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

Optimal. Leaf size=592 \[ \frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^2 \sqrt{a^2-b^2}}-\frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^2 \sqrt{a^2-b^2}}+\frac{2 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^3 \sqrt{a^2-b^2}}-\frac{2 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^3 \sqrt{a^2-b^2}}+\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d \sqrt{a^2-b^2}}-\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d \sqrt{a^2-b^2}}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac{f^2 \sin (c+d x) \cos (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 b d}-\frac{f^2 x}{4 b d^2}+\frac{(e+f x)^3}{6 b f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.18028, antiderivative size = 592, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {4515, 3311, 32, 2635, 8, 3296, 2638, 3323, 2264, 2190, 2531, 2282, 6589} \[ \frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^2 \sqrt{a^2-b^2}}-\frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^2 \sqrt{a^2-b^2}}+\frac{2 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^3 \sqrt{a^2-b^2}}-\frac{2 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^3 \sqrt{a^2-b^2}}+\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d \sqrt{a^2-b^2}}-\frac{i a^3 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d \sqrt{a^2-b^2}}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac{f^2 \sin (c+d x) \cos (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 b d}-\frac{f^2 x}{4 b d^2}+\frac{(e+f x)^3}{6 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4515

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

Rule 3311

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3296

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

Rule 2638

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

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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^2 \sin ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{b}\\ &=-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac{a \int (e+f x)^2 \sin (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac{\int (e+f x)^2 \, dx}{2 b}-\frac{f^2 \int \sin ^2(c+d x) \, dx}{2 b d^2}\\ &=\frac{(e+f x)^3}{6 b f}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac{a^2 \int (e+f x)^2 \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^2}{a+b \sin (c+d x)} \, dx}{b^3}-\frac{(2 a f) \int (e+f x) \cos (c+d x) \, dx}{b^2 d}-\frac{f^2 \int 1 \, dx}{4 b d^2}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{(e+f x)^3}{6 b f}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\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^3}+\frac{\left (2 a f^2\right ) \int \sin (c+d x) \, dx}{b^2 d^2}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^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}{b^2 \sqrt{a^2-b^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}{b^2 \sqrt{a^2-b^2}}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\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^3 \sqrt{a^2-b^2} d}-\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^3 \sqrt{a^2-b^2} d}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^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^3 \sqrt{a^2-b^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^3 \sqrt{a^2-b^2} d}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\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^3 \sqrt{a^2-b^2} d}-\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^3 \sqrt{a^2-b^2} d}+\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^3 \sqrt{a^2-b^2} d^2}-\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^3 \sqrt{a^2-b^2} d^2}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b 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^3 \sqrt{a^2-b^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^3 \sqrt{a^2-b^2} d^2}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\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^3 \sqrt{a^2-b^2} d}-\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^3 \sqrt{a^2-b^2} d}+\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^3 \sqrt{a^2-b^2} d^2}-\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^3 \sqrt{a^2-b^2} d^2}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\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^3 \sqrt{a^2-b^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^3 \sqrt{a^2-b^2} d^3}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\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^3 \sqrt{a^2-b^2} d}-\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^3 \sqrt{a^2-b^2} d}+\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^3 \sqrt{a^2-b^2} d^2}-\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^3 \sqrt{a^2-b^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^3 \sqrt{a^2-b^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^3 \sqrt{a^2-b^2} d^3}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}\\ \end{align*}

