3.6.20 \(\int \frac {1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx\) [520]

Optimal. Leaf size=423 \[ -\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{30 a^3 (c-d)^4 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 c^2-21 c d+65 d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{30 a^3 (c-d)^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-1/30*d*(4*c^3-21*c^2*d+62*c*d^2+147*d^3)*cos(f*x+e)/a^3/(c-d)^4/(c+d)/f/(c+d*sin(f*x+e))^(1/2)-1/5*cos(f*x+e)
/(c-d)/f/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2)-2/15*(c-4*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2/(c+d
*sin(f*x+e))^(1/2)-1/30*(4*c^2-21*c*d+65*d^2)*cos(f*x+e)/(c-d)^3/f/(a^3+a^3*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2)
+1/30*(4*c^3-21*c^2*d+62*c*d^2+147*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)*Elliptic
E(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)^4/(c+d)/f/((c+d*sin(f*x+
e))/(c+d))^(1/2)-1/30*(4*c^2-21*c*d+65*d^2)*(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)*Elli
pticF(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)^3/f/(c+d*sin
(f*x+e))^(1/2)

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Rubi [A]
time = 0.66, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 9, 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^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\left (4 c^2-21 c d+65 d^2\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)^3 \sqrt {c+d \sin (e+f x)}}-\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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)^4 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(d*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Cos[e + f*x])/(a^3*(c - d)^4*(c + d)*f*Sqrt[c + d*Sin[e + f*x
]]) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]) - (2*(c - 4*d)*Cos[e + f*x])/
(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x])/
(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*
EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[(c + d
*Sin[e + f*x])/(c + d)]) + ((4*c^2 - 21*c*d + 65*d^2)*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)^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 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))^{3/2}} \, dx &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a (4 c-11 d)-\frac {5}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx}{5 a^2 (c-d)}\\ &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {\int \frac {\frac {1}{2} a^2 \left (4 c^2-15 c d+41 d^2\right )+3 a^2 (c-4 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-\frac {3}{4} a^3 (c-49 d) d^2-\frac {1}{4} a^3 d \left (4 c^2-21 c d+65 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}+\frac {2 \int \frac {-\frac {1}{8} a^3 d^2 \left (c^2+126 c d+65 d^2\right )-\frac {1}{8} a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 a^6 (c-d)^4 (c+d)}\\ &=-\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{60 a^3 (c-d)^3}-\frac {\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{60 a^3 (c-d)^4 (c+d)}\\ &=-\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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)^4 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (\left (4 c^2-21 c d+65 d^2\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)^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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)^4 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 c^2-21 c d+65 d^2\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)^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 6.39, size = 745, normalized size = 1.76 \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^3-21 c^2 d+62 c d^2+117 d^3}{15 (c-d)^4 (c+d)}+\frac {2 \sin \left (\frac {1}{2} (e+f x)\right )}{5 (c-d)^2 \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)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {-2 c+11 d}{15 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 \left (2 c \sin \left (\frac {1}{2} (e+f x)\right )-11 d \sin \left (\frac {1}{2} (e+f x)\right )\right )}{15 (c-d)^3 \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 )-25 c d \sin \left (\frac {1}{2} (e+f x)\right )+87 d^2 \sin \left (\frac {1}{2} (e+f x)\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 )}-\frac {2 d^4 \cos (e+f x)}{(c-d)^4 (c+d) (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^2 d-126 c d^2-65 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}}}{\sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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^3+21 c^2 d-62 c d^2-147 d^3\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)^4 (c+d) 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])^(3/2)),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]]*(-1/15*(4*c^3 - 21*c^2*d + 62*c*d^2 + 117*d^
3)/((c - d)^4*(c + d)) + (2*Sin[(e + f*x)/2])/(5*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) - 1/(5*(c
- d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (-2*c + 11*d)/(15*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^2) + (2*(2*c*Sin[(e + f*x)/2] - 11*d*Sin[(e + f*x)/2]))/(15*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/
2])^3) + (4*c^2*Sin[(e + f*x)/2] - 25*c*d*Sin[(e + f*x)/2] + 87*d^2*Sin[(e + f*x)/2])/(15*(c - d)^4*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2])) - (2*d^4*Cos[e + f*x])/((c - d)^4*(c + d)*(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^2*d) - 126*c*d^2 - 65*d^3)*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^3 - 21*c^
2*d + 62*c*d^2 + 147*d^3)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Sin[e + f*x]^2)) - ((-4*c^3 + 21*c^
2*d - 62*c*d^2 - 147*d^3)*((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)^4*(c + d)*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. \(1850\) vs. \(2(461)=922\).
time = 32.72, size = 1851, normalized size = 4.38

method	result	size
default	\(\text {Expression too large to display}\)	\(1851\)

