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

Optimal. Leaf size=382 \[ -\frac {3 e f^2 x}{4 a d^2}-\frac {3 f^3 x^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2} \]

[Out]

-3/4*e*f^2*x/a/d^2-3/8*f^3*x^2/a/d^2+12*I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3+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+I*(f*x+e)^3/a/d-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]
time = 0.41, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 19, 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, 3399, 4269, 3798, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {12 f^3 \text {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\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 {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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\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) \text {Li}_2\left (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 \text {Li}_2\left (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) \text {Li}_2\left (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 {\text {Li}_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) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (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]
time = 3.37, size = 391, normalized size = 1.02 \begin {gather*} \frac {24 e^3 x+36 e^2 f x^2+24 e f^2 x^3+6 f^3 x^4-\frac {96 f^2 (e+f x) \cos (c+d x)}{d^3}+\frac {16 (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos (2 (c+d x))}{d^4}-\frac {6 f (e+f x)^2 \cos (2 (c+d x))}{d^2}-\frac {96 f (e+f x)^2 \log (1-i \cos (c+d x)+\sin (c+d x))}{d^2}+\frac {192 i f^2 (e+f x) \text {Li}_2(i \cos (c+d x)-\sin (c+d x))}{d^3}-\frac {192 f^3 \text {Li}_3(i \cos (c+d x)-\sin (c+d x))}{d^4}+\frac {32 i f x \left (3 e^2+3 e f x+f^2 x^2\right ) (\cos (c)+i \sin (c))}{d (\cos (c)+i (1+\sin (c)))}-\frac {32 (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 {96 f^3 \sin (c+d x)}{d^4}-\frac {48 f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {6 f^2 (e+f x) \sin (2 (c+d x))}{d^3}-\frac {4 (e+f x)^3 \sin (2 (c+d x))}{d}}{16 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*e^3*x + 36*e^2*f*x^2 + 24*e*f^2*x^3 + 6*f^3*x^4 - (96*f^2*(e + f*x)*Cos[c + d*x])/d^3 + (16*(e + f*x)^3*Co
s[c + d*x])/d + (3*f^3*Cos[2*(c + d*x)])/d^4 - (6*f*(e + f*x)^2*Cos[2*(c + d*x)])/d^2 - (96*f*(e + f*x)^2*Log[
1 - I*Cos[c + d*x] + Sin[c + d*x]])/d^2 + ((192*I)*f^2*(e + f*x)*PolyLog[2, I*Cos[c + d*x] - Sin[c + d*x]])/d^
3 - (192*f^3*PolyLog[3, I*Cos[c + d*x] - Sin[c + d*x]])/d^4 + ((32*I)*f*x*(3*e^2 + 3*e*f*x + f^2*x^2)*(Cos[c]
+ I*Sin[c]))/(d*(Cos[c] + I*(1 + Sin[c]))) - (32*(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])) + (96*f^3*Sin[c + d*x])/d^4 - (48*f*(e + f*x)^2*Sin[c + d*x])/d^2 + (6*f^2*(e + f
*x)*Sin[2*(c + d*x)])/d^3 - (4*(e + f*x)^3*Sin[2*(c + d*x)])/d)/(16*a)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (351 ) = 702\).
time = 0.21, size = 881, normalized size = 2.31

