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

Optimal. Leaf size=280 \[ \frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a x^2 \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}} \]

[Out]

(-2*a*x)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/
(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f^3*Sqrt[a
- a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^2*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*S
in[e + f*x]]) + (2*a*x*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x^2*Tan[e
+ f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])

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Rubi [A]  time = 2.1552, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 14, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.424, Rules used = {4604, 6741, 12, 6742, 4186, 3770, 4181, 2531, 2282, 6589, 3757, 4184, 3475, 4413} \[ \frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a x^2 \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*x)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/
(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f^3*Sqrt[a
- a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^2*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*S
in[e + f*x]]) + (2*a*x*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x^2*Tan[e
+ f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])

Rule 4604

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 6741

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

Rule 12

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

Rule 6742

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

Rule 4186

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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4181

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 2531

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 2282

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 6589

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 3757

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 4184

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4413

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]

Rubi steps

\begin{align*} \int \frac{x^2 \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx &=\frac{\cos (e+f x) \int x^2 \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^2 \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^2 \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^2 \sec ^3(e+f x)-2 x^2 \sec ^2(e+f x) \tan (e+f x)+x^2 \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^2 \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^2 \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^2 \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{a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a x^2 \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^2 \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^2 \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^2 \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 \sec (e+f x) \, dx}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{(2 a \cos (e+f x)) \int x \sec ^2(e+f x) \, dx}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{i a x^2 \tan ^{-1}\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 \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a x^2 \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^2 \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 \sec (e+f x) \, dx}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(2 a \cos (e+f x)) \int \tan (e+f x) \, dx}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(a \cos (e+f x)) \int x \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{(a \cos (e+f x)) \int x \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{(2 a \cos (e+f x)) \int x \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{(2 a \cos (e+f x)) \int x \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{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \cos (e+f x) \log (\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 \cos (e+f x) \text{Li}_2\left (-i e^{i (e+f x)}\right )}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{i a x \cos (e+f x) \text{Li}_2\left (i e^{i (e+f x)}\right )}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a x^2 \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(i a \cos (e+f x)) \int \text{Li}_2\left (-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{(i a \cos (e+f x)) \int \text{Li}_2\left (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{(2 i a \cos (e+f x)) \int \text{Li}_2\left (-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{(2 i a \cos (e+f x)) \int \text{Li}_2\left (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{(a \cos (e+f x)) \int x \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{(a \cos (e+f x)) \int x \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{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a x^2 \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)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,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 \cos (e+f x)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{(2 a \cos (e+f x)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(2 a \cos (e+f x)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(i a \cos (e+f x)) \int \text{Li}_2\left (-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{(i a \cos (e+f x)) \int \text{Li}_2\left (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{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a \cos (e+f x) \text{Li}_3\left (-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 \cos (e+f x) \text{Li}_3\left (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^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a x^2 \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)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,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 \cos (e+f x)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a x^2 \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 1.59998, size = 154, normalized size = 0.55 \[ -\frac{\sqrt{a-a \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (-4 \log \left (e^{i (e+f x)}+i\right )+2 f x \cos (e+f x)+\left (2 i f x-4 \log \left (e^{i (e+f x)}+i\right )\right ) \sin (e+f x)+f^2 x^2+2 i f x\right )}{f^3 (c (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*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]]*((2*I)*f*x + f^2*x^2 + 2*f*x*Cos[e + f*x] -
4*Log[I + E^(I*(e + f*x))] + ((2*I)*f*x - 4*Log[I + E^(I*(e + f*x))])*Sin[e + f*x]))/(f^3*(Cos[(e + f*x)/2] -
Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^(3/2)))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}\sqrt{a-a\sin \left ( fx+e \right ) } \left ( c+c\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a} x^{2}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a} x^{2}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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