3.249 \(\int \frac{(e+f x)^2 \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=1584 \[ \text{result too large to display} \]

[Out]

(((-3*I)/2)*a^2*(e + f*x)^2)/(b*(a^2 - b^2)^2*d) + (I*(e + f*x)^2)/(b*(a^2 - b^2)*d) + (2*a*f^2*ArcTan[(b + a*
Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)*d^3) + (3*a^2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)
))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^2) - (2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2
- b^2])])/(b*(a^2 - b^2)*d^2) + (((3*I)/2)*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])
])/(b*(a^2 - b^2)^(5/2)*d) - (((3*I)/2)*a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b
*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2
)^2*d^2) - (2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) - (((3*I)/
2)*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*
a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - ((3*I)*a^2*f^2*P
olyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^3) + ((2*I)*f^2*PolyLog[2, (I*b*E^(
I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (3*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x))
)/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - S
qrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - ((3*I)*a^2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 -
b^2])])/(b*(a^2 - b^2)^2*d^3) + ((2*I)*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 -
b^2)*d^3) - (3*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d
^2) + (3*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) + ((
3*I)*a^3*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) - ((3*I)*a*f^2
*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((3*I)*a^3*f^2*PolyLog[3
, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) + ((3*I)*a*f^2*PolyLog[3, (I*b*E^(I*
(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - (a*(e + f*x)^2*Cos[c + d*x])/(2*(a^2 - b^2)*d*
(a + b*Sin[c + d*x])^2) - (a*f*(e + f*x))/(b*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)^2*Cos[c
+ d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + ((e + f*x)^2*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*
x]))

________________________________________________________________________________________

Rubi [A]  time = 5.94317, antiderivative size = 1584, normalized size of antiderivative = 1., number of steps used = 73, number of rules used = 16, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {6742, 3325, 3324, 3323, 2264, 2190, 2531, 2282, 6589, 4519, 2279, 2391, 4422, 2660, 618, 204} \[ \frac{3 i (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a^3}{2 b \left (a^2-b^2\right )^{5/2} d}-\frac{3 i (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a^3}{2 b \left (a^2-b^2\right )^{5/2} d}+\frac{3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a^3}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a^3}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{3 i f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a^3}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{3 i f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a^3}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{3 i (e+f x)^2 a^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{3 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a^2}{b \left (a^2-b^2\right )^2 d^2}+\frac{3 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a^2}{b \left (a^2-b^2\right )^2 d^2}-\frac{3 i f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a^2}{b \left (a^2-b^2\right )^2 d^3}-\frac{3 i f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a^2}{b \left (a^2-b^2\right )^2 d^3}-\frac{3 (e+f x)^2 \cos (c+d x) a^2}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{2 f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right ) a}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{3 i (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a}{2 b \left (a^2-b^2\right )^{3/2} d}+\frac{3 i (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a}{b \left (a^2-b^2\right )^{3/2} d^2}+\frac{3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) a}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 i f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) a}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{f (e+f x) a}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{(e+f x)^2 \cos (c+d x) a}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{2 i f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{2 i f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/2)*a^2*(e + f*x)^2)/(b*(a^2 - b^2)^2*d) + (I*(e + f*x)^2)/(b*(a^2 - b^2)*d) + (2*a*f^2*ArcTan[(b + a*
Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)*d^3) + (3*a^2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)
))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^2) - (2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2
- b^2])])/(b*(a^2 - b^2)*d^2) + (((3*I)/2)*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])
])/(b*(a^2 - b^2)^(5/2)*d) - (((3*I)/2)*a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b
*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2
)^2*d^2) - (2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) - (((3*I)/
2)*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*
a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - ((3*I)*a^2*f^2*P
olyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^3) + ((2*I)*f^2*PolyLog[2, (I*b*E^(
I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (3*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x))
)/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - S
qrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - ((3*I)*a^2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 -
b^2])])/(b*(a^2 - b^2)^2*d^3) + ((2*I)*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 -
b^2)*d^3) - (3*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d
^2) + (3*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) + ((
3*I)*a^3*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) - ((3*I)*a*f^2
*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((3*I)*a^3*f^2*PolyLog[3
, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) + ((3*I)*a*f^2*PolyLog[3, (I*b*E^(I*
(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - (a*(e + f*x)^2*Cos[c + d*x])/(2*(a^2 - b^2)*d*
(a + b*Sin[c + d*x])^2) - (a*f*(e + f*x))/(b*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)^2*Cos[c
+ d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + ((e + f*x)^2*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*
x]))

