3.3.28 \(\int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) [228]

Optimal. Leaf size=802 \[ -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 i a^3 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 a^3 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a^3 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2} \]

[Out]

-3/4*e*f^2*x/b/d^2-3/8*f^3*x^2/b/d^2+1/4*a^2*(f*x+e)^4/b^3/f+1/8*(f*x+e)^4/b/f-6*a*f^2*(f*x+e)*cos(d*x+c)/b^2/
d^3+a*(f*x+e)^3*cos(d*x+c)/b^2/d+6*a*f^3*sin(d*x+c)/b^2/d^4-3*a*f*(f*x+e)^2*sin(d*x+c)/b^2/d^2+3/4*f^2*(f*x+e)
*cos(d*x+c)*sin(d*x+c)/b/d^3-1/2*(f*x+e)^3*cos(d*x+c)*sin(d*x+c)/b/d-3/8*f^3*sin(d*x+c)^2/b/d^4+3/4*f*(f*x+e)^
2*sin(d*x+c)^2/b/d^2+6*I*a^3*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^3/d^3/(a^2-b^2)^(
1/2)-6*I*a^3*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3/d^3/(a^2-b^2)^(1/2)+3*a^3*f*(f*
x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^3/d^2/(a^2-b^2)^(1/2)-3*a^3*f*(f*x+e)^2*polylog(2,I
*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3/d^2/(a^2-b^2)^(1/2)+I*a^3*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2
-b^2)^(1/2)))/b^3/d/(a^2-b^2)^(1/2)-I*a^3*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3/d/(a^2-b^
2)^(1/2)-6*a^3*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^3/d^4/(a^2-b^2)^(1/2)+6*a^3*f^3*polylog
(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3/d^4/(a^2-b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 802, normalized size of antiderivative = 1.00, 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 = {4611, 3392, 32, 3391, 3377, 2717, 3404, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \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} \end {gather*}

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 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 2221

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

Rule 2296

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 2320

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

Rule 2611

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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 3391

