3.3.9 \(\int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [209]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.74, 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 = {4631, 4271, 4268, 2317, 2438, 2611, 6744, 2320, 6724, 4269, 3798, 2221, 3399} \begin {gather*} \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 {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 {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} \end {gather*}

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 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 2317

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 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 2438

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 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 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 4268

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 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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4631

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 6724

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

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \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 {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] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1370\) vs. \(2(600)=1200\).
time = 26.70, size = 1370, normalized size = 2.28 \begin {gather*} \frac {3 e^3 \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d}+\frac {3 e f^2 \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d^3}+\frac {9 e^2 f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )-c \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+i \left (\text {Li}_2\left (-e^{i (c+d x)}\right )-\text {Li}_2\left (e^{i (c+d x)}\right )\right )\right )}{2 a d^2}+\frac {3 f^3 \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )-c \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+i \left (\text {Li}_2\left (-e^{i (c+d x)}\right )-\text {Li}_2\left (e^{i (c+d x)}\right )\right )\right )}{a d^4}+\frac {e^{-i c} f^3 \csc (c) \left (2 d^2 x^2 \left (2 d e^{2 i c} x+3 i \left (-1+e^{2 i c}\right ) \log \left (1-e^{2 i (c+d x)}\right )\right )+6 d \left (-1+e^{2 i c}\right ) x \text {Li}_2\left (e^{2 i (c+d x)}\right )+3 i \left (-1+e^{2 i c}\right ) \text {Li}_3\left (e^{2 i (c+d x)}\right )\right )}{4 a d^4}-\frac {9 e f^2 \left (d^2 x^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))-i d x \text {Li}_2(-\cos (c+d x)-i \sin (c+d x))+i d x \text {Li}_2(\cos (c+d x)+i \sin (c+d x))+\text {Li}_3(-\cos (c+d x)-i \sin (c+d x))-\text {Li}_3(\cos (c+d x)+i \sin (c+d x))\right )}{a d^3}+\frac {3 f^3 \left (-2 d^3 x^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))+3 i d^2 x^2 \text {Li}_2(-\cos (c+d x)-i \sin (c+d x))-3 i d^2 x^2 \text {Li}_2(\cos (c+d x)+i \sin (c+d x))-6 d x \text {Li}_3(-\cos (c+d x)-i \sin (c+d x))+6 d x \text {Li}_3(\cos (c+d x)+i \sin (c+d x))-6 i \text {Li}_4(-\cos (c+d x)-i \sin (c+d x))+6 i \text {Li}_4(\cos (c+d x)+i \sin (c+d x))\right )}{2 a d^4}-\frac {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 \left (\cos ^2(c)+\sin ^2(c)\right )}+\frac {2 f \left (-3 d^2 (e+f x)^2 \log (1-i \cos (c+d x)+\sin (c+d x))+6 i d f (e+f x) \text {Li}_2(i \cos (c+d x)-\sin (c+d x))-6 f^2 \text {Li}_3(i \cos (c+d x)-\sin (c+d x))+\frac {i d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) (\cos (c)+i \sin (c))}{\cos (c)+i (1+\sin (c))}\right )}{a d^4}+\frac {\csc (c) \csc ^2(c+d x) \left (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)\right )}{2 a d}+\frac {\csc (c) \csc (c+d x) \left (-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)\right )}{2 a d^2}-\frac {2 \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {3 e f^2 \csc (c) \sec (c) \left (d^2 e^{i \tan ^{-1}(\tan (c))} x^2+\frac {\left (i d x \left (-\pi +2 \tan ^{-1}(\tan (c))\right )-\pi \log \left (1+e^{-2 i d x}\right )-2 \left (d x+\tan ^{-1}(\tan (c))\right ) \log \left (1-e^{2 i \left (d x+\tan ^{-1}(\tan (c))\right )}\right )+\pi \log (\cos (d x))+2 \tan ^{-1}(\tan (c)) \log \left (\sin \left (d x+\tan ^{-1}(\tan (c))\right )\right )+i \text {Li}_2\left (e^{2 i \left (d x+\tan ^{-1}(\tan (c))\right )}\right )\right ) \tan (c)}{\sqrt {1+\tan ^2(c)}}\right )}{a d^3 \sqrt {\sec ^2(c) \left (\cos ^2(c)+\sin ^2(c)\right )}} \end {gather*}

