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

Optimal. Leaf size=802 \[ \frac{(e+f x)^4}{8 b f}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{a \cos (c+d x) (e+f x)^3}{b^2 d}+\frac{i a^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)^3}{b^3 \sqrt{a^2-b^2} d}-\frac{i a^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)^3}{b^3 \sqrt{a^2-b^2} d}-\frac{\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}+\frac{3 f \sin ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)^2}{b^3 \sqrt{a^2-b^2} d^2}-\frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)^2}{b^3 \sqrt{a^2-b^2} d^2}-\frac{3 a f \sin (c+d x) (e+f x)^2}{b^2 d^2}-\frac{6 a f^2 \cos (c+d x) (e+f x)}{b^2 d^3}+\frac{6 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)}{b^3 \sqrt{a^2-b^2} d^3}-\frac{6 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)}{b^3 \sqrt{a^2-b^2} d^3}+\frac{3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}-\frac{3 f^3 x^2}{8 b d^2}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}-\frac{3 e f^2 x}{4 b d^2}-\frac{6 a^3 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a^3 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a f^3 \sin (c+d x)}{b^2 d^4} \]

[Out]

(-3*e*f^2*x)/(4*b*d^2) - (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) + (e + f*x)^4/(8*b*f) - (6*a*f^2*
(e + f*x)*Cos[c + d*x])/(b^2*d^3) + (a*(e + f*x)^3*Cos[c + d*x])/(b^2*d) + (I*a^3*(e + f*x)^3*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)^3*Log[1 - (I*b*E^(I*(c + d*x)
))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) + (3*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a
- Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sq
rt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) + ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sq
rt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sq
rt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - (6*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])
])/(b^3*Sqrt[a^2 - b^2]*d^4) + (6*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a
^2 - b^2]*d^4) + (6*a*f^3*Sin[c + d*x])/(b^2*d^4) - (3*a*f*(e + f*x)^2*Sin[c + d*x])/(b^2*d^2) + (3*f^2*(e + f
*x)*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*b*d) - (3*f^3*Sin[c + d*
x]^2)/(8*b*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x]^2)/(4*b*d^2)

________________________________________________________________________________________

Rubi [A]  time = 1.3412, antiderivative size = 802, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {4515, 3311, 32, 3310, 3296, 2637, 3323, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{(e+f x)^4}{8 b f}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{a \cos (c+d x) (e+f x)^3}{b^2 d}+\frac{i a^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)^3}{b^3 \sqrt{a^2-b^2} d}-\frac{i a^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)^3}{b^3 \sqrt{a^2-b^2} d}-\frac{\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}+\frac{3 f \sin ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)^2}{b^3 \sqrt{a^2-b^2} d^2}-\frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)^2}{b^3 \sqrt{a^2-b^2} d^2}-\frac{3 a f \sin (c+d x) (e+f x)^2}{b^2 d^2}-\frac{6 a f^2 \cos (c+d x) (e+f x)}{b^2 d^3}+\frac{6 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)}{b^3 \sqrt{a^2-b^2} d^3}-\frac{6 i a^3 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)}{b^3 \sqrt{a^2-b^2} d^3}+\frac{3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}-\frac{3 f^3 x^2}{8 b d^2}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}-\frac{3 e f^2 x}{4 b d^2}-\frac{6 a^3 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a^3 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a f^3 \sin (c+d x)}{b^2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*e*f^2*x)/(4*b*d^2) - (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) + (e + f*x)^4/(8*b*f) - (6*a*f^2*
(e + f*x)*Cos[c + d*x])/(b^2*d^3) + (a*(e + f*x)^3*Cos[c + d*x])/(b^2*d) + (I*a^3*(e + f*x)^3*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)^3*Log[1 - (I*b*E^(I*(c + d*x)
))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) + (3*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a
- Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sq
rt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) + ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sq
rt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sq
rt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - (6*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])
])/(b^3*Sqrt[a^2 - b^2]*d^4) + (6*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a
^2 - b^2]*d^4) + (6*a*f^3*Sin[c + d*x])/(b^2*d^4) - (3*a*f*(e + f*x)^2*Sin[c + d*x])/(b^2*d^2) + (3*f^2*(e + f
*x)*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*b*d) - (3*f^3*Sin[c + d*
x]^2)/(8*b*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x]^2)/(4*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 3310

