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

Optimal. Leaf size=600 \[ \frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}-\frac {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}-\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {2 i (e+f x)^3}{a d} \]

[Out]

3*I*f^2*(f*x+e)*polylog(2,exp(2*I*(d*x+c)))/a/d^3-6*f^2*(f*x+e)*arctanh(exp(I*(d*x+c)))/a/d^3-3*(f*x+e)^3*arct
anh(exp(I*(d*x+c)))/a/d+(f*x+e)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d+(f*x+e)^3*cot(d*x+c)/a/d-3/2*f*(f*x+e)^2*csc(d
*x+c)/a/d^2-1/2*(f*x+e)^3*cot(d*x+c)*csc(d*x+c)/a/d-6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2-3*f*(f*x+e)^2*l
n(1-exp(2*I*(d*x+c)))/a/d^2+2*I*(f*x+e)^3/a/d-9/2*I*f*(f*x+e)^2*polylog(2,exp(I*(d*x+c)))/a/d^2+9/2*I*f*(f*x+e
)^2*polylog(2,-exp(I*(d*x+c)))/a/d^2+9*I*f^3*polylog(4,exp(I*(d*x+c)))/a/d^4+12*I*f^2*(f*x+e)*polylog(2,I*exp(
I*(d*x+c)))/a/d^3-9*I*f^3*polylog(4,-exp(I*(d*x+c)))/a/d^4-9*f^2*(f*x+e)*polylog(3,-exp(I*(d*x+c)))/a/d^3-12*f
^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+9*f^2*(f*x+e)*polylog(3,exp(I*(d*x+c)))/a/d^3-3/2*f^3*polylog(3,exp(2*I*(
d*x+c)))/a/d^4+3*I*f^3*polylog(2,-exp(I*(d*x+c)))/a/d^4-3*I*f^3*polylog(2,exp(I*(d*x+c)))/a/d^4

________________________________________________________________________________________

Rubi [A]  time = 1.11, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {4535, 4186, 4183, 2279, 2391, 2531, 6609, 2282, 6589, 4184, 3717, 2190, 3318} \[ \frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f^2 (e+f x) \text {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {9 i f (e+f x)^2 \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {9 i f (e+f x)^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^3 \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}-\frac {3 i f^3 \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {12 f^3 \text {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {3 f^3 \text {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \text {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \text {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {2 i (e+f x)^3}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d^3) - (3*(e + f*x)^3*ArcTanh[E^(I*(
c + d*x))])/(a*d) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)^3*Cot[c + d*x])/(a*d) - (3*f*(e
 + f*x)^2*Csc[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (6*f*(e + f*x)^2*Log[1 -
 I*E^(I*(c + d*x))])/(a*d^2) - (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) + ((3*I)*f^3*PolyLog[2,
-E^(I*(c + d*x))])/(a*d^4) + (((9*I)/2)*f*(e + f*x)^2*PolyLog[2, -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) - ((3*I)*f^3*PolyLog[2, E^(I*(c + d*x))])/(a*d^4) - (((9*I)/2)*f*
(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^
3) - (9*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) +
 (9*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) - (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) - (
(9*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((9*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4)

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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
 e, n}, x] && EqQ[c*d, 1]

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 3318

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*(e + (Pi*a)/(2*b)))/2 + (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 3717

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 4183

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

Rule 4184

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 4186

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

Rule 4535

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

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]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \csc ^3(c+d x) \, dx}{a}-\int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{2 a}-\frac {\int (e+f x)^3 \csc ^2(c+d x) \, dx}{a}+\frac {\left (3 f^2\right ) \int (e+f x) \csc (c+d x) \, dx}{a d^2}+\int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{2 a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \cot (c+d x) \, dx}{a d}-\frac {\left (3 f^3\right ) \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (3 f^3\right ) \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d^3}-\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx\\ &=\frac {i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 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 {(6 i f) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1-e^{2 i (c+d x)}} \, dx}{a d}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}-\frac {\left (3 i f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (3 i f^2\right ) \int (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (3 i f^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (3 i f^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}\\ &=\frac {i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\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 i f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (3 f^3\right ) \int \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (3 f^3\right ) \int \text {Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a d^3}\\ &=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\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 {\left (3 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (3 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (3 i f^3\right ) \int \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a d^3}\\ &=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\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 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {3 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\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 {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4}\\ &=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\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 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 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 {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\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 {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\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 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 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 {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {\left (12 f^3\right ) \operatorname {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 {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\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 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{2 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 {3 i f^3 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-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 {9 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}\\ \end {align*}

