sqrt(2)*(8*c^7 - 22*c^6*d + 27*c^5*d^2 + 711*c^4*d^3 - 54*c^3*d^4 - 3300*c^2*d^5 - 4077*c*d^6 - 1485*d^7)*
cos(f*x + e)^3 - sqrt(2)*(24*c^7 - 82*c^6*d + 173*c^5*d^2 + 1803*c^4*d^3 - 594*c^3*d^4 - 8496*c^2*d^5 - 9843*c
*d^6 - 3465*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4*d^3 - 486*c^3*d^4 - 1896
*c^2*d^5 - 1689*c*d^6 - 495*d^7)*cos(f*x + e) + (sqrt(2)*(8*c^5*d^2 - 54*c^4*d^3 + 219*c^3*d^4 - 3*c^2*d^5 - 6
99*c*d^6 - 495*d^7)*cos(f*x + e)^4 - 2*sqrt(2)*(8*c^6*d - 46*c^5*d^2 + 165*c^4*d^3 + 216*c^3*d^4 - 702*c^2*d^5
- 1194*c*d^6 - 495*d^7)*cos(f*x + e)^3 - sqrt(2)*(8*c^7 - 6*c^6*d - 65*c^5*d^2 + 1041*c^4*d^3 + 378*c^3*d^4 -
4704*c^2*d^5 - 6465*c*d^6 - 2475*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4*d^
3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7)*cos(f*x + e) + 4*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^
2 + 381*c^4*d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7))*sin(f*x + e) + 4*sqrt(2)*(8*c^7 - 38*c^6
*d + 119*c^5*d^2 + 381*c^4*d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7))*sqrt(-I*d)*weierstrassPIn
verse(-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*c^4*d^3 - 27*I*c^3*d^4 + 111*I*c^2*d^5 + 579*I*c*d^6 + 357*I*d^7)*cos(f*x + e)^5
+ sqrt(2)*(8*I*c^5*d^2 - 42*I*c^4*d^3 + 141*I*c^3*d^4 + 1491*I*c^2*d^5 + 2451*I*c*d^6 + 1071*I*d^7)*cos(f*x +
e)^4 + sqrt(2)*(-4*I*c^6*d + 11*I*c^5*d^2 - 15*I*c^4*d^3 - 942*I*c^3*d^4 - 3006*I*c^2*d^5 - 3165*I*c*d^6 - 107
1*I*d^7)*cos(f*x + e)^3 + sqrt(2)*(-12*I*c^6*d + 41*I*c^5*d^2 - 91*I*c^4*d^3 - 2658*I*c^3*d^4 - 7638*I*c^2*d^5
- 7623*I*c*d^6 - 2499*I*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(4*I*c^6*d - 19*I*c^5*d^2 + 61*I*c^4*d^3 + 774*I*c^3*
d^4 + 1626*I*c^2*d^5 + 1293*I*c*d^6 + 357*I*d^7)*cos(f*x + e) + (sqrt(2)*(4*I*c^4*d^3 - 27*I*c^3*d^4 + 111*I*c
^2*d^5 + 579*I*c*d^6 + 357*I*d^7)*cos(f*x + e)^4 + 2*sqrt(2)*(-4*I*c^5*d^2 + 23*I*c^4*d^3 - 84*I*c^3*d^4 - 690
*I*c^2*d^5 - 936*I*c*d^6 - 357*I*d^7)*cos(f*x + e)^3 + sqrt(2)*(-4*I*c^6*d + 3*I*c^5*d^2 + 31*I*c^4*d^3 - 1110
*I*c^3*d^4 - 4386*I*c^2*d^5 - 5037*I*c*d^6 - 1785*I*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(4*I*c^6*d - 19*I*c^5*d^2
+ 61*I*c^4*d^3 + 774*I*c^3*d^4 + 1626*I*c^2*d^5 + 1293*I*c*d^6 + 357*I*d^7)*cos(f*x + e) + 4*sqrt(2)*(4*I*c^6*
d - 19*I*c^5*d^2 + 61*I*c^4*d^3 + 774*I*c^3*d^4 + 1626*I*c^2*d^5 + 1293*I*c*d^6 + 357*I*d^7))*sin(f*x + e) + 4
*sqrt(2)*(4*I*c^6*d - 19*I*c^5*d^2 + 61*I*c^4*d^3 + 774*I*c^3*d^4 + 1626*I*c^2*d^5 + 1293*I*c*d^6 + 357*I*d^7)
)*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*c^4*d^3 + 27*I*c^3*d^4 - 111*I*c^2*d^5 - 579*I*c*d^6 - 357*I*d^7)*cos(f*x + e)^5 + sqrt(2
)*(-8*I*c^5*d^2 + 42*I*c^4*d^3 - 141*I*c^3*d^4 - 1491*I*c^2*d^5 - 2451*I*c*d^6 - 1071*I*d^7)*cos(f*x + e)^4 +
sqrt(2)*(4*I*c^6*d - 11*I*c^5*d^2 + 15*I*c^4*d^3 + 942*I*c^3*d^4 + 3006*I*c^2*d^5 + 3165*I*c*d^6 + 1071*I*d^7)
*cos(f*x + e)^3 + sqrt(2)*(12*I*c^6*d - 41*I*c^5*d^2 + 91*I*c^4*d^3 + 2658*I*c^3*d^4 + 7638*I*c^2*d^5 + 7623*I
*c*d^6 + 2499*I*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^6*d + 19*I*c^5*d^2 - 61*I*c^4*d^3 - 774*I*c^3*d^4 - 16
26*I*c^2*d^5 - 1293*I*c*d^6 - 357*I*d^7)*cos(f*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )} + 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{5}{\left (e + f x \right )} + 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )} + 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a^{3}} \end {gather*}

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]

Integral(1/(c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2 +
3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + c**2*sqrt(c + d*sin(e + f*x)) + 2*c*d*sqrt(c + d*sin(e + f*x))*
sin(e + f*x)**4 + 6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)
**2 + 2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**5 + 3*d**2*sqr
t(c + d*sin(e + f*x))*sin(e + f*x)**4 + 3*d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + d**2*sqrt(c + d*sin(
e + f*x))*sin(e + f*x)**2), x)/a**3

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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