Rule 6742

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

Rule 3325

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(c + d*x)^m*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(a^2 - b^2)), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m*
(a + b*Sin[e + f*x])^(n + 1), x], x] - Dist[(b*(n + 2))/((n + 1)*(a^2 - b^2)), Int[(c + d*x)^m*Sin[e + f*x]*(a
+ b*Sin[e + f*x])^(n + 1), x], x] + Dist[(b*d*m)/(f*(n + 1)*(a^2 - b^2)), Int[(c + d*x)^(m - 1)*Cos[e + f*x]*
(a + b*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] && I
GtQ[m, 0]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3323

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 2279

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 2391

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 4422

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

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \left (-\frac{a (e+f x)^2}{b (a+b \sin (c+d x))^3}+\frac{(e+f x)^2}{b (a+b \sin (c+d x))^2}\right ) \, dx\\ &=\frac{\int \frac{(e+f x)^2}{(a+b \sin (c+d x))^2} \, dx}{b}-\frac{a \int \frac{(e+f x)^2}{(a+b \sin (c+d x))^3} \, dx}{b}\\ &=-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{a \int \frac{(e+f x)^2 \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )}+\frac{a \int \frac{(e+f x)^2}{a+b \sin (c+d x)} \, dx}{b \left (a^2-b^2\right )}-\frac{a^2 \int \frac{(e+f x)^2}{(a+b \sin (c+d x))^2} \, dx}{b \left (a^2-b^2\right )}-\frac{(2 f) \int \frac{(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right ) d}+\frac{(a f) \int \frac{(e+f x) \cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{\left (a^2-b^2\right ) d}\\ &=\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{a^2 (e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{a^3 \int \frac{(e+f x)^2}{a+b \sin (c+d x)} \, dx}{b \left (a^2-b^2\right )^2}+\frac{a \int \left (-\frac{a (e+f x)^2}{b (a+b \sin (c+d x))^2}+\frac{(e+f x)^2}{b (a+b \sin (c+d x))}\right ) \, dx}{2 \left (a^2-b^2\right )}+\frac{(2 a) \int \frac{e^{i (c+d x)} (e+f x)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b \left (a^2-b^2\right )}+\frac{\left (2 a^2 f\right ) \int \frac{(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2 d}-\frac{(2 f) \int \frac{e^{i (c+d x)} (e+f x)}{a-\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right ) d}-\frac{(2 f) \int \frac{e^{i (c+d x)} (e+f x)}{a+\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right ) d}+\frac{\left (a f^2\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b \left (a^2-b^2\right ) d^2}\\ &=-\frac{i a^2 (e+f x)^2}{b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{a^2 (e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (2 a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b \left (a^2-b^2\right )^2}-\frac{(2 i a) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{(2 i a) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{a \int \frac{(e+f x)^2}{a+b \sin (c+d x)} \, dx}{2 b \left (a^2-b^2\right )}-\frac{a^2 \int \frac{(e+f x)^2}{(a+b \sin (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}+\frac{\left (2 a^2 f\right ) \int \frac{e^{i (c+d x)} (e+f x)}{a-\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^2 d}+\frac{\left (2 a^2 f\right ) \int \frac{e^{i (c+d x)} (e+f x)}{a+\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^2 d}+\frac{\left (2 a f^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b \left (a^2-b^2\right ) d^3}+\frac{\left (2 f^2\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right ) d^2}+\frac{\left (2 f^2\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right ) d^2}\\ &=-\frac{i a^2 (e+f x)^2}{b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d}+\frac{2 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (2 i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{\left (2 i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{a^3 \int \frac{(e+f x)^2}{a+b \sin (c+d x)} \, dx}{2 b \left (a^2-b^2\right )^2}+\frac{a \int \frac{e^{i (c+d x)} (e+f x)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b \left (a^2-b^2\right )}+\frac{\left (a^2 f\right ) \int \frac{(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2 d}+\frac{(2 i a f) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d}-\frac{(2 i a f) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d}-\frac{\left (2 i f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{\left (2 i f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{\left (4 a f^2\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b \left (a^2-b^2\right ) d^3}-\frac{\left (2 a^2 f^2\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^2 d^2}-\frac{\left (2 a^2 f^2\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^2 d^2}\\ &=-\frac{3 i a^2 (e+f x)^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{2 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d}-\frac{i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d}+\frac{2 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d}+\frac{i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{2 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{2 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{a^3 \int \frac{e^{i (c+d x)} (e+f x)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b \left (a^2-b^2\right )^2}-\frac{(i a) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{(i a) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}-\frac{\left (2 i a^3 f\right ) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d}+\frac{\left (2 i a^3 f\right ) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d}+\frac{\left (a^2 f\right ) \int \frac{e^{i (c+d x)} (e+f x)}{a-\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^2 d}+\frac{\left (a^2 f\right ) \int \frac{e^{i (c+d x)} (e+f x)}{a+\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^2 d}+\frac{\left (2 i a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{\left (2 i a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{\left (2 a f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{\left (2 a f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^2}\\ &=-\frac{3 i a^2 (e+f x)^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d}-\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d}+\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{2 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{2 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{2 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{2 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{\left (i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}+\frac{(i a f) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d}-\frac{(i a f) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d}-\frac{\left (2 i a f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{\left (2 i a f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{\left (2 a^3 f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{\left (2 a^3 f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{\left (a^2 f^2\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^2 d^2}-\frac{\left (a^2 f^2\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^2 d^2}\\ &=-\frac{3 i a^2 (e+f x)^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}-\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}+\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{2 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{2 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{2 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{2 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (i a^3 f\right ) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d}+\frac{\left (i a^3 f\right ) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d}+\frac{\left (2 i a^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{\left (2 i a^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}+\frac{\left (i a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{\left (i a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{\left (a f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{\left (a f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^2}\\ &=-\frac{3 i a^2 (e+f x)^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}-\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}+\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{3 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}+\frac{2 i a^3 f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{2 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{2 i a^3 f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}+\frac{2 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (i a f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{\left (i a f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{\left (a^3 f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{\left (a^3 f^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{5/2} d^2}\\ &=-\frac{3 i a^2 (e+f x)^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}-\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}+\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{3 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}+\frac{2 i a^3 f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{3 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{2 i a^3 f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}+\frac{3 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (i a^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{\left (i a^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}\\ &=-\frac{3 i a^2 (e+f x)^2}{2 b \left (a^2-b^2\right )^2 d}+\frac{i (e+f x)^2}{b \left (a^2-b^2\right ) d}+\frac{2 a f^2 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}+\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}-\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}+\frac{3 a^2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^2}-\frac{2 f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^2}-\frac{3 i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d}+\frac{3 i a (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{3 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}-\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i a^2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^2 d^3}+\frac{2 i f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 a^3 f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{3 a f (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^2}+\frac{3 i a^3 f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}-\frac{3 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{3 i a^3 f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d^3}+\frac{3 i a f^2 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{a (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f (e+f x)}{b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x)^2 \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 25.016, size = 13567, normalized size = 8.57 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 3.147, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}\sin \left ( dx+c \right ) }{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 7.4687, size = 12407, normalized size = 7.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*(a^6 - 2*a^4*b^2 + a^2*b^4)*d*f^2*x + 8*(a^6 - 2*a^4*b^2 + a^2*b^4)*d*e*f + 12*(a*b^5*f^2*cos(d*x + c)^
2 - 2*a^2*b^4*f^2*sin(d*x + c) - (a^3*b^3 + a*b^5)*f^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(2*I*a*cos(d*x +
c) - 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 12*(a*b^5*f^2*cos(
d*x + c)^2 - 2*a^2*b^4*f^2*sin(d*x + c) - (a^3*b^3 + a*b^5)*f^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(2*I*a*
cos(d*x + c) - 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 12*(a*b^5
*f^2*cos(d*x + c)^2 - 2*a^2*b^4*f^2*sin(d*x + c) - (a^3*b^3 + a*b^5)*f^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(
I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 12*(a*b^5
*f^2*cos(d*x + c)^2 - 2*a^2*b^4*f^2*sin(d*x + c) - (a^3*b^3 + a*b^5)*f^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(
I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 4*((2*a^5
*b - a^3*b^3 - a*b^5)*d^2*f^2*x^2 + 2*(2*a^5*b - a^3*b^3 - a*b^5)*d^2*e*f*x + (2*a^5*b - a^3*b^3 - a*b^5)*d^2*