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Mathematica [B]  time = 33.96, size = 1485, normalized size = 2.48 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*e^3*Log[Tan[(c + d*x)/2]])/(2*a*d) + (3*e*f^2*Log[Tan[(c + d*x)/2]])/(a*d^3) + (9*e^2*f*((c + d*x)*(Log[1 -
 E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - Po
lyLog[2, E^(I*(c + d*x))])))/(2*a*d^2) + (3*f^3*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))
]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^4) + (E^(
I*c)*f^3*Csc[c]*((2*d^3*x^3)/E^((2*I)*c) + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 - E^((-I)*(c + d*x))] + (3*I
)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 + E^((-I)*(c + d*x))] - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, -E^((-I)*(c +
 d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, E^((-I)*(c +
 d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(2*a*d^4) - (9*e*f^2*(d^2*x^2*ArcTanh[Cos[c + d*x]
+ I*Sin[c + d*x]] - I*d*x*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + I*d*x*PolyLog[2, Cos[c + d*x] + I*Sin[c
 + d*x]] + PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]]))/(a*d^3) +
(3*f^3*(-2*d^3*x^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + (3*I)*d^2*x^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c +
 d*x]] - (3*I)*d^2*x^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] - 6*d*x*PolyLog[3, -Cos[c + d*x] - I*Sin[c +
d*x]] + 6*d*x*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - (6*I)*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]] + (
6*I)*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/(2*a*d^4) - (3*e^2*f*Csc[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin
[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d^2*(Cos[c]^2 + Sin[c]^2)) + (6*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c]
 - I*Sin[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]])*(C
os[c] - I*(1 + Sin[c])))/d^3))/(a*d*(Cos[c] + I*(1 + Sin[c]))) + (Csc[c]*Csc[c + d*x]^2*(e^3*Sin[d*x] + 3*e^2*
f*x*Sin[d*x] + 3*e*f^2*x^2*Sin[d*x] + f^3*x^3*Sin[d*x]))/(2*a*d) + (Csc[c]*Csc[c + d*x]*(-(d*e^3*Cos[c]) - 3*d
*e^2*f*x*Cos[c] - 3*d*e*f^2*x^2*Cos[c] - d*f^3*x^3*Cos[c] - 3*e^2*f*Sin[c] - 6*e*f^2*x*Sin[c] - 3*f^3*x^2*Sin[
c] - 2*d*e^3*Sin[d*x] - 6*d*e^2*f*x*Sin[d*x] - 6*d*e*f^2*x^2*Sin[d*x] - 2*d*f^3*x^3*Sin[d*x]))/(2*a*d^2) - (2*
(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(a*d*(Cos[c/2]
+ Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])) + (3*e*f^2*Csc[c]*Sec[c]*(d^2*E^(I*ArcTan[Tan[c]])*x^2
+ ((I*d*x*(-Pi + 2*ArcTan[Tan[c]]) - Pi*Log[1 + E^((-2*I)*d*x)] - 2*(d*x + ArcTan[Tan[c]])*Log[1 - E^((2*I)*(d
*x + ArcTan[Tan[c]]))] + Pi*Log[Cos[d*x]] + 2*ArcTan[Tan[c]]*Log[Sin[d*x + ArcTan[Tan[c]]]] + I*PolyLog[2, E^(
(2*I)*(d*x + ArcTan[Tan[c]]))])*Tan[c])/Sqrt[1 + Tan[c]^2]))/(a*d^3*Sqrt[Sec[c]^2*(Cos[c]^2 + Sin[c]^2)])