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) + (f^3
*Csc[c]*(2*d^2*x^2*(2*d*E^((2*I)*c)*x + (3*I)*(-1 + E^((2*I)*c))*Log[1 - E^((2*I)*(c + d*x))]) + 6*d*(-1 + E^(
(2*I)*c))*x*PolyLog[2, E^((2*I)*(c + d*x))] + (3*I)*(-1 + E^((2*I)*c))*PolyLog[3, E^((2*I)*(c + d*x))]))/(4*a*
d^4*E^(I*c)) - (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*S
in[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)) + (2*f*(-3*d^2*(e + f*x)^2*Log[1 - I*Cos[c + d*x] + Sin[c + d*x]] + (6*I)*d*f*(e + f*x)*PolyLog[2, I*Cos
[c + d*x] - Sin[c + d*x]] - 6*f^2*PolyLog[3, I*Cos[c + d*x] - Sin[c + d*x]] + (I*d^3*x*(3*e^2 + 3*e*f*x + f^2*
x^2)*(Cos[c] + I*Sin[c]))/(Cos[c] + I*(1 + Sin[c]))))/(a*d^4) + (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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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2256 vs. \(2 (540 ) = 1080\).
time = 0.27, size = 2257, normalized size = 3.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(2257\)

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,method=_RETURNVERBOSE)

[Out]

9/d^3/a*e*f^2*polylog(3,exp(I*(d*x+c)))-9/d^3/a*e*f^2*polylog(3,-exp(I*(d*x+c)))-3/2/d^4/a*f^3*c^3*ln(exp(I*(d
*x+c))-1)+9/d^3/a*f^3*polylog(3,exp(I*(d*x+c)))*x-9/d^3/a*f^3*polylog(3,-exp(I*(d*x+c)))*x-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+3/2/d/a*e^3*ln(exp
(I*(d*x+c))-1)-3/2/d/a*e^3*ln(exp(I*(d*x+c))+1)-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+(-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))+4*d*f^3*x^3+3*I*e^2*f*exp(2*I*(d*x+c))+3*I*f^3*x^2*exp(2*I*(d*x+c))-5*d*f^3*x^3*exp(2*I*(d*x+c))-3*f^
3*x^2*exp(I*(d*x+c))-3*exp(I*(d*x+c))*e^2*f-I*d*e^3*exp(I*(d*x+c))-6*e*f^2*x*exp(I*(d*x+c))-I*d*f^3*x^3*exp(I*
(d*x+c))+9*I*d*e*f^2*x^2*exp(3*I*(d*x+c))+4*d*e^3+12*d*e*f^2*x^2+12*d*e^2*f*x+3*d*f^3*x^3*exp(4*I*(d*x+c))-3*I
*f^3*x^2*exp(4*I*(d*x+c))-3*I*e^2*f*exp(4*I*(d*x+c))+3*I*d*e^3*exp(3*I*(d*x+c))+6*e*f^2*x*exp(3*I*(d*x+c))+9*I
*d*e^2*f*x*exp(3*I*(d*x+c))-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))+9*d*e^2*f*x*exp(4*I*(d
*x+c))+9*d*e*f^2*x^2*exp(4*I*(d*x+c))-6*I*e*f^2*x*exp(4*I*(d*x+c))+3*I*d*f^3*x^3*exp(3*I*(d*x+c))-15*d*e*f^2*x
^2*exp(2*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)))/(exp(2*I*(d*x+c))-1)^2/d^2/(ex
p(I*(d*x+c))+I)/a-6/a/d^2*f^3*ln(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+9*I/a/d^2*poly
log(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/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)+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^4-6/a/d^2*f*ln(exp(I*(d*x+c))+I)*e^2-24/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+12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+
c)))+9/2/d^2/a*ln(1-exp(I*(d*x+c)))*c*e^2*f-9/2/d/a*e*f^2*ln(exp(I*(d*x+c))+1)*x^2+9/2/d/a*ln(1-exp(I*(d*x+c))
)*e^2*f*x-9/2/d/a*ln(exp(I*(d*x+c))+1)*e^2*f*x+9/2/d^3/a*e*f^2*c^2*ln(exp(I*(d*x+c))-1)-9/2/d^2/a*e^2*f*c*ln(e
xp(I*(d*x+c))-1)+3/2/d/a*f^3*ln(1-exp(I*(d*x+c)))*x^3+3/2/d^4/a*f^3*ln(1-exp(I*(d*x+c)))*c^3-3/2/d/a*f^3*ln(ex
p(I*(d*x+c))+1)*x^3-9/2/d^3/a*e*f^2*c^2*ln(1-exp(I*(d*x+c)))+9/2/d/a*e*f^2*ln(1-exp(I*(d*x+c)))*x^2+6*I/a/d^3*
f^3*polylog(2,-exp(I*(d*x+c)))*x+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+12*I/a/d*e*f^2*x^2+12*I/a/d^3*c^2*e*f^2-12*I/a/d^3*f^3*c^2*x+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)))-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-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)-8*I/a/d^4*f^3*c^3+4*I/a/d*f^3*x^3-9/2*I/a/d^2*e^2*f*polylog(2,exp(I*
(d*x+c)))+9/2*I/a/d^2*e^2*f*polylog(2,-exp(I*(d*x+c)))+6*I/a/d^3*f^3*polylog(2,exp(I*(d*x+c)))*x-12*f^3*polylo
g(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)+6/a/d^3*e*f^2*c*ln(exp(I*(d*x+c))-1)-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^4-12/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-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)-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*f^3*ln(exp(I*(d*x+c))+1)*x^2