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

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 6609

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

Rule 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)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \sin ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{b}\\ &=-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac{a \int (e+f x)^3 \sin (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac{\int (e+f x)^3 \, dx}{2 b}-\frac{\left (3 f^2\right ) \int (e+f x) \sin ^2(c+d x) \, dx}{2 b d^2}\\ &=\frac{(e+f x)^4}{8 b f}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}+\frac{a^2 \int (e+f x)^3 \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^3}{a+b \sin (c+d x)} \, dx}{b^3}-\frac{(3 a f) \int (e+f x)^2 \cos (c+d x) \, dx}{b^2 d}-\frac{\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2}\\ &=-\frac{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac{\left (2 a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b^3}+\frac{\left (6 a f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{b^2 d^2}\\ &=-\frac{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}-\frac{6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}+\frac{\left (2 i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{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)^3}{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 (6 a f^3\right ) \int \cos (c+d x) \, dx}{b^2 d^3}\\ &=-\frac{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}-\frac{6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac{i a^3 (e+f x)^3 \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)^3 \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{6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac{\left (3 i a^3 f\right ) \int (e+f x)^2 \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 (3 i a^3 f\right ) \int (e+f x)^2 \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{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}-\frac{6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac{i a^3 (e+f x)^3 \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)^3 \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{3 a^3 f (e+f x)^2 \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{3 a^3 f (e+f x)^2 \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{6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac{\left (6 a^3 f^2\right ) \int (e+f x) \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 (6 a^3 f^2\right ) \int (e+f x) \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{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}-\frac{6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac{i a^3 (e+f x)^3 \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)^3 \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{3 a^3 f (e+f x)^2 \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{3 a^3 f (e+f x)^2 \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{6 i a^3 f^2 (e+f x) \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{6 i a^3 f^2 (e+f x) \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{6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac{\left (6 i a^3 f^3\right ) \int \text{Li}_3\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^3}+\frac{\left (6 i a^3 f^3\right ) \int \text{Li}_3\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^3}\\ &=-\frac{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}-\frac{6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac{i a^3 (e+f x)^3 \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)^3 \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{3 a^3 f (e+f x)^2 \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{3 a^3 f (e+f x)^2 \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{6 i a^3 f^2 (e+f x) \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{6 i a^3 f^2 (e+f x) \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{6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac{\left (6 a^3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^4}+\frac{\left (6 a^3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^4}\\ &=-\frac{3 e f^2 x}{4 b d^2}-\frac{3 f^3 x^2}{8 b d^2}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{(e+f x)^4}{8 b f}-\frac{6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac{a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac{i a^3 (e+f x)^3 \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)^3 \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{3 a^3 f (e+f x)^2 \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{3 a^3 f (e+f x)^2 \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{6 i a^3 f^2 (e+f x) \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{6 i a^3 f^2 (e+f x) \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{6 a^3 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a^3 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac{3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}\\ \end{align*}

Mathematica [B]  time = 5.50605, size = 1923, normalized size = 2.4 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(16*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e^3*x + 8*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*e^3*x + 24*a^2*Sqrt[-(a^2 - b^2)^2]*d
^4*e^2*f*x^2 + 12*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*e^2*f*x^2 + 16*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e*f^2*x^3 + 8*b^2*
Sqrt[-(-a^2 + b^2)^2]*d^4*e*f^2*x^3 + 4*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*f^3*x^4 + 2*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4
*f^3*x^4 - 32*a^3*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 16*a*b*Sqrt[-(a
^2 - b^2)^2]*d^3*e^3*Cos[c + d*x] - 96*a*b*Sqrt[-(a^2 - b^2)^2]*d*e*f^2*Cos[c + d*x] + 48*a*b*Sqrt[-(a^2 - b^2
)^2]*d^3*e^2*f*x*Cos[c + d*x] - 96*a*b*Sqrt[-(a^2 - b^2)^2]*d*f^3*x*Cos[c + d*x] + 48*a*b*Sqrt[-(a^2 - b^2)^2]
*d^3*e*f^2*x^2*Cos[c + d*x] + 16*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*f^3*x^3*Cos[c + d*x] - 6*b^2*Sqrt[-(a^2 - b^2)^2
]*d^2*e^2*f*Cos[2*(c + d*x)] + 3*b^2*Sqrt[-(a^2 - b^2)^2]*f^3*Cos[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d
^2*e*f^2*x*Cos[2*(c + d*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*f^3*x^2*Cos[2*(c + d*x)] - 48*a^3*Sqrt[a^2 - b^2]
*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 48*a^3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*L
og[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 16*a^3*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(
c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a
+ Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2
])] + 16*a^3*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + (48*I)*a^3*Sq
rt[a^2 - b^2]*d^2*f*(e + f*x)^2*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^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 96*a^3*Sqrt[a^2 - b
^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 96*a^3*Sqrt[a^2 - b^2]*d*f^3*x*PolyL
og[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 96*a^3*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(
c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 96*a^3*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a +
Sqrt[-a^2 + b^2]))] - (96*I)*a^3*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2
])] + (96*I)*a^3*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 48*a*b*Sqrt
[-(a^2 - b^2)^2]*d^2*e^2*f*Sin[c + d*x] + 96*a*b*Sqrt[-(a^2 - b^2)^2]*f^3*Sin[c + d*x] - 96*a*b*Sqrt[-(a^2 - b
^2)^2]*d^2*e*f^2*x*Sin[c + d*x] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*f^3*x^2*Sin[c + d*x] - 4*b^2*Sqrt[-(a^2 - b^
2)^2]*d^3*e^3*Sin[2*(c + d*x)] + 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*e*f^2*Sin[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2
)^2]*d^3*e^2*f*x*Sin[2*(c + d*x)] + 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*f^3*x*Sin[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 -
b^2)^2]*d^3*e*f^2*x^2*Sin[2*(c + d*x)] - 4*b^2*Sqrt[-(a^2 - b^2)^2]*d^3*f^3*x^3*Sin[2*(c + d*x)])/(16*b^3*Sqrt
[-(a^2 - b^2)^2]*d^4)