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))^(3/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(-d/(c-d)^2*(-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)*EllipticE
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1
/2))))+1/(c-d)*(-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^3-180*c
^2*d+180*c*d^2-60*d^3)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-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^2/(c-d
)^3*(-(-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))))-d^3/(c-d)^3*(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*si
n(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)))))/c
os(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))^(3/2),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 = 0.32, size = 3124, normalized size = 7.39 \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))^(3/2),x, algorithm="fricas")

[Out]

1/180*((sqrt(2)*(8*c^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)*cos(f*x + e)^4 - sqrt(2)*(8*c^5 - 26
*c^4*d + 37*c^3*d^2 + 158*c^2*d^3 - 363*c*d^4 - 390*d^5)*cos(f*x + e)^3 - sqrt(2)*(24*c^5 - 86*c^4*d + 153*c^3
*d^2 + 353*c^2*d^3 - 1005*c*d^4 - 975*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*
d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e) - (sqrt(2)*(8*c^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)*c
os(f*x + e)^3 + sqrt(2)*(8*c^5 - 18*c^4*d - 5*c^3*d^2 + 279*c^2*d^3 - 447*c*d^4 - 585*d^5)*cos(f*x + e)^2 - 2*
sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e) - 4*sqrt(2)*(8*c^5 - 3
4*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5))*sin(f*x + e) + 4*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*
d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5))*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*c^4*d - 42*c^3*d^2 +
121*c^2*d^3 - 84*c*d^4 - 195*d^5)*cos(f*x + e)^4 - sqrt(2)*(8*c^5 - 26*c^4*d + 37*c^3*d^2 + 158*c^2*d^3 - 363*
c*d^4 - 390*d^5)*cos(f*x + e)^3 - sqrt(2)*(24*c^5 - 86*c^4*d + 153*c^3*d^2 + 353*c^2*d^3 - 1005*c*d^4 - 975*d^
5)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e)
- (sqrt(2)*(8*c^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)*cos(f*x + e)^3 + sqrt(2)*(8*c^5 - 18*c^4*
d - 5*c^3*d^2 + 279*c^2*d^3 - 447*c*d^4 - 585*d^5)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 +
37*c^2*d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e) - 4*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*
c*d^4 - 195*d^5))*sin(f*x + e) + 4*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5))
*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*(sqrt(2)*(4*I*c^3*d^2 - 21*I*c^2*d^3 + 62*I*c*d^4 + 147*I*d^5)*cos(f*
x + e)^4 + sqrt(2)*(-4*I*c^4*d + 13*I*c^3*d^2 - 20*I*c^2*d^3 - 271*I*c*d^4 - 294*I*d^5)*cos(f*x + e)^3 + sqrt(
2)*(-12*I*c^4*d + 43*I*c^3*d^2 - 81*I*c^2*d^3 - 751*I*c*d^4 - 735*I*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(4*I*c^4*d
- 17*I*c^3*d^2 + 41*I*c^2*d^3 + 209*I*c*d^4 + 147*I*d^5)*cos(f*x + e) + (sqrt(2)*(-4*I*c^3*d^2 + 21*I*c^2*d^3