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fricas [C]  time = 0.92, size = 7842, normalized size = 13.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*d^3*f^3*x^3 + 4*d^3*e^3 - 6*d^2*e^2*f - 8*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*co
s(d*x + c)^3 + 6*(2*d^3*e*f^2 - d^2*f^3)*x^2 - 6*(d^3*f^3*x^3 + d^3*e^3 - d^2*e^2*f + (3*d^3*e*f^2 - d^2*f^3)*
x^2 + (3*d^3*e^2*f - 2*d^2*e*f^2)*x)*cos(d*x + c)^2 + 12*(d^3*e^2*f - d^2*e*f^2)*x + 6*(d^3*f^3*x^3 + 3*d^3*e*
f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*cos(d*x + c) - (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + (-9*I*d^2*
f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^3 + 6*I*f^3 + (
-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 + 6*
I*(3*d^2*e*f^2 - 2*d*f^3)*x + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2
*d*f^3)*x)*cos(d*x + c) + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + (-9*I*d^2*f^3*x^2 - 9*I*
d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 + 6*I*(3*d^2*e*f^2 - 2*d*f^
3)*x)*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 +
(9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^3 - 6*
I*f^3 + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c
)^2 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*
e*f^2 - 2*d*f^3)*x)*cos(d*x + c) + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 + (9*I*d^2*f^3*x
^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 - 6*I*(3*d^2*e*f^2
 - 2*d*f^3)*x)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) - (-24*I*d*f^3*x - 24*I*d*e*f^2 + (24*I*d*f^
3*x + 24*I*d*e*f^2)*cos(d*x + c)^3 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c)^2 + (-24*I*d*f^3*x - 24*I*d*e*
f^2)*cos(d*x + c) + (-24*I*d*f^3*x - 24*I*d*e*f^2 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c)^2)*sin(d*x + c)
)*dilog(I*cos(d*x + c) - sin(d*x + c)) - (24*I*d*f^3*x + 24*I*d*e*f^2 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*cos(d*x
 + c)^3 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*cos(d*x + c)^2 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c) + (24*I*d
*f^3*x + 24*I*d*e*f^2 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*cos(d*x + c)^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - s
in(d*x + c)) - (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2 + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*
f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^3 + 6*I*f^3 + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f -
12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x + (9*I*
d^2*f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c) + (9*I*d^2*
f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2 + 6*I*f^3 + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 - 6*I*f^3
- 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*dilog(-cos(d*x
+ c) + I*sin(d*x + c)) - (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f +
 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^3 - 6*I*f^3 + (9*I*d^2*f^3*x^2 + 9*I*d^2
*e^2*f + 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*
x + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c) +
 (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 - 6*I*f^3 + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2
+ 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*dilog
(-cos(d*x + c) - I*sin(d*x + c)) - 3*(d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 - (d^3*f^3*x^3 + d^3*e^3
 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x
+ c)^3 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d