Mathematica [A]  time = 4.43864, size = 1166, normalized size = 1.97 \[ \frac{-48 \sqrt{b^2-a^2} d^2 e^2 \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right ) a^3-24 \sqrt{a^2-b^2} d^2 f^2 x^2 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) a^3-48 \sqrt{a^2-b^2} d^2 e f x \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) a^3+24 \sqrt{a^2-b^2} d^2 f^2 x^2 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) a^3+48 \sqrt{a^2-b^2} d^2 e f x \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) a^3+48 i \sqrt{a^2-b^2} d f (e+f x) \text{PolyLog}\left (2,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) a^3-48 i \sqrt{a^2-b^2} d f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) a^3-48 \sqrt{a^2-b^2} f^2 \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) a^3+48 \sqrt{a^2-b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) a^3+8 \sqrt{-\left (a^2-b^2\right )^2} d^3 f^2 x^3 a^2+24 \sqrt{-\left (a^2-b^2\right )^2} d^3 e f x^2 a^2+24 \sqrt{-\left (a^2-b^2\right )^2} d^3 e^2 x a^2+24 b \sqrt{-\left (a^2-b^2\right )^2} d^2 e^2 \cos (c+d x) a-48 b \sqrt{-\left (a^2-b^2\right )^2} f^2 \cos (c+d x) a+24 b \sqrt{-\left (a^2-b^2\right )^2} d^2 f^2 x^2 \cos (c+d x) a+48 b \sqrt{-\left (a^2-b^2\right )^2} d^2 e f x \cos (c+d x) a-48 b \sqrt{-\left (a^2-b^2\right )^2} d e f \sin (c+d x) a-48 b \sqrt{-\left (a^2-b^2\right )^2} d f^2 x \sin (c+d x) a+4 b^2 \sqrt{-\left (b^2-a^2\right )^2} d^3 f^2 x^3+12 b^2 \sqrt{-\left (b^2-a^2\right )^2} d^3 e f x^2+12 b^2 \sqrt{-\left (b^2-a^2\right )^2} d^3 e^2 x-6 b^2 \sqrt{-\left (a^2-b^2\right )^2} d e f \cos (2 (c+d x))-6 b^2 \sqrt{-\left (a^2-b^2\right )^2} d f^2 x \cos (2 (c+d x))-6 b^2 \sqrt{-\left (a^2-b^2\right )^2} d^2 e^2 \sin (2 (c+d x))+3 b^2 \sqrt{-\left (a^2-b^2\right )^2} f^2 \sin (2 (c+d x))-6 b^2 \sqrt{-\left (a^2-b^2\right )^2} d^2 f^2 x^2 \sin (2 (c+d x))-12 b^2 \sqrt{-\left (a^2-b^2\right )^2} d^2 e f x \sin (2 (c+d x))}{24 b^3 \sqrt{-\left (a^2-b^2\right )^2} d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(24*a^2*Sqrt[-(a^2 - b^2)^2]*d^3*e^2*x + 12*b^2*Sqrt[-(-a^2 + b^2)^2]*d^3*e^2*x + 24*a^2*Sqrt[-(a^2 - b^2)^2]*
d^3*e*f*x^2 + 12*b^2*Sqrt[-(-a^2 + b^2)^2]*d^3*e*f*x^2 + 8*a^2*Sqrt[-(a^2 - b^2)^2]*d^3*f^2*x^3 + 4*b^2*Sqrt[-
(-a^2 + b^2)^2]*d^3*f^2*x^3 - 48*a^3*Sqrt[-a^2 + b^2]*d^2*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]
] + 24*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*Cos[c + d*x] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*f^2*Cos[c + d*x] + 48*a*b*S
qrt[-(a^2 - b^2)^2]*d^2*e*f*x*Cos[c + d*x] + 24*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*f^2*x^2*Cos[c + d*x] - 6*b^2*Sqrt
[-(a^2 - b^2)^2]*d*e*f*Cos[2*(c + d*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*f^2*x*Cos[2*(c + d*x)] - 48*a^3*Sqrt[a^
2 - b^2]*d^2*e*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 24*a^3*Sqrt[a^2 - b^2]*d^2*f^2*x
^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^2*e*f*x*Log[1 + (b*E^(I
*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + 24*a^3*Sqrt[a^2 - b^2]*d^2*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a +
Sqrt[-a^2 + b^2])] + (48*I)*a^3*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-
a^2 + b^2])] - (48*I)*a^3*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^
2]))] - 48*a^3*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a
^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d*e*f*
Sin[c + d*x] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d*f^2*x*Sin[c + d*x] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*Sin[2*(c
+ d*x)] + 3*b^2*Sqrt[-(a^2 - b^2)^2]*f^2*Sin[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e*f*x*Sin[2*(c + d
*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*f^2*x^2*Sin[2*(c + d*x)])/(24*b^3*Sqrt[-(a^2 - b^2)^2]*d^3)

________________________________________________________________________________________

Maple [F]  time = 0.388, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 4.03307, size = 4691, normalized size = 7.92 \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)^3/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(2*(2*a^4 - a^2*b^2 - b^4)*d^3*f^2*x^3 + 6*(2*a^4 - a^2*b^2 - b^4)*d^3*e*f*x^2 + 12*a^3*b*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^3*b*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^3*b*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^3*b*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) - 6*((a^2*b^2 - b^4)*d*f^2*x + (a^2*b^2 - b^4)*d*e*f)*cos(d*x + c)^2
- 2*(6*I*a^3*b*d*f^2*x + 6*I*a^3*b*d*e*f)*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*(-6*I*a^3*b*d*f^2*x - 6
*I*a^3*b*d*e*f)*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*(-6*I*a^3*b*d*f^2*x - 6*I*a^3*b*d*e*f)*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*(6*I*a^3*b*d*f^2*x + 6*I*a^3*b*d*e*f)*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) - 6*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x + c) + 2*I*b*s
in(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - 6*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*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) + 6*(a^3*b*d^
2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*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) + 6*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*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) - 6*(a^3*b*d^2*f^2*x^2 + 2*
a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*sqrt(-(a^2 - b^2)/b^2)*log(1/2*(2*I*a*cos(d*x + c) + 2*a*si
n(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) + 6*(a^3*b*d^2*f^2*x^2 + 2
*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*sqrt(-(a^2 - b^2)/b^2)*log(1/2*(2*I*a*cos(d*x + c) + 2*a*s
in(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) - 6*(a^3*b*d^2*f^2*x^2 +
2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*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) + 6*(a^3*b*d^2*f^2*x^2
+ 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*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) + 3*(2*(2*a^4 - a^2*b
^2 - b^4)*d^3*e^2 + (a^2*b^2 - b^4)*d*f^2)*x + 12*((a^3*b - a*b^3)*d^2*f^2*x^2 + 2*(a^3*b - a*b^3)*d^2*e*f*x +
(a^3*b - a*b^3)*d^2*e^2 - 2*(a^3*b - a*b^3)*f^2)*cos(d*x + c) - 3*(8*(a^3*b - a*b^3)*d*f^2*x + 8*(a^3*b - a*b
^3)*d*e*f + (2*(a^2*b^2 - b^4)*d^2*f^2*x^2 + 4*(a^2*b^2 - b^4)*d^2*e*f*x + 2*(a^2*b^2 - b^4)*d^2*e^2 - (a^2*b^
2 - b^4)*f^2)*cos(d*x + c))*sin(d*x + c))/((a^2*b^3 - b^5)*d^3)

________________________________________________________________________________________

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)**3/(a+b*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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 )^{3}}{b \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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