\(\int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [191]

Optimal result

Integrand size = 28, antiderivative size = 382 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2} \]

[Out]

-3/4*e*f^2*x/a/d^2-3/8*f^3*x^2/a/d^2+I*(f*x+e)^3/a/d+3/8*(f*x+e)^4/a/f-6*f^2*(f*x+e)*cos(d*x+c)/a/d^3+(f*x+e)^
3*cos(d*x+c)/a/d+(f*x+e)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d-6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2+12*I*f^2*(
f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+6*f^3*sin(d*x+c)/a/d^4-3*f*(
f*x+e)^2*sin(d*x+c)/a/d^2+3/4*f^2*(f*x+e)*cos(d*x+c)*sin(d*x+c)/a/d^3-1/2*(f*x+e)^3*cos(d*x+c)*sin(d*x+c)/a/d-
3/8*f^3*sin(d*x+c)^2/a/d^4+3/4*f*(f*x+e)^2*sin(d*x+c)^2/a/d^2

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {4611, 3392, 32, 3391, 3377, 2717, 3399, 4269, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {3 f^2 (e+f x) \sin (c+d x) \cos (c+d x)}{4 a d^3}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f} \]

[In]

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

[Out]

(-3*e*f^2*x)/(4*a*d^2) - (3*f^3*x^2)/(8*a*d^2) + (I*(e + f*x)^3)/(a*d) + (3*(e + f*x)^4)/(8*a*f) - (6*f^2*(e +
f*x)*Cos[c + d*x])/(a*d^3) + ((e + f*x)^3*Cos[c + d*x])/(a*d) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d)
- (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])
/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (6*f^3*Sin[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Si
n[c + d*x])/(a*d^2) + (3*f^2*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(4*a*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c
+ d*x])/(2*a*d) - (3*f^3*Sin[c + d*x]^2)/(8*a*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x]^2)/(4*a*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 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 3399

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

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \sin ^2(c+d x) \, dx}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx \\ & = -\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}+\frac {\int (e+f x)^3 \, dx}{2 a}-\frac {\int (e+f x)^3 \sin (c+d x) \, dx}{a}-\frac {\left (3 f^2\right ) \int (e+f x) \sin ^2(c+d x) \, dx}{2 a d^2}+\int \frac {(e+f x)^3 \sin (c+d x)}{a+a \sin (c+d x)} \, dx \\ & = \frac {(e+f x)^4}{8 a f}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}+\frac {\int (e+f x)^3 \, dx}{a}-\frac {(3 f) \int (e+f x)^2 \cos (c+d x) \, dx}{a d}-\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2}-\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {3 (e+f x)^4}{8 a f}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac {\int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a}+\frac {\left (6 f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{a d^2} \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac {(3 f) \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}+\frac {\left (6 f^3\right ) \int \cos (c+d x) \, dx}{a d^3} \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac {(6 f) \int \frac {e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d} \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}+\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2} \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac {\left (12 i f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3} \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac {\left (12 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4} \\ & = -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.61 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.41 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {48 e^3 x+72 e^2 f x^2+48 e f^2 x^3+12 f^3 x^4+\frac {192 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 (\cos (c)-i \sin (c))}{3 f}-\frac {(e+f x)^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {2 f (d (e+f x) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i f \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (1+\sin (c)))}{d^3}\right )}{d (\cos (c)+i (1+\sin (c)))}-\frac {64 (e+f x)^3 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {16 \left (6 i f^3-6 d f^2 (e+f x)-3 i d^2 f (e+f x)^2+d^3 (e+f x)^3\right ) (\cos (c+d x)-i \sin (c+d x))}{d^4}+\frac {16 \left (-6 i f^3-6 d f^2 (e+f x)+3 i d^2 f (e+f x)^2+d^3 (e+f x)^3\right ) (\cos (c+d x)+i \sin (c+d x))}{d^4}+\frac {\left (3 f^3+6 i d f^2 (e+f x)-6 d^2 f (e+f x)^2-4 i d^3 (e+f x)^3\right ) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{d^4}+\frac {\left (3 f^3-6 i d f^2 (e+f x)-6 d^2 f (e+f x)^2+4 i d^3 (e+f x)^3\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{d^4}}{32 a} \]

[In]

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

[Out]

(48*e^3*x + 72*e^2*f*x^2 + 48*e*f^2*x^3 + 12*f^3*x^4 + (192*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Si
n[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f
*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] -
I*(1 + Sin[c])))/d^3))/(d*(Cos[c] + I*(1 + Sin[c]))) - (64*(e + f*x)^3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (16*((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*((-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 + ((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 + ((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*a)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (351 ) = 702\).