^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*
x + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*
f^2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*
x^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*
e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*log(cos(d*x + c) +
 I*sin(d*x + c) + 1) - 12*(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
)^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^2 + (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 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(cos
(d*x + c) + I*sin(d*x + c) + I) - 3*(d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 - (d^3*f^3*x^3 + d^3*e^3
+ 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x +
 c)^3 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^
2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x
 + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f
^2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x
^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e
*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*log(cos(d*x + c) -
I*sin(d*x + c) + 1) - 12*(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)^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)^2
+ (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 - (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)^2)*sin(d*x + c))*log(I*cos(
d*x + c) + sin(d*x + c) + 1) - 12*(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)^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)^2 + (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 - (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)^2)*sin(d*x + c))*l
og(-I*cos(d*x + c) + sin(d*x + c) + 1) + 3*(d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 +
 2*c^2 + 2*c)*f^3 - (d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(
d*x + c)^3 - (d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c
)^2 + (d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c) + (d^
3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3 - (d^3*e^3 - (3*c + 2)*d^2*e
^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c
) + 1/2*I*sin(d*x + c) + 1/2) + 3*(d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 +
2*c)*f^3 - (d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c)^
3 - (d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c)^2 + (d^
3*e^3 - (3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c) + (d^3*e^3 - (
3*c + 2)*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3 - (d^3*e^3 - (3*c + 2)*d^2*e^2*f + (3
*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I
*sin(d*x + c) + 1/2) + 3*(d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 - (d^3
*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (
3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + 3*c*d^
2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d
^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x + (d^3*f^3*x^3 + 3*c*d^2*e^2*f
 - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^
2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3
+ (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f
^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f -
4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + 1) - 12*(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)^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x +
 c)^2 + (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 - (d^2*e^2*f - 2
*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I) + 3*(d^3*f^3*x^3 +
 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 - (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c
