\(\int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx\) [181]

Optimal result

Integrand size = 33, antiderivative size = 536 \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=-\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \]

[Out]

-3*a*x^2/c/f^2/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)-3*I*a*x^2*cos(f*x+e)/c/f^2/(a-a*sin(f*x+e))^(1/2)
/(c+c*sin(f*x+e))^(1/2)-12*I*a*x*arctan(exp(I*(f*x+e)))*cos(f*x+e)/c/f^3/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e
))^(1/2)+6*a*x*cos(f*x+e)*ln(1+exp(2*I*(f*x+e)))/c/f^3/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)+6*I*a*cos
(f*x+e)*polylog(2,-I*exp(I*(f*x+e)))/c/f^4/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)-6*I*a*cos(f*x+e)*poly
log(2,I*exp(I*(f*x+e)))/c/f^4/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)-3*I*a*cos(f*x+e)*polylog(2,-exp(2*
I*(f*x+e)))/c/f^4/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)-a*x^3*sec(f*x+e)/c/f/(a-a*sin(f*x+e))^(1/2)/(c
+c*sin(f*x+e))^(1/2)+3*a*x^2*sin(f*x+e)/c/f^2/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)+a*x^3*tan(f*x+e)/c
/f/(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 4.17 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 51, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {4700, 6873, 12, 6874, 4271, 4266, 2317, 2438, 2611, 6744, 2320, 6724, 3842, 4269, 3800, 2221, 4498} \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {6 i a \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \cos (e+f x)}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {6 i a \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \cos (e+f x)}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {3 i a \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right ) \cos (e+f x)}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {6 a x \log \left (1+e^{2 i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]

[In]

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

[Out]

(-3*a*x^2)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((3*I)*a*x^2*Cos[e + f*x])/(c*f^2*Sqrt[
a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((12*I)*a*x*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(c*f^3*Sqrt[
a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*a*x*Cos[e + f*x]*Log[1 + E^((2*I)*(e + f*x))])/(c*f^3*Sqrt[
a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + ((6*I)*a*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(c*f^4
*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((6*I)*a*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(c*
f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((3*I)*a*Cos[e + f*x]*PolyLog[2, -E^((2*I)*(e + f*x))
])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^3*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*
x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*a*x^2*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]
]) + (a*x^3*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 3842

Int[(x_)^(m_.)*Sec[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tan[(a_.) + (b_.)*(x_)^(n_.)]^(q_.), x_Symbol] :> Simp[x^(m
- n + 1)*(Sec[a + b*x^n]^p/(b*n*p)), x] - Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Sec[a + b*x^n]^p, x], x] /;
FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4498

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]*Tan[(a_.) + (b_.)*(x_)]^(p_), x_Symbol] :> -Int[(c + d*
x)^m*Sec[a + b*x]*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^3*Tan[a + b*x]^(p - 2), x] /; FreeQ[
{a, b, c, d, m}, x] && IGtQ[p/2, 0]

Rule 4700

Int[((g_.) + (h_.)*(x_))^(p_.)*((a_) + (b_.)*Sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*Sin[(e_.) + (f_.)*(x_
)])^(n_), x_Symbol] :> Dist[a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*((c + d*Sin[e + f*x])^F
racPart[m]/Cos[e + f*x]^(2*FracPart[m])), Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x],
x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ
[2*m] && IGeQ[n - m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int x^3 \sec ^3(e+f x) (a-a \sin (e+f x))^2 \, dx}{a c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \int a^2 x^3 \sec ^3(e+f x) (1-\sin (e+f x))^2 \, dx}{a c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int x^3 \sec ^3(e+f x) (1-\sin (e+f x))^2 \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int \left (x^3 \sec ^3(e+f x)-2 x^3 \sec ^2(e+f x) \tan (e+f x)+x^3 \sec (e+f x) \tan ^2(e+f x)\right ) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int x^3 \sec ^3(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \tan ^2(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 a \cos (e+f x)) \int x^3 \sec ^2(e+f x) \tan (e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \, dx}{2 c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec ^3(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x \sec (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \sec ^2(e+f x) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a x^3 \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \, dx}{2 c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x \sec (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 a \cos (e+f x)) \int x \tan (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 i a x^2 \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(6 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(12 i a \cos (e+f x)) \int \frac {e^{2 i (e+f x)} x}{1+e^{2 i (e+f x)}} \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x \cos (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 a x \cos (e+f x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 a \cos (e+f x)) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(6 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(6 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.36 \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a-a \sin (e+f x)} \left (12 i \operatorname {PolyLog}\left (2,-i e^{-i (e+f x)}\right ) (1+\sin (e+f x))+f x \left (3 i f x-f^2 x^2-3 f x \cos (e+f x)+12 \log \left (1+i e^{-i (e+f x)}\right )+3 \left (i f x+4 \log \left (1+i e^{-i (e+f x)}\right )\right ) \sin (e+f x)\right )\right )}{f^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c (1+\sin (e+f x)))^{3/2}} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a - a*Sin[e + f*x]]*((12*I)*PolyLog[2, (-I)/E^(I*(e + f*x))]*(1 +
Sin[e + f*x]) + f*x*((3*I)*f*x - f^2*x^2 - 3*f*x*Cos[e + f*x] + 12*Log[1 + I/E^(I*(e + f*x))] + 3*(I*f*x + 4*L
og[1 + I/E^(I*(e + f*x))])*Sin[e + f*x])))/(f^4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^(
3/2))

Maple [F]

\[\int \frac {x^{3} \sqrt {a -\sin \left (f x +e \right ) a}}{\left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(x^3*(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\int \frac {x^{3} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(x**3*sqrt(-a*(sin(e + f*x) - 1))/(c*(sin(e + f*x) + 1))**(3/2), x)

Maxima [F]

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

[In]

integrate(x^3*(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(x^3*(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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