Time = 0.57 (sec) , antiderivative size = 1054, normalized size of antiderivative = 2.76

method	result	size
risch	\(\frac {3 f^{2} e \,x^{3}}{2 a}+\frac {9 f \,e^{2} x^{2}}{4 a}+\frac {3 e^{3} x}{2 a}+\frac {6 f \,e^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {3 f \,e^{2} \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{a \,d^{2}}+\frac {2 i f^{3} x^{3}}{a d}-\frac {4 i f^{3} c^{3}}{a \,d^{4}}+\frac {12 i f^{3} \operatorname {Li}_{2}\left (i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{d^{3} a}+\frac {6 i f^{2} e \,x^{2}}{a d}+\frac {6 i f^{2} e \,c^{2}}{a \,d^{3}}+\frac {12 i f^{2} e \,\operatorname {Li}_{2}\left (i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {6 i f^{3} c^{2} x}{a \,d^{3}}-\frac {12 f^{2} c e \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}+\frac {i \left (4 d^{3} x^{3} f^{3}+12 e \,f^{2} x^{2} d^{3}+6 i d^{2} f^{3} x^{2}+12 e^{2} f x \,d^{3}+12 i d^{2} e \,f^{2} x +4 d^{3} e^{3}+6 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}-3 i f^{3}\right ) {\mathrm e}^{2 i \left (d x +c \right )}}{32 a \,d^{4}}-\frac {i \left (4 d^{3} x^{3} f^{3}+12 e \,f^{2} x^{2} d^{3}-6 i d^{2} f^{3} x^{2}+12 e^{2} f x \,d^{3}-12 i d^{2} e \,f^{2} x +4 d^{3} e^{3}-6 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}+3 i f^{3}\right ) {\mathrm e}^{-2 i \left (d x +c \right )}}{32 a \,d^{4}}+\frac {6 i f^{3} c^{2} \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}+\frac {6 i f \,e^{2} \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {3 f^{3} x^{4}}{8 a}+\frac {\left (d^{3} x^{3} f^{3}+3 e \,f^{2} x^{2} d^{3}-3 i d^{2} f^{3} x^{2}+3 e^{2} f x \,d^{3}-6 i d^{2} e \,f^{2} x +d^{3} e^{3}-3 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}+6 i f^{3}\right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 a \,d^{4}}+\frac {\left (d^{3} x^{3} f^{3}+3 e \,f^{2} x^{2} d^{3}+3 i d^{2} f^{3} x^{2}+3 e^{2} f x \,d^{3}+6 i d^{2} e \,f^{2} x +d^{3} e^{3}+3 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}-6 i f^{3}\right ) {\mathrm e}^{i \left (d x +c \right )}}{2 a \,d^{4}}-\frac {12 i f^{2} e c \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {12 i f^{2} e c x}{a \,d^{2}}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}+\frac {3 e^{4}}{8 a f}+\frac {6 f^{2} e c \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 f^{3} x^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {6 f^{3} c^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {3 f^{3} c^{2} \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {12 f^{3} \operatorname {Li}_{3}\left (i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}\)	\(1054\)

[In]

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

[Out]

-3/a/d^2*f*e^2*ln(1+exp(2*I*(d*x+c)))-6/a/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2+6/a/d^2*f*e^2*ln(exp(I*(d*x+c)))+
6/a/d^4*f^3*c^2*ln(1-I*exp(I*(d*x+c)))+6/a/d^4*f^3*c^2*ln(exp(I*(d*x+c)))-3/a/d^4*f^3*c^2*ln(1+exp(2*I*(d*x+c)
))+2*I/a/d*f^3*x^3-4*I/a/d^4*f^3*c^3+3/2/a*f^2*e*x^3+9/4/a*f*e^2*x^2+3/2/a*e^3*x+1/32*I*(4*d^3*x^3*f^3+6*I*d^2
*f^3*x^2+12*e*f^2*x^2*d^3+12*I*d^2*e*f^2*x+12*e^2*f*x*d^3+6*I*d^2*e^2*f+4*d^3*e^3-6*d*f^3*x-3*I*f^3-6*d*e*f^2)
/a/d^4*exp(2*I*(d*x+c))-1/32*I*(4*d^3*x^3*f^3-6*I*d^2*f^3*x^2+12*e*f^2*x^2*d^3-12*I*d^2*e*f^2*x+12*e^2*f*x*d^3
-6*I*d^2*e^2*f+4*d^3*e^3-6*d*f^3*x+3*I*f^3-6*d*e*f^2)/a/d^4*exp(-2*I*(d*x+c))-6*I/a/d^3*f^3*c^2*x+12*I/a/d^3*f
^3*polylog(2,I*exp(I*(d*x+c)))*x+6*I/a/d*f^2*e*x^2+6*I/a/d^3*f^2*e*c^2+12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x
+c)))+6*I/a/d^4*f^3*c^2*arctan(exp(I*(d*x+c)))+6*I/a/d^2*f*e^2*arctan(exp(I*(d*x+c)))+1/2*(d^3*x^3*f^3-3*I*d^2
*f^3*x^2+3*e*f^2*x^2*d^3-6*I*d^2*e*f^2*x+3*e^2*f*x*d^3-3*I*d^2*e^2*f+d^3*e^3-6*d*f^3*x+6*I*f^3-6*d*e*f^2)/a/d^
4*exp(-I*(d*x+c))+3/8/a*f^3*x^4+3/8/a/f*e^4+1/2*(d^3*x^3*f^3+3*I*d^2*f^3*x^2+3*e*f^2*x^2*d^3+6*I*d^2*e*f^2*x+3
*e^2*f*x*d^3+3*I*d^2*e^2*f+d^3*e^3-6*d*f^3*x-6*I*f^3-6*d*e*f^2)/a/d^4*exp(I*(d*x+c))-12/a/d^3*f^2*e*ln(1-I*exp
(I*(d*x+c)))*c-12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c)))-12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x+6/a/d^3*f^2*e*c*ln(
1+exp(2*I*(d*x+c)))+2*(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e^3)/d/a/(exp(I*(d*x+c))+I)-12*I/a/d^3*f^2*e*c*arctan(exp
(I*(d*x+c)))+12*I/a/d^2*f^2*e*c*x-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (345) = 690\).