method	result	size
risch	\(-\frac {3 f \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}-f^{2}\right ) \cos \left (2 d x +2 c \right )}{16 a \,d^{4}}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}-3 i d^{2} f^{3} x^{2}+3 d^{3} e^{2} f x -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 {3 f^{3} x^{4}}{8 a}+\frac {3 e^{4}}{8 a f}-\frac {6 i f^{3} c^{2} x}{a \,d^{3}}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 i d^{2} f^{3} x^{2}+3 d^{3} e^{2} f x +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 {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{a \,d^{2}}+\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c^{2}}{a \,d^{4}}+\frac {6 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{4}}-\frac {6 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) e^{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^{2} e c x}{a \,d^{2}}+\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 i f^{2} e \,x^{2}}{a d}+\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 {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {12 f^{3} \polylog \left (3, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {\left (2 d^{2} x^{3} f^{3}+6 d^{2} e \,f^{2} x^{2}+6 d^{2} e^{2} f x +2 d^{2} e^{3}-3 f^{3} x -3 e \,f^{2}\right ) \sin \left (2 d x +2 c \right )}{8 d^{3} a}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}+\frac {12 i f^{2} e \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {6 i f^{2} e \,c^{2}}{a \,d^{3}}+\frac {12 i f^{3} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{3}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{3}}\)	\(881\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*f*(2*d^2*f^2*x^2+4*d^2*e*f*x+2*d^2*e^2-f^2)/a/d^4*cos(2*d*x+2*c)+3/8/a*f^3*x^4+3/8/a/f*e^4-6/a/d^2*f^3*l
n(1-I*exp(I*(d*x+c)))*x^2+6/a/d^4*f^3*ln(1-I*exp(I*(d*x+c)))*c^2+6/a/d^2*f*ln(exp(I*(d*x+c)))*e^2+6/a/d^4*f^3*
c^2*ln(exp(I*(d*x+c)))-6/a/d^4*f^3*c^2*ln(exp(I*(d*x+c))+I)-6/a/d^2*f*ln(exp(I*(d*x+c))+I)*e^2+2*I/a/d*f^3*x^3
-4*I/a/d^4*f^3*c^3+12*I/a/d^2*f^2*e*c*x+3/2/a*f^2*e*x^3+9/4/a*f*e^2*x^2+3/2/a*e^3*x-12/a/d^3*f^2*e*c*ln(exp(I*
(d*x+c)))+12*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x-6*I/a/d^3*f^3*c^2*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)))+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)-1
/8/d^3*(2*d^2*f^3*x^3+6*d^2*e*f^2*x^2+6*d^2*e^2*f*x+2*d^2*e^3-3*f^3*x-3*e*f^2)/a*sin(2*d*x+2*c)-12*f^3*polylog
(3,I*exp(I*(d*x+c)))/a/d^4+12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c))+I)-12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x-12/a/
d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c+1/2*(d^3*x^3*f^3+3*I*d^2*f^3*x^2+3*d^3*e*f^2*x^2+6*I*d^2*e*f^2*x+3*d^3*e^2*
f*x+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))+1/2*(d^3*x^3*f^3-3*I*d^2*f^3*x^2+3
*d^3*e*f^2*x^2-6*I*d^2*e*f^2*x+3*d^3*e^2*f*x-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))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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.