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 3392

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 3404

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 4611

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 6724

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

Rule 6744

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

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 ) \text {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 ) \text {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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1851\) vs. \(2(802)=1604\).
time = 3.08, size = 1851, normalized size = 2.31 \begin {gather*} \frac {16 \left (2 a^2+b^2\right ) e^3 x+24 \left (2 a^2+b^2\right ) e^2 f x^2+16 \left (2 a^2+b^2\right ) e f^2 x^3+4 \left (2 a^2+b^2\right ) f^3 x^4-\frac {32 i a^3 \left (3 i \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1+\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))+3 i \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1+\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))+i \sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1+\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))+3 \sqrt {a^2-b^2} d^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))-3 \sqrt {a^2-b^2} d^2 f (e+f x)^2 \text {Li}_2\left (\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (\cos (c)+i \sin (c))+6 i \sqrt {a^2-b^2} d e f^2 \text {Li}_3\left (-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))+6 i \sqrt {a^2-b^2} d f^3 x \text {Li}_3\left (-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))-6 \sqrt {a^2-b^2} f^3 \text {Li}_4\left (-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))+6 \sqrt {a^2-b^2} f^3 \text {Li}_4\left (\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (\cos (c)+i \sin (c))+3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (-i \cos (c)+\sin (c))+3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (-i \cos (c)+\sin (c))+\sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (-i \cos (c)+\sin (c))+6 \sqrt {a^2-b^2} d e f^2 \text {Li}_3\left (\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (-i \cos (c)+\sin (c))+6 \sqrt {a^2-b^2} d f^3 x \text {Li}_3\left (\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (-i \cos (c)+\sin (c))-2 i d^3 e^3 \tan ^{-1}\left (\frac {b \cos (c+d x)+i (a+b \sin (c+d x))}{\sqrt {a^2-b^2}}\right ) \sqrt {\left (-a^2+b^2\right ) (\cos (2 c)+i \sin (2 c))}\right )}{\sqrt {a^2-b^2} d^4 \sqrt {\left (-a^2+b^2\right ) (\cos (2 c)+i \sin (2 c))}}+\frac {16 a b \left (6 i f^3-6 d f^2 (e+f x)-3 i d^2 f (e+f x)^2+d^3 (e+f x)^3\right ) (\cos (c+d x)-i \sin (c+d x))}{d^4}+\frac {16 a b \left (-6 i f^3-6 d f^2 (e+f x)+3 i d^2 f (e+f x)^2+d^3 (e+f x)^3\right ) (\cos (c+d x)+i \sin (c+d x))}{d^4}+\frac {b^2 \left (3 f^3+6 i d f^2 (e+f x)-6 d^2 f (e+f x)^2-4 i d^3 (e+f x)^3\right ) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{d^4}+\frac {b^2 \left (3 f^3-6 i d f^2 (e+f x)-6 d^2 f (e+f x)^2+4 i d^3 (e+f x)^3\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{d^4}}{32 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(2*a^2 + b^2)*e^3*x + 24*(2*a^2 + b^2)*e^2*f*x^2 + 16*(2*a^2 + b^2)*e*f^2*x^3 + 4*(2*a^2 + b^2)*f^3*x^4 -
((32*I)*a^3*((3*I)*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + S
qrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + (3*I)*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2
*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Si
n[c])]*(Cos[c] + I*Sin[c]) + I*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*
a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + 3*Sqrt[a^2 - b^2]*d^2*f
*(e + f*x)^2*PolyLog[2, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*
Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) - 3*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*(Cos[2*c + d*
x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*(Cos[c] + I*Sin
[c]) + (6*I)*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(
-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) + (6*I)*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3
, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])
)]*(Cos[c] + I*Sin[c]) - 6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c
] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) + 6*Sqrt[a^2 - b^2]*f^3*PolyLog
[4, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[
c])]*(Cos[c] + I*Sin[c]) + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)
*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + 3*Sqrt[a^2 - b^2]*d
^3*e*f^2*x^2*Log[1 - (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Si
n[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*(Cos[2*c + d*x] + I*Sin[
2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + 6
*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)
*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*(Cos[2*c
+ d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[
c] + Sin[c]) - (2*I)*d^3*e^3*ArcTan[(b*Cos[c + d*x] + I*(a + b*Sin[c + d*x]))/Sqrt[a^2 - b^2]]*Sqrt[(-a^2 + b^
2)*(Cos[2*c] + I*Sin[2*c])]))/(Sqrt[a^2 - b^2]*d^4*Sqrt[(-a^2 + b^2)*(Cos[2*c] + I*Sin[2*c])]) + (16*a*b*((6*I
)*f^3 - 6*d*f^2*(e + f*x) - (3*I)*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(Cos[c + d*x] - I*Sin[c + d*x]))/d^4 +
(16*a*b*((-6*I)*f^3 - 6*d*f^2*(e + f*x) + (3*I)*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(Cos[c + d*x] + I*Sin[c +
d*x]))/d^4 + (b^2*(3*f^3 + (6*I)*d*f^2*(e + f*x) - 6*d^2*f*(e + f*x)^2 - (4*I)*d^3*(e + f*x)^3)*(Cos[2*(c + d
*x)] - I*Sin[2*(c + d*x)]))/d^4 + (b^2*(3*f^3 - (6*I)*d*f^2*(e + f*x) - 6*d^2*f*(e + f*x)^2 + (4*I)*d^3*(e + f
*x)^3)*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))/d^4)/(32*b^3)

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3002 vs. \(2 (727) = 1454\).
time = 0.68, size = 3002, normalized size = 3.74 \begin {gather*} \text {Too large to display} \end {gather*}