Time = 0.33 (sec) , antiderivative size = 1565, normalized size of antiderivative = 4.10 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/16*(6*d^4*f^3*x^4 + 16*d^3*e^3 - 42*d^2*e^2*f + 8*(3*d^4*e*f^2 + 2*d^3*f^3)*x^3 + 2*(4*d^3*f^3*x^3 + 4*d^3*e
^3 - 6*d^2*e^2*f - 6*d*e*f^2 + 3*f^3 + 6*(2*d^3*e*f^2 - d^2*f^3)*x^2 + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 - d*f^3)*x
)*cos(d*x + c)^3 + 93*f^3 + 6*(6*d^4*e^2*f + 8*d^3*e*f^2 - 7*d^2*f^3)*x^2 + 2*(8*d^3*f^3*x^3 + 8*d^3*e^3 + 18*
d^2*e^2*f - 48*d*e*f^2 - 45*f^3 + 6*(4*d^3*e*f^2 + 3*d^2*f^3)*x^2 + 12*(2*d^3*e^2*f + 3*d^2*e*f^2 - 4*d*f^3)*x
)*cos(d*x + c)^2 + 12*(2*d^4*e^3 + 4*d^3*e^2*f - 7*d^2*e*f^2)*x + 3*(2*d^4*f^3*x^4 + 8*d^3*e^3 + 2*d^2*e^2*f -
28*d*e*f^2 + 8*(d^4*e*f^2 + d^3*f^3)*x^3 - f^3 + 2*(6*d^4*e^2*f + 12*d^3*e*f^2 + d^2*f^3)*x^2 + 4*(2*d^4*e^3
+ 6*d^3*e^2*f + d^2*e*f^2 - 7*d*f^3)*x)*cos(d*x + c) - 96*(-I*d*f^3*x - I*d*e*f^2 + (-I*d*f^3*x - I*d*e*f^2)*c
os(d*x + c) + (-I*d*f^3*x - I*d*e*f^2)*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - 96*(I*d*f^3*x + I*
d*e*f^2 + (I*d*f^3*x + I*d*e*f^2)*cos(d*x + c) + (I*d*f^3*x + I*d*e*f^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) -
sin(d*x + c)) - 48*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) + (d
^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - 48*(d^2*f^3*x^2 + 2*d
^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c) + (d^2
*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*sin(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) - 48*(d
^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos
(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*sin(d*x + c))*log(-I*cos(d*x + c) + sin(d*x
+ c) + 1) - 48*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) + (d^2*e^
2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I) - 96*(f^3*cos(d*x + c) + f^
3*sin(d*x + c) + f^3)*polylog(3, I*cos(d*x + c) - sin(d*x + c)) - 96*(f^3*cos(d*x + c) + f^3*sin(d*x + c) + f^
3)*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) + (6*d^4*f^3*x^4 - 16*d^3*e^3 - 42*d^2*e^2*f + 8*(3*d^4*e*f^2 -
2*d^3*f^3)*x^3 + 93*f^3 + 6*(6*d^4*e^2*f - 8*d^3*e*f^2 - 7*d^2*f^3)*x^2 - 2*(4*d^3*f^3*x^3 + 4*d^3*e^3 + 6*d^2
*e^2*f - 6*d*e*f^2 - 3*f^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*x^2 + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 - d*f^3)*x)*cos(d*x
+ c)^2 + 12*(2*d^4*e^3 - 4*d^3*e^2*f - 7*d^2*e*f^2)*x + 4*(2*d^3*f^3*x^3 + 2*d^3*e^3 - 12*d^2*e^2*f - 21*d*e*f
^2 + 24*f^3 + 6*(d^3*e*f^2 - 2*d^2*f^3)*x^2 + 3*(2*d^3*e^2*f - 8*d^2*e*f^2 - 7*d*f^3)*x)*cos(d*x + c))*sin(d*x
+ c))/(a*d^4*cos(d*x + c) + a*d^4*sin(d*x + c) + a*d^4)

Sympy [F]

\[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sin \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+a\,\sin \left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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