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Maple [F]  time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \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)^3*sin(d*x+c)^3/(a+b*sin(d*x+c)),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 5.39924, size = 6724, normalized size = 8.38 \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)^3*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8*((2*a^4 - a^2*b^2 - b^4)*d^4*f^3*x^4 + 4*(2*a^4 - a^2*b^2 - b^4)*d^4*e*f^2*x^3 + 24*I*a^3*b*f^3*sqrt(-(a^2
- b^2)/b^2)*polylog(4, 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))*sqr
t(-(a^2 - b^2)/b^2))/b) - 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, 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) + 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/
b^2)*polylog(4, -(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) - 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(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) + 3*(2*(2*a^4 - a^2*b^2 - b^4)*d^4*e^2*f + (a^2*b^2 - b^4)*d^
2*f^3)*x^2 - 3*(2*(a^2*b^2 - b^4)*d^2*f^3*x^2 + 4*(a^2*b^2 - b^4)*d^2*e*f^2*x + 2*(a^2*b^2 - b^4)*d^2*e^2*f -
(a^2*b^2 - b^4)*f^3)*cos(d*x + c)^2 - 2*(6*I*a^3*b*d^2*f^3*x^2 + 12*I*a^3*b*d^2*e*f^2*x + 6*I*a^3*b*d^2*e^2*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^2*f^3*x^2 - 12*I*a^3*b*d^2*e*f^2*x - 6*I*a^3*b*d^2*
e^2*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^2*f^3*x^2 - 12*I*a^3*b*d^2*e*f^2*x - 6*I*a^3*
b*d^2*e^2*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^2*f^3*x^2 + 12*I*a^3*b*d^2*e*f^2*x + 6*
I*a^3*b*d^2*e^2*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) - 4*(a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b
*c^2*d*e*f^2 - a^3*b*c^3*f^3)*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) - 4*(a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*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) + 4*(a^3*b*d^3
*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*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) + 4*(a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b
*c^2*d*e*f^2 - a^3*b*c^3*f^3)*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) - 4*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e
^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*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*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*
d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*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*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f
- 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*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*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3
*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*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) + 24*(a^3*b*d*f^3*x + a^3*b*d*e*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) - 24*(a^3*b*d*f^3*x +
a^3*b*d*e*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) - 24*(a^3*b*d*f^3*x + a^3*b*d*e*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
) + 24*(a^3*b*d*f^3*x + a^3*b*d*e*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) + 2*(2*(2*a^4 - a^2*b^2 - b^4)*d^4*e^3 + 3*(a^
2*b^2 - b^4)*d^2*e*f^2)*x + 8*((a^3*b - a*b^3)*d^3*f^3*x^3 + 3*(a^3*b - a*b^3)*d^3*e*f^2*x^2 + (a^3*b - a*b^3)
*d^3*e^3 - 6*(a^3*b - a*b^3)*d*e*f^2 + 3*((a^3*b - a*b^3)*d^3*e^2*f - 2*(a^3*b - a*b^3)*d*f^3)*x)*cos(d*x + c)
- 2*(12*(a^3*b - a*b^3)*d^2*f^3*x^2 + 24*(a^3*b - a*b^3)*d^2*e*f^2*x + 12*(a^3*b - a*b^3)*d^2*e^2*f - 24*(a^3
*b - a*b^3)*f^3 + (2*(a^2*b^2 - b^4)*d^3*f^3*x^3 + 6*(a^2*b^2 - b^4)*d^3*e*f^2*x^2 + 2*(a^2*b^2 - b^4)*d^3*e^3
- 3*(a^2*b^2 - b^4)*d*e*f^2 + 3*(2*(a^2*b^2 - b^4)*d^3*e^2*f - (a^2*b^2 - b^4)*d*f^3)*x)*cos(d*x + c))*sin(d*
x + c))/((a^2*b^3 - b^5)*d^4)

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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)**3*sin(d*x+c)**3/(a+b*sin(d*x+c)),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 )}^{3} \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)^3*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

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