)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)
*cos(d*x + c)^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3
+ 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 +
 (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x + (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2
 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3
*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*
f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3
*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))
*log(-cos(d*x + c) - I*sin(d*x + c) + 1) - (18*I*f^3*cos(d*x + c)^3 + 18*I*f^3*cos(d*x + c)^2 - 18*I*f^3*cos(d
*x + c) - 18*I*f^3 + (18*I*f^3*cos(d*x + c)^2 - 18*I*f^3)*sin(d*x + c))*polylog(4, cos(d*x + c) + I*sin(d*x +
c)) - (-18*I*f^3*cos(d*x + c)^3 - 18*I*f^3*cos(d*x + c)^2 + 18*I*f^3*cos(d*x + c) + 18*I*f^3 + (-18*I*f^3*cos(
d*x + c)^2 + 18*I*f^3)*sin(d*x + c))*polylog(4, cos(d*x + c) - I*sin(d*x + c)) - (18*I*f^3*cos(d*x + c)^3 + 18
*I*f^3*cos(d*x + c)^2 - 18*I*f^3*cos(d*x + c) - 18*I*f^3 + (18*I*f^3*cos(d*x + c)^2 - 18*I*f^3)*sin(d*x + c))*
polylog(4, -cos(d*x + c) + I*sin(d*x + c)) - (-18*I*f^3*cos(d*x + c)^3 - 18*I*f^3*cos(d*x + c)^2 + 18*I*f^3*co
s(d*x + c) + 18*I*f^3 + (-18*I*f^3*cos(d*x + c)^2 + 18*I*f^3)*sin(d*x + c))*polylog(4, -cos(d*x + c) - I*sin(d
*x + c)) + 6*(3*d*f^3*x + 3*d*e*f^2 - (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^3 - 2*f^3 - (3*d*f^3*x + 3*
d*e*f^2 - 2*f^3)*cos(d*x + c)^2 + (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 - 2*f^
3 - (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*polylog(3, cos(d*x + c) + I*sin(d*x + c)) +
6*(3*d*f^3*x + 3*d*e*f^2 - (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^3 - 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 - 2
*f^3)*cos(d*x + c)^2 + (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 - 2*f^3 - (3*d*f^
3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*polylog(3, cos(d*x + c) - I*sin(d*x + c)) + 24*(f^3*cos
(d*x + c)^3 + f^3*cos(d*x + c)^2 - f^3*cos(d*x + c) - f^3 + (f^3*cos(d*x + c)^2 - f^3)*sin(d*x + c))*polylog(3
, I*cos(d*x + c) - sin(d*x + c)) + 24*(f^3*cos(d*x + c)^3 + f^3*cos(d*x + c)^2 - f^3*cos(d*x + c) - f^3 + (f^3
*cos(d*x + c)^2 - f^3)*sin(d*x + c))*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) - 6*(3*d*f^3*x + 3*d*e*f^2 - (
3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^3 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^2 + (3*d*
f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos
(d*x + c)^2)*sin(d*x + c))*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) - 6*(3*d*f^3*x + 3*d*e*f^2 - (3*d*f^3*x
+ 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^3 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^2 + (3*d*f^3*x + 3*
d*e*f^2 + 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^
2)*sin(d*x + c))*polylog(3, -cos(d*x + c) - I*sin(d*x + c)) - 2*(2*d^3*f^3*x^3 + 2*d^3*e^3 + 3*d^2*e^2*f + 3*(
2*d^3*e*f^2 + d^2*f^3)*x^2 - 4*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*cos(d*x + c)^2 + 6*(d
^3*e^2*f + d^2*e*f^2)*x - (d^3*f^3*x^3 + d^3*e^3 - 3*d^2*e^2*f + 3*(d^3*e*f^2 - d^2*f^3)*x^2 + 3*(d^3*e^2*f -
2*d^2*e*f^2)*x)*cos(d*x + c))*sin(d*x + c))/(a*d^4*cos(d*x + c)^3 + a*d^4*cos(d*x + c)^2 - a*d^4*cos(d*x + c)
- a*d^4 + (a*d^4*cos(d*x + c)^2 - a*d^4)*sin(d*x + c))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.44, size = 2257, normalized size = 3.76 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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-24/a/d^3*f^2*e*c*ln(exp(I*(d*