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (357) = 714\).
time = 0.44, size = 1569, normalized size = 4.11 \begin {gather*} \frac {6 \, d^{4} f^{3} x^{4} + 16 \, d^{3} f^{3} x^{3} - 42 \, d^{2} f^{3} x^{2} + 2 \, {\left (4 \, d^{3} f^{3} x^{3} - 6 \, d^{2} f^{3} x^{2} - 6 \, d f^{3} x + 4 \, d^{3} e^{3} + 3 \, f^{3} + 6 \, {\left (2 \, d^{3} f x - d^{2} f\right )} e^{2} + 6 \, {\left (2 \, d^{3} f^{2} x^{2} - 2 \, d^{2} f^{2} x - d f^{2}\right )} e\right )} \cos \left (d x + c\right )^{3} + 93 \, f^{3} + 2 \, {\left (8 \, d^{3} f^{3} x^{3} + 18 \, d^{2} f^{3} x^{2} - 48 \, d f^{3} x + 8 \, d^{3} e^{3} - 45 \, f^{3} + 6 \, {\left (4 \, d^{3} f x + 3 \, d^{2} f\right )} e^{2} + 12 \, {\left (2 \, d^{3} f^{2} x^{2} + 3 \, d^{2} f^{2} x - 4 \, d f^{2}\right )} e\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, d^{4} f^{3} x^{4} + 8 \, d^{3} f^{3} x^{3} + 2 \, d^{2} f^{3} x^{2} - 28 \, d f^{3} x - f^{3} + 8 \, {\left (d^{4} x + d^{3}\right )} e^{3} + 2 \, {\left (6 \, d^{4} f x^{2} + 12 \, d^{3} f x + d^{2} f\right )} e^{2} + 4 \, {\left (2 \, d^{4} f^{2} x^{3} + 6 \, d^{3} f^{2} x^{2} + d^{2} f^{2} x - 7 \, d f^{2}\right )} e\right )} \cos \left (d x + c\right ) - 96 \, {\left (-i \, d f^{3} x - i \, d f^{2} e + {\left (-i \, d f^{3} x - i \, d f^{2} e\right )} \cos \left (d x + c\right ) + {\left (-i \, d f^{3} x - i \, d f^{2} e\right )} \sin \left (d x + c\right )\right )} {\rm Li}_2\left (i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 96 \, {\left (i \, d f^{3} x + i \, d f^{2} e + {\left (i \, d f^{3} x + i \, d f^{2} e\right )} \cos \left (d x + c\right ) + {\left (i \, d f^{3} x + i \, d f^{2} e\right )} \sin \left (d x + c\right )\right )} {\rm Li}_2\left (-i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) + 8 \, {\left (3 \, d^{4} x + 2 \, d^{3}\right )} e^{3} + 6 \, {\left (6 \, d^{4} f x^{2} + 8 \, d^{3} f x - 7 \, d^{2} f\right )} e^{2} + 12 \, {\left (2 \, d^{4} f^{2} x^{3} + 4 \, d^{3} f^{2} x^{2} - 7 \, d^{2} f^{2} x\right )} e - 48 \, {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2} + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \cos \left (d x + c\right ) + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) - 48 \, {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \cos \left (d x + c\right ) + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \log \left (i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - 48 \, {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \cos \left (d x + c\right ) + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \log \left (-i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - 48 \, {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2} + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \cos \left (d x + c\right ) + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) - 96 \, {\left (f^{3} \cos \left (d x + c\right ) + f^{3} \sin \left (d x + c\right ) + f^{3}\right )} {\rm polylog}\left (3, i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 96 \, {\left (f^{3} \cos \left (d x + c\right ) + f^{3} \sin \left (d x + c\right ) + f^{3}\right )} {\rm polylog}\left (3, -i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) + {\left (6 \, d^{4} f^{3} x^{4} - 16 \, d^{3} f^{3} x^{3} - 42 \, d^{2} f^{3} x^{2} + 93 \, f^{3} - 2 \, {\left (4 \, d^{3} f^{3} x^{3} + 6 \, d^{2} f^{3} x^{2} - 6 \, d f^{3} x + 4 \, d^{3} e^{3} - 3 \, f^{3} + 6 \, {\left (2 \, d^{3} f x + d^{2} f\right )} e^{2} + 6 \, {\left (2 \, d^{3} f^{2} x^{2} + 2 \, d^{2} f^{2} x - d f^{2}\right )} e\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, d^{3} f^{3} x^{3} - 12 \, d^{2} f^{3} x^{2} - 21 \, d f^{3} x + 2 \, d^{3} e^{3} + 24 \, f^{3} + 6 \, {\left (d^{3} f x - 2 \, d^{2} f\right )} e^{2} + 3 \, {\left (2 \, d^{3} f^{2} x^{2} - 8 \, d^{2} f^{2} x - 7 \, d f^{2}\right )} e\right )} \cos \left (d x + c\right ) + 8 \, {\left (3 \, d^{4} x - 2 \, d^{3}\right )} e^{3} + 6 \, {\left (6 \, d^{4} f x^{2} - 8 \, d^{3} f x - 7 \, d^{2} f\right )} e^{2} + 12 \, {\left (2 \, d^{4} f^{2} x^{3} - 4 \, d^{3} f^{2} x^{2} - 7 \, d^{2} f^{2} x\right )} e\right )} \sin \left (d x + c\right )}{16 \, {\left (a d^{4} \cos \left (d x + c\right ) + a d^{4} \sin \left (d x + c\right ) + a d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

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+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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________