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 12815 vs. \(2 (522) = 1044\).
time = 12.48, size = 12815, normalized size = 21.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*c*e^2*f*((3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)/(a*d*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + a*d*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) - (4*sin(d*x + c)/(cos(d*x + c) + 1) -
sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a*d) + 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/(a*d)) + e^3*((4*sin(d*x
+ c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a - (3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*si
n(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)/(a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^3/(cos(d*x + c)
+ 1)^3) - 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/a) + 8*(48*I*c^2*d*e*f^2 - 16*I*c^3*f^3 - 24*(-I*d^2*e^2*f
+ 2*I*c*d*e*f^2 - I*c^2*f^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(5*d*x + 5*c) + (-I*d^2*e^2*f + 2*I*c*d*e
*f^2 - I*c^2*f^3)*cos(4*d*x + 4*c) + 2*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(3*d*x + 3*c) + 2*(I*d^2*e^2*f -
2*I*c*d*e*f^2 + I*c^2*f^3)*cos(2*d*x + 2*c) - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) + (-I*d^2*e^2*
f + 2*I*c*d*e*f^2 - I*c^2*f^3)*sin(5*d*x + 5*c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(4*d*x + 4*c) + 2*(I*
d^2*e^2*f - 2*I*c*d*e*f^2 + I*c^2*f^3)*sin(3*d*x + 3*c) - 2*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(2*d*x + 2*
c) + (-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 24*(I*
(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c) + ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(5*
d*x + 5*c) + (I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos(4*d*x + 4*c) - 2*((d*x + c)^2*f^3 + 2
*(d*e*f^2 - c*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2*(-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*c
os(2*d*x + 2*c) + ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) + (I*(d*x + c)^2*f^3 + 2*(I*d
*e*f^2 - I*c*f^3)*(d*x + c))*sin(5*d*x + 5*c) - ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(4*d*x +
4*c) + 2*(-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 2*((d*x + c)^2*f^3 + 2*(
d*e*f^2 - c*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*sin(d*x
+ c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) - 6*(-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2
*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f
+ 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c) - ((d*x + c)^3*f^3 + 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 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*
c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*cos(5*d*x + 5*c) + (-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-
3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^
2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c))*cos(4*d*x + 4*c) + 2*
((d*x + c)^3*f^3 + 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 + (3*d*e*f^2 - (3*c - 2)*
f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2