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 + 6*(2*a^4 - a^2*b^2 - b^4)*d^4*f*x^2*e^2 + 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) - 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*(a^2*b^2 - b^4)*d^2*f^3*x^2 + 4*(2*a^4 - a^2*b^2 - b^4)*d^4*x*e^3 - 3*(2*
(a^2*b^2 - b^4)*d^2*f^3*x^2 + 4*(a^2*b^2 - b^4)*d^2*f^2*x*e + 2*(a^2*b^2 - b^4)*d^2*f*e^2 - (a^2*b^2 - b^4)*f^
3)*cos(d*x + c)^2 + 12*(-I*a^3*b*d^2*f^3*x^2 - 2*I*a^3*b*d^2*f^2*x*e - I*a^3*b*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^
2)*dilog((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)/
b + 1) + 12*(I*a^3*b*d^2*f^3*x^2 + 2*I*a^3*b*d^2*f^2*x*e + I*a^3*b*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((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)/b + 1) + 12*
(I*a^3*b*d^2*f^3*x^2 + 2*I*a^3*b*d^2*f^2*x*e + I*a^3*b*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((-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)/b + 1) + 12*(-I*a^3*b*d
^2*f^3*x^2 - 2*I*a^3*b*d^2*f^2*x*e - I*a^3*b*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x + c) - a*si
n(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) + 4*(a^3*b*c^3*f^3 - 3*a^3
*b*c^2*d*f^2*e + 3*a^3*b*c*d^2*f*e^2 - a^3*b*d^3*e^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*c^3*f^3 - 3*a^3*b*c^2*d*f^2*e + 3*a^3*b*c*d^2*f*e^2
- a^3*b*d^3*e^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*c^3*f^3 - 3*a^3*b*c^2*d*f^2*e + 3*a^3*b*c*d^2*f*e^2 - a^3*b*d^3*e^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*c^3*f^3 - 3*a^3
*b*c^2*d*f^2*e + 3*a^3*b*c*d^2*f*e^2 - a^3*b*d^3*e^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 + a^3*b*c^3*f^3 + 3*(a^3*b*d^3*f*x + a^
3*b*c*d^2*f)*e^2 + 3*(a^3*b*d^3*f^2*x^2 - a^3*b*c^2*d*f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(-(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)/b) - 4*(a^3*b*d^3*f^3*x^3 + a
^3*b*c^3*f^3 + 3*(a^3*b*d^3*f*x + a^3*b*c*d^2*f)*e^2 + 3*(a^3*b*d^3*f^2*x^2 - a^3*b*c^2*d*f^2)*e)*sqrt(-(a^2 -
b^2)/b^2)*log(-(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)/b) + 4*(a^3*b*d^3*f^3*x^3 + a^3*b*c^3*f^3 + 3*(a^3*b*d^3*f*x + a^3*b*c*d^2*f)*e^2 + 3*(a^3*b*d^3*f^2*x^
2 - a^3*b*c^2*d*f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(-(-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)/b) - 4*(a^3*b*d^3*f^3*x^3 + a^3*b*c^3*f^3 + 3*(a^3*b*d^3*f*x + a^3
*b*c*d^2*f)*e^2 + 3*(a^3*b*d^3*f^2*x^2 - a^3*b*c^2*d*f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(-(-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)/b) - 24*(a^3*b*d*f^3*x + a^3*
b*d*f^2*e)*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*f^2*e)*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*f^2*e)*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*f^2*e)*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) + 8*((a^3*b - a*b^3)*d^3*f^3*x^3 + 3*(a^3*b - a*b^3)*d^3*f*x*e^2 - 6*(a^3*b - a*b^3)*d*f^3*x + (a^3*b - a*
b^3)*d^3*e^3 + 3*((a^3*b - a*b^3)*d^3*f^2*x^2 - 2*(a^3*b - a*b^3)*d*f^2)*e)*cos(d*x + c) + 2*(2*(2*a^4 - a^2*b
^2 - b^4)*d^4*f^2*x^3 + 3*(a^2*b^2 - b^4)*d^2*f^2*x)*e - 2*(12*(a^3*b - a*b^3)*d^2*f^3*x^2 + 24*(a^3*b - a*b^3
)*d^2*f^2*x*e + 12*(a^3*b - a*b^3)*d^2*f*e^2 - 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*f*x*e^2 - 3*(a^2*b^2 - b^4)*d*f^3*x + 2*(a^2*b^2 - b^4)*d^3*e^3 + 3*(2*(a^2*b^2 - b^4)*d^3*f^
2*x^2 - (a^2*b^2 - b^4)*d*f^2)*e)*cos(d*x + c))*sin(d*x + c))/((a^2*b^3 - b^5)*d^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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