e^2)*cos(d*x + c) + (4*I*(a^4*b^2 + a^2*b^4 - 2*b^6)*f^2*cos(d*x + c)^2 - 8*I*(a^5*b + a^3*b^3 - 2*a*b^5)*f^2*
sin(d*x + c) - 4*I*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f^2 + 2*(6*I*(a^3*b^3 + a*b^5)*d*f^2*x + 6*I*(a^3*b^3 +
a*b^5)*d*e*f + (-6*I*a*b^5*d*f^2*x - 6*I*a*b^5*d*e*f)*cos(d*x + c)^2 + (12*I*a^2*b^4*d*f^2*x + 12*I*a^2*b^4*d
*e*f)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x +
c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (4*I*(a^4*b^2 + a^2*b^4 - 2*b^6)*f^2*cos(d*x +
c)^2 - 8*I*(a^5*b + a^3*b^3 - 2*a*b^5)*f^2*sin(d*x + c) - 4*I*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f^2 + 2*(-6*
I*(a^3*b^3 + a*b^5)*d*f^2*x - 6*I*(a^3*b^3 + a*b^5)*d*e*f + (6*I*a*b^5*d*f^2*x + 6*I*a*b^5*d*e*f)*cos(d*x + c)
^2 + (-12*I*a^2*b^4*d*f^2*x - 12*I*a^2*b^4*d*e*f)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(2*I*a*cos(
d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (-4
*I*(a^4*b^2 + a^2*b^4 - 2*b^6)*f^2*cos(d*x + c)^2 + 8*I*(a^5*b + a^3*b^3 - 2*a*b^5)*f^2*sin(d*x + c) + 4*I*(a^
6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f^2 + 2*(-6*I*(a^3*b^3 + a*b^5)*d*f^2*x - 6*I*(a^3*b^3 + a*b^5)*d*e*f + (6*I*
a*b^5*d*f^2*x + 6*I*a*b^5*d*e*f)*cos(d*x + c)^2 + (-12*I*a^2*b^4*d*f^2*x - 12*I*a^2*b^4*d*e*f)*sin(d*x + c))*s
qrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x +
c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (-4*I*(a^4*b^2 + a^2*b^4 - 2*b^6)*f^2*cos(d*x + c)^2 + 8*I*(a^5*b +
a^3*b^3 - 2*a*b^5)*f^2*sin(d*x + c) + 4*I*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f^2 + 2*(6*I*(a^3*b^3 + a*b^5)*
d*f^2*x + 6*I*(a^3*b^3 + a*b^5)*d*e*f + (-6*I*a*b^5*d*f^2*x - 6*I*a*b^5*d*e*f)*cos(d*x + c)^2 + (12*I*a^2*b^4*
d*f^2*x + 12*I*a^2*b^4*d*e*f)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(
d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - 2*(2*(a^6 + 2*a^4*b^2
- a^2*b^4 - 2*b^6)*d*e*f - 2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^4*b^2 + a^2*b^4 - 2*b^6)*d*e*f
- (a^4*b^2 + a^2*b^4 - 2*b^6)*c*f^2)*cos(d*x + c)^2 + 4*((a^5*b + a^3*b^3 - 2*a*b^5)*d*e*f - (a^5*b + a^3*b^3
- 2*a*b^5)*c*f^2)*sin(d*x + c) - (3*(a^3*b^3 + a*b^5)*d^2*e^2 - 6*(a^3*b^3 + a*b^5)*c*d*e*f - (2*a^5*b - 2*a*b
^5 - 3*(a^3*b^3 + a*b^5)*c^2)*f^2 - (3*a*b^5*d^2*e^2 - 6*a*b^5*c*d*e*f + (3*a*b^5*c^2 - 2*a^3*b^3 + 2*a*b^5)*f
^2)*cos(d*x + c)^2 + 2*(3*a^2*b^4*d^2*e^2 - 6*a^2*b^4*c*d*e*f + (3*a^2*b^4*c^2 - 2*a^4*b^2 + 2*a^2*b^4)*f^2)*s
in(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) +
2*I*a) - 2*(2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*d*e*f - 2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^
4*b^2 + a^2*b^4 - 2*b^6)*d*e*f - (a^4*b^2 + a^2*b^4 - 2*b^6)*c*f^2)*cos(d*x + c)^2 + 4*((a^5*b + a^3*b^3 - 2*a
*b^5)*d*e*f - (a^5*b + a^3*b^3 - 2*a*b^5)*c*f^2)*sin(d*x + c) - (3*(a^3*b^3 + a*b^5)*d^2*e^2 - 6*(a^3*b^3 + a*
b^5)*c*d*e*f - (2*a^5*b - 2*a*b^5 - 3*(a^3*b^3 + a*b^5)*c^2)*f^2 - (3*a*b^5*d^2*e^2 - 6*a*b^5*c*d*e*f + (3*a*b
^5*c^2 - 2*a^3*b^3 + 2*a*b^5)*f^2)*cos(d*x + c)^2 + 2*(3*a^2*b^4*d^2*e^2 - 6*a^2*b^4*c*d*e*f + (3*a^2*b^4*c^2
- 2*a^4*b^2 + 2*a^2*b^4)*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c)
+ 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) - 2*(2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*d*e*f - 2*(a^6 + 2*a^4*b^2 -
a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^4*b^2 + a^2*b^4 - 2*b^6)*d*e*f - (a^4*b^2 + a^2*b^4 - 2*b^6)*c*f^2)*cos(d*x + c
)^2 + 4*((a^5*b + a^3*b^3 - 2*a*b^5)*d*e*f - (a^5*b + a^3*b^3 - 2*a*b^5)*c*f^2)*sin(d*x + c) + (3*(a^3*b^3 + a
*b^5)*d^2*e^2 - 6*(a^3*b^3 + a*b^5)*c*d*e*f - (2*a^5*b - 2*a*b^5 - 3*(a^3*b^3 + a*b^5)*c^2)*f^2 - (3*a*b^5*d^2
*e^2 - 6*a*b^5*c*d*e*f + (3*a*b^5*c^2 - 2*a^3*b^3 + 2*a*b^5)*f^2)*cos(d*x + c)^2 + 2*(3*a^2*b^4*d^2*e^2 - 6*a^
2*b^4*c*d*e*f + (3*a^2*b^4*c^2 - 2*a^4*b^2 + 2*a^2*b^4)*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(-2*b*co
s(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - 2*(2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^
6)*d*e*f - 2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^4*b^2 + a^2*b^4 - 2*b^6)*d*e*f - (a^4*b^2 + a^2