*(I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (3*I*c^2 - 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*f^3 + (3*I*
d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 - 4*I*c + 2*I
)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - ((d*x + c)^3*f^3 + 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 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)
*f^3)*(d*x + c))*cos(d*x + c) + (-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^
3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*
d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c))*sin(5*d*x + 5*c) + ((d*x + c)^3*f^3 + 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 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c
- 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*(I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (3
*I*c^2 - 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*f^3 + (3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c)^
2 + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 - 4*I*c + 2*I)*f^3)*(d*x + c))*sin(3*d*x + 3*c) - 2*(
(d*x + c)^3*f^3 + 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 + (3*d*e*f^2 - (3*c - 2)*f
^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (-
I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d
*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)
*f^3)*(d*x + c))*sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) - 6*(-2*I*d^2*e^2*f + (3*I*c^2 + 4*I*c
+ 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c)*f^3 - (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(5*d*x + 5*c) + (-2*I*d^2*e^2*f + (3...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 7847 vs. \(2 (537) = 1074\).
time = 0.65, size = 7847, normalized size = 13.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*d^3*f^3*x^3 - 6*d^2*f^3*x^2 - 8*(d^3*f^3*x^3 + 3*d^3*f^2*x^2*e + 3*d^3*f*x*e^2 + d^3*e^3)*cos(d*x + c)
^3 + 4*d^3*e^3 - 6*(d^3*f^3*x^3 - d^2*f^3*x^2 + d^3*e^3 + (3*d^3*f*x - d^2*f)*e^2 + (3*d^3*f^2*x^2 - 2*d^2*f^2
*x)*e)*cos(d*x + c)^2 + 6*(d^3*f^3*x^3 + 3*d^3*f^2*x^2*e + 3*d^3*f*x*e^2 + d^3*e^3)*cos(d*x + c) + 3*(-3*I*d^2
*f^3*x^2 + 4*I*d*f^3*x + (3*I*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 + 2*I*(3*d^2*f^2*x - 2*d*f^2
)*e)*cos(d*x + c)^3 - 3*I*d^2*f*e^2 - 2*I*f^3 + (3*I*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 + 2*I
*(3*d^2*f^2*x - 2*d*f^2)*e)*cos(d*x + c)^2 + (-3*I*d^2*f^3*x^2 + 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3 - 2*I*(
3*d^2*f^2*x - 2*d*f^2)*e)*cos(d*x + c) - 2*I*(3*d^2*f^2*x - 2*d*f^2)*e + (-3*I*d^2*f^3*x^2 + 4*I*d*f^3*x - 3*I
*d^2*f*e^2 - 2*I*f^3 + (3*I*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 + 2*I*(3*d^2*f^2*x - 2*d*f^2)*
e)*cos(d*x + c)^2 - 2*I*(3*d^2*f^2*x - 2*d*f^2)*e)*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) + 3*(3*I
*d^2*f^3*x^2 - 4*I*d*f^3*x + (-3*I*d^2*f^3*x^2 + 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3 - 2*I*(3*d^2*f^2*x - 2*
d*f^2)*e)*cos(d*x + c)^3 + 3*I*d^2*f*e^2 + 2*I*f^3 + (-3*I*d^2*f^3*x^2 + 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3
- 2*I*(3*d^2*f^2*x - 2*d*f^2)*e)*cos(d*x + c)^2 + (3*I*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 +