- 62*I*c*d^4 - 147*I*d^5)*cos(f*x + e)^3 + sqrt(2)*(-4*I*c^4*d + 9*I*c^3*d^2 + I*c^2*d^3 - 333*I*c*d^4 - 441*
I*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(4*I*c^4*d - 17*I*c^3*d^2 + 41*I*c^2*d^3 + 209*I*c*d^4 + 147*I*d^5)*cos(f*x
+ e) + 4*sqrt(2)*(4*I*c^4*d - 17*I*c^3*d^2 + 41*I*c^2*d^3 + 209*I*c*d^4 + 147*I*d^5))*sin(f*x + e) + 4*sqrt(2)
*(4*I*c^4*d - 17*I*c^3*d^2 + 41*I*c^2*d^3 + 209*I*c*d^4 + 147*I*d^5))*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^3*d^2 + 21*I*c^2*d^
3 - 62*I*c*d^4 - 147*I*d^5)*cos(f*x + e)^4 + sqrt(2)*(4*I*c^4*d - 13*I*c^3*d^2 + 20*I*c^2*d^3 + 271*I*c*d^4 +
294*I*d^5)*cos(f*x + e)^3 + sqrt(2)*(12*I*c^4*d - 43*I*c^3*d^2 + 81*I*c^2*d^3 + 751*I*c*d^4 + 735*I*d^5)*cos(f
*x + e)^2 + 2*sqrt(2)*(-4*I*c^4*d + 17*I*c^3*d^2 - 41*I*c^2*d^3 - 209*I*c*d^4 - 147*I*d^5)*cos(f*x + e) + (sqr
t(2)*(4*I*c^3*d^2 - 21*I*c^2*d^3 + 62*I*c*d^4 + 147*I*d^5)*cos(f*x + e)^3 + sqrt(2)*(4*I*c^4*d - 9*I*c^3*d^2 -
I*c^2*d^3 + 333*I*c*d^4 + 441*I*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^4*d + 17*I*c^3*d^2 - 41*I*c^2*d^3 - 2
09*I*c*d^4 - 147*I*d^5)*cos(f*x + e) + 4*sqrt(2)*(-4*I*c^4*d + 17*I*c^3*d^2 - 41*I*c^2*d^3 - 209*I*c*d^4 - 147
*I*d^5))*sin(f*x + e) + 4*sqrt(2)*(-4*I*c^4*d + 17*I*c^3*d^2 - 41*I*c^2*d^3 - 209*I*c*d^4 - 147*I*d^5))*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)) - 6
*(6*c^4*d - 12*c^3*d^2 + 12*c*d^4 - 6*d^5 - (4*c^3*d^2 - 21*c^2*d^3 + 62*c*d^4 + 147*d^5)*cos(f*x + e)^4 - (4*
c^4*d - 9*c^3*d^2 - 2*c^2*d^3 + 207*c*d^4 + 376*d^5)*cos(f*x + e)^3 + (8*c^4*d - 33*c^3*d^2 + 31*c^2*d^3 + 165
*c*d^4 + 213*d^5)*cos(f*x + e)^2 + 2*(9*c^4*d - 29*c^3*d^2 + 25*c^2*d^3 + 161*c*d^4 + 218*d^5)*cos(f*x + e) -
(6*c^4*d - 12*c^3*d^2 + 12*c*d^4 - 6*d^5 + (4*c^3*d^2 - 21*c^2*d^3 + 62*c*d^4 + 147*d^5)*cos(f*x + e)^3 - (4*c
^4*d - 13*c^3*d^2 + 19*c^2*d^3 + 145*c*d^4 + 229*d^5)*cos(f*x + e)^2 - 2*(6*c^4*d - 23*c^3*d^2 + 25*c^2*d^3 +
155*c*d^4 + 221*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/((a^3*c^5*d^2 - 3*a^3*c^4*d^3 + 2*a
^3*c^3*d^4 + 2*a^3*c^2*d^5 - 3*a^3*c*d^6 + a^3*d^7)*f*cos(f*x + e)^4 - (a^3*c^6*d - a^3*c^5*d^2 - 4*a^3*c^4*d^
3 + 6*a^3*c^3*d^4 + a^3*c^2*d^5 - 5*a^3*c*d^6 + 2*a^3*d^7)*f*cos(f*x + e)^3 - (3*a^3*c^6*d - 4*a^3*c^5*d^2 - 9
*a^3*c^4*d^3 + 16*a^3*c^3*d^4 + a^3*c^2*d^5 - 12*a^3*c*d^6 + 5*a^3*d^7)*f*cos(f*x + e)^2 + 2*(a^3*c^6*d - 2*a^
3*c^5*d^2 - a^3*c^4*d^3 + 4*a^3*c^3*d^4 - a^3*c...

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

[Out]

Integral(1/(c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2 + 3*c*sq
rt(c + d*sin(e + f*x))*sin(e + f*x) + c*sqrt(c + d*sin(e + f*x)) + d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4
+ 3*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2 + d*sqrt(c + d*s
in(e + f*x))*sin(e + f*x)), 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))^(3/2),x, algorithm="giac")

[Out]

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

[Out]

\text{Hanged}

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