*b^4 - 2*b^6)*c*f^2)*cos(d*x + c)^2 + 4*((a^5*b + a^3*b^3 - 2*a*b^5)*d*e*f - (a^5*b + a^3*b^3 - 2*a*b^5)*c*f^2
)*sin(d*x + c) + (3*(a^3*b^3 + a*b^5)*d^2*e^2 - 6*(a^3*b^3 + a*b^5)*c*d*e*f - (2*a^5*b - 2*a*b^5 - 3*(a^3*b^3
+ a*b^5)*c^2)*f^2 - (3*a*b^5*d^2*e^2 - 6*a*b^5*c*d*e*f + (3*a*b^5*c^2 - 2*a^3*b^3 + 2*a*b^5)*f^2)*cos(d*x + c)
^2 + 2*(3*a^2*b^4*d^2*e^2 - 6*a^2*b^4*c*d*e*f + (3*a^2*b^4*c^2 - 2*a^4*b^2 + 2*a^2*b^4)*f^2)*sin(d*x + c))*sqr
t(-(a^2 - b^2)/b^2))*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) - 2*(2*(
a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*d*f^2*x + 2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^4*b^2 + a^2*b
^4 - 2*b^6)*d*f^2*x + (a^4*b^2 + a^2*b^4 - 2*b^6)*c*f^2)*cos(d*x + c)^2 + 4*((a^5*b + a^3*b^3 - 2*a*b^5)*d*f^2
*x + (a^5*b + a^3*b^3 - 2*a*b^5)*c*f^2)*sin(d*x + c) - 3*((a^3*b^3 + a*b^5)*d^2*f^2*x^2 + 2*(a^3*b^3 + a*b^5)*
d^2*e*f*x + 2*(a^3*b^3 + a*b^5)*c*d*e*f - (a^3*b^3 + a*b^5)*c^2*f^2 - (a*b^5*d^2*f^2*x^2 + 2*a*b^5*d^2*e*f*x +
2*a*b^5*c*d*e*f - a*b^5*c^2*f^2)*cos(d*x + c)^2 + 2*(a^2*b^4*d^2*f^2*x^2 + 2*a^2*b^4*d^2*e*f*x + 2*a^2*b^4*c*
d*e*f - a^2*b^4*c^2*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c)
+ 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) - 2*(2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2
*b^6)*d*f^2*x + 2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^4*b^2 + a^2*b^4 - 2*b^6)*d*f^2*x + (a^4*b^
2 + a^2*b^4 - 2*b^6)*c*f^2)*cos(d*x + c)^2 + 4*((a^5*b + a^3*b^3 - 2*a*b^5)*d*f^2*x + (a^5*b + a^3*b^3 - 2*a*b
^5)*c*f^2)*sin(d*x + c) + 3*((a^3*b^3 + a*b^5)*d^2*f^2*x^2 + 2*(a^3*b^3 + a*b^5)*d^2*e*f*x + 2*(a^3*b^3 + a*b^
5)*c*d*e*f - (a^3*b^3 + a*b^5)*c^2*f^2 - (a*b^5*d^2*f^2*x^2 + 2*a*b^5*d^2*e*f*x + 2*a*b^5*c*d*e*f - a*b^5*c^2*
f^2)*cos(d*x + c)^2 + 2*(a^2*b^4*d^2*f^2*x^2 + 2*a^2*b^4*d^2*e*f*x + 2*a^2*b^4*c*d*e*f - a^2*b^4*c^2*f^2)*sin(
d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin
(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) - 2*(2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*d*f^2*x + 2*(a^6 + 2*a^
4*b^2 - a^2*b^4 - 2*b^6)*c*f^2 - 2*((a^4*b^2 + a^2*b^4 - 2*b^6)*d*f^2*x + (a^4*b^2 + a^2*b^4 - 2*b^6)*c*f^2)*c
os(d*x + c)^2 + 4*((a^5*b + a^3*b^3 - 2*a*b^5)*d*f^2*x + (a^5*b + a^3*b^3 - 2*a*b^5)*c*f^2)*sin(d*x + c) - 3*(
(a^3*b^3 + a*b^5)*d^2*f^2*x^2 + 2*(a^3*b^3 + a*b^5)*d^2*e*f*x + 2*(a^3*b^3 + a*b^5)*c*d*e*f - (a^3*b^3 + a*b^5
)*c^2*f^2 - (a*b^5*d^2*f^2*x^2 + 2*a*b^5*d^2*e*f*x + 2*a*b^5*c*d*e*f - a*b^5*c^2*f^2)*cos(d*x + c)^2 + 2*(a^2*
b^4*d^2*f^2*x^2 + 2*a^2*b^4*d^2*e*f*x + 2*a^2*b^4*c*d*e*f - a^2*b^4*c^2*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b
^2))*log(1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)
/b^2) + 2*b)/b) - 2*(2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*d*f^2*x + 2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*c*f
^2 - 2*((a^4*b^2 + a^2*b^4 - 2*b^6)*d*f^2*x + (a^4*b^2 + a^2*b^4 - 2*b^6)*c*f^2)*cos(d*x + c)^2 + 4*((a^5*b +
a^3*b^3 - 2*a*b^5)*d*f^2*x + (a^5*b + a^3*b^3 - 2*a*b^5)*c*f^2)*sin(d*x + c) + 3*((a^3*b^3 + a*b^5)*d^2*f^2*x^
2 + 2*(a^3*b^3 + a*b^5)*d^2*e*f*x + 2*(a^3*b^3 + a*b^5)*c*d*e*f - (a^3*b^3 + a*b^5)*c^2*f^2 - (a*b^5*d^2*f^2*x
^2 + 2*a*b^5*d^2*e*f*x + 2*a*b^5*c*d*e*f - a*b^5*c^2*f^2)*cos(d*x + c)^2 + 2*(a^2*b^4*d^2*f^2*x^2 + 2*a^2*b^4*
d^2*e*f*x + 2*a^2*b^4*c*d*e*f - a^2*b^4*c^2*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(-2*I*a*cos(d*x
+ c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 4*(2*(a^5*
b - 2*a^3*b^3 + a*b^5)*d*f^2*x + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*e*f + ((a^4*b^2 + a^2*b^4 - 2*b^6)*d^2*f^2*x^
2 + 2*(a^4*b^2 + a^2*b^4 - 2*b^6)*d^2*e*f*x + (a^4*b^2 + a^2*b^4 - 2*b^6)*d^2*e^2)*cos(d*x + c))*sin(d*x + c))
/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^3*cos(d*x + c)^2 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*d^3
*sin(d*x + c) - (a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9)*d^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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