x+c)))+9*I/a/d^2*polylog(2,-exp(I*(d*x+c)))*e*f^2*x-9*I/a/d^2*polylog(2,exp(I*(d*x+c)))*e*f^2*x+24*I/a/d^2*c*e
*f^2*x+12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+c)))+6/a/d^3*e*f^2*c*ln(exp(I*(d*x+c))-1)+(-3*I*f^3*x^2*exp(4*I
*(d*x+c))-3*I*e^2*f*exp(4*I*(d*x+c))-5*d*f^3*x^3*exp(2*I*(d*x+c))+3*I*e^2*f*exp(2*I*(d*x+c))+3*I*f^3*x^2*exp(2
*I*(d*x+c))-I*d*e^3*exp(I*(d*x+c))+4*d*f^3*x^3+12*d*e*f^2*x^2+12*d*e^2*f*x-3*f^3*x^2*exp(I*(d*x+c))-3*exp(I*(d
*x+c))*e^2*f+4*d*e^3-3*I*d*e*f^2*x^2*exp(I*(d*x+c))-3*I*d*e^2*f*x*exp(I*(d*x+c))-5*d*e^3*exp(2*I*(d*x+c))+3*d*
e^3*exp(4*I*(d*x+c))+3*f^3*x^2*exp(3*I*(d*x+c))+3*e^2*f*exp(3*I*(d*x+c))+6*e*f^2*x*exp(3*I*(d*x+c))+3*d*f^3*x^
3*exp(4*I*(d*x+c))+3*I*d*e^3*exp(3*I*(d*x+c))-6*e*f^2*x*exp(I*(d*x+c))-15*d*e^2*f*x*exp(2*I*(d*x+c))+6*I*e*f^2
*x*exp(2*I*(d*x+c))+9*d*e*f^2*x^2*exp(4*I*(d*x+c))+9*d*e^2*f*x*exp(4*I*(d*x+c))+3*I*d*f^3*x^3*exp(3*I*(d*x+c))
-I*d*f^3*x^3*exp(I*(d*x+c))-15*d*e*f^2*x^2*exp(2*I*(d*x+c))+9*I*d*e*f^2*x^2*exp(3*I*(d*x+c))+9*I*d*e^2*f*x*exp
(3*I*(d*x+c))-6*I*e*f^2*x*exp(4*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/a-3/a/d^2*f^3*ln(exp
(I*(d*x+c))+1)*x^2-3/a/d^2*f^3*ln(1-exp(I*(d*x+c)))*x^2+3/a/d^4*f^3*ln(1-exp(I*(d*x+c)))*c^2-3/a/d^2*e^2*f*ln(
exp(I*(d*x+c))+1)-3/a/d^2*e^2*f*ln(exp(I*(d*x+c))-1)-3/a/d^4*f^3*c^2*ln(exp(I*(d*x+c))-1)+12/a/d^2*f*ln(exp(I*
(d*x+c)))*e^2+12/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+3/2/a/d*e^3*ln(exp(I*(d*x+c))-1)-3/2/a/d*e^3*ln(exp(I*(d*x+c))+1)-3/a/d^3*f^3*ln(exp(I*(d*x+c))+1)
*x+3/a/d^3*f^3*ln(1-exp(I*(d*x+c)))*x+3/a/d^4*f^3*ln(1-exp(I*(d*x+c)))*c+4*I/a/d*f^3*x^3-8*I/a/d^4*f^3*c^3-6/a
/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2+3/a/d^3*e*f^2*ln(exp(I*(d*x+c))-1)-3/a/d^3*e*f^2*ln(exp(I*(d*x+c))+1)-3/a/
d^4*f^3*c*ln(exp(I*(d*x+c))-1)-6/a/d^2*e*f^2*ln(1-exp(I*(d*x+c)))*x-6/a/d^3*e*f^2*ln(1-exp(I*(d*x+c)))*c-6/a/d
^2*e*f^2*ln(exp(I*(d*x+c))+1)*x-6*f^3*polylog(3,-exp(I*(d*x+c)))/a/d^4-6*f^3*polylog(3,exp(I*(d*x+c)))/a/d^4-1
2*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+9*I*f^3*polylog(4,exp(I*(d*x+c)))/a/d^4+3*I*f^3*polylog(2,-exp(I*(d*x+
c)))/a/d^4-3*I*f^3*polylog(2,exp(I*(d*x+c)))/a/d^4-9*I*f^3*polylog(4,-exp(I*(d*x+c)))/a/d^4-3/2/a/d^4*f^3*c^3*
ln(exp(I*(d*x+c))-1)+9/a/d^3*e*f^2*polylog(3,exp(I*(d*x+c)))-9/a/d^3*e*f^2*polylog(3,-exp(I*(d*x+c)))+9/a/d^3*
f^3*polylog(3,exp(I*(d*x+c)))*x-9/a/d^3*f^3*polylog(3,-exp(I*(d*x+c)))*x+6/a/d^4*f^3*ln(1-I*exp(I*(d*x+c)))*c^
2-3/2/a/d*f^3*ln(exp(I*(d*x+c))+1)*x^3+3/2/a/d*f^3*ln(1-exp(I*(d*x+c)))*x^3+3/2/a/d^4*f^3*ln(1-exp(I*(d*x+c)))
*c^3+9/2/a/d^3*e*f^2*c^2*ln(exp(I*(d*x+c))-1)+12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c))+I)+12*I/a/d^3*f^3*polylog(2,I
*exp(I*(d*x+c)))*x-9/2/a/d*e*f^2*ln(exp(I*(d*x+c))+1)*x^2+9/2/a/d*e*f^2*ln(1-exp(I*(d*x+c)))*x^2+9/2/a/d*ln(1-
exp(I*(d*x+c)))*e^2*f*x-9/2/a/d*ln(exp(I*(d*x+c))+1)*e^2*f*x-9/2/a/d^3*e*f^2*c^2*ln(1-exp(I*(d*x+c)))+9/2/a/d^
2*ln(1-exp(I*(d*x+c)))*c*e^2*f-9/2/a/d^2*e^2*f*c*ln(exp(I*(d*x+c))-1)+12*I/a/d*e*f^2*x^2-9/2*I/a/d^2*e^2*f*pol
ylog(2,exp(I*(d*x+c)))+9/2*I/a/d^2*e^2*f*polylog(2,-exp(I*(d*x+c)))-12*I/a/d^3*f^3*c^2*x+12*I/a/d^3*c^2*e*f^2+
6*I/a/d^3*e*f^2*polylog(2,exp(I*(d*x+c)))+6*I/a/d^3*e*f^2*polylog(2,-exp(I*(d*x+c)))-9/2*I/a/d^2*f^3*polylog(2
,exp(I*(d*x+c)))*x^2+9/2*I/a/d^2*f^3*polylog(2,-exp(I*(d*x+c)))*x^2+6*I/a/d^3*f^3*polylog(2,-exp(I*(d*x+c)))*x
+6*I/a/d^3*f^3*polylog(2,exp(I*(d*x+c)))*x

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {e^{3} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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