2*I*(3*d^2*f^2*x - 2*d*f^2)*e)*cos(d*x + c) + 2*I*(3*d^2*f^2*x - 2*d*f^2)*e + (3*I*d^2*f^3*x^2 - 4*I*d*f^3*x +
3*I*d^2*f*e^2 + 2*I*f^3 + (-3*I*d^2*f^3*x^2 + 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3 - 2*I*(3*d^2*f^2*x - 2*d*
f^2)*e)*cos(d*x + c)^2 + 2*I*(3*d^2*f^2*x - 2*d*f^2)*e)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) + 2
4*(I*d*f^3*x + (-I*d*f^3*x - I*d*f^2*e)*cos(d*x + c)^3 + I*d*f^2*e + (-I*d*f^3*x - I*d*f^2*e)*cos(d*x + c)^2 +
(I*d*f^3*x + I*d*f^2*e)*cos(d*x + c) + (I*d*f^3*x + I*d*f^2*e + (-I*d*f^3*x - I*d*f^2*e)*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 + (I*d*f^3*x + I*d*f^2*e)*cos(d*x + c)^3 - I*d
*f^2*e + (I*d*f^3*x + I*d*f^2*e)*cos(d*x + c)^2 + (-I*d*f^3*x - I*d*f^2*e)*cos(d*x + c) + (-I*d*f^3*x - I*d*f^
2*e + (I*d*f^3*x + I*d*f^2*e)*cos(d*x + c)^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) + 3*(-3*I*d^
2*f^3*x^2 - 4*I*d*f^3*x + (3*I*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 + 2*I*(3*d^2*f^2*x + 2*d*f^
2)*e)*cos(d*x + c)^3 - 3*I*d^2*f*e^2 - 2*I*f^3 + (3*I*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 + 2*
I*(3*d^2*f^2*x + 2*d*f^2)*e)*cos(d*x + c)^2 + (-3*I*d^2*f^3*x^2 - 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3 - 2*I*
(3*d^2*f^2*x + 2*d*f^2)*e)*cos(d*x + c) - 2*I*(3*d^2*f^2*x + 2*d*f^2)*e + (-3*I*d^2*f^3*x^2 - 4*I*d*f^3*x - 3*
I*d^2*f*e^2 - 2*I*f^3 + (3*I*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3 + 2*I*(3*d^2*f^2*x + 2*d*f^2)
*e)*cos(d*x + c)^2 - 2*I*(3*d^2*f^2*x + 2*d*f^2)*e)*sin(d*x + c))*dilog(-cos(d*x + c) + I*sin(d*x + c)) + 3*(3
*I*d^2*f^3*x^2 + 4*I*d*f^3*x + (-3*I*d^2*f^3*x^2 - 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3 - 2*I*(3*d^2*f^2*x +
2*d*f^2)*e)*cos(d*x + c)^3 + 3*I*d^2*f*e^2 + 2*I*f^3 + (-3*I*d^2*f^3*x^2 - 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f
^3 - 2*I*(3*d^2*f^2*x + 2*d*f^2)*e)*cos(d*x + c)^2 + (3*I*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*I*d^2*f*e^2 + 2*I*f^3
+ 2*I*(3*d^2*f^2*x + 2*d*f^2)*e)*cos(d*x + c) + 2*I*(3*d^2*f^2*x + 2*d*f^2)*e + (3*I*d^2*f^3*x^2 + 4*I*d*f^3*x
+ 3*I*d^2*f*e^2 + 2*I*f^3 + (-3*I*d^2*f^3*x^2 - 4*I*d*f^3*x - 3*I*d^2*f*e^2 - 2*I*f^3 - 2*I*(3*d^2*f^2*x + 2*
d*f^2)*e)*cos(d*x + c)^2 + 2*I*(3*d^2*f^2*x + 2*d*f^2)*e)*sin(d*x + c))*dilog(-cos(d*x + c) - I*sin(d*x + c))
+ 6*(2*d^3*f*x - d^2*f)*e^2 + 12*(d^3*f^2*x^2 - d^2*f^2*x)*e - 3*(d^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x - (d
^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x + d^3*e^3 + (3*d^3*f*x + 2*d^2*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^2*x +
2*d*f^2)*e)*cos(d*x + c)^3 + d^3*e^3 - (d^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x + d^3*e^3 + (3*d^3*f*x + 2*d^2
*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^2*x + 2*d*f^2)*e)*cos(d*x + c)^2 + (d^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x
+ d^3*e^3 + (3*d^3*f*x + 2*d^2*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^2*x + 2*d*f^2)*e)*cos(d*x + c) + (3*d^3*f*x
+ 2*d^2*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^2*x + 2*d*f^2)*e + (d^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x + d^3*e^
3 - (d^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x + d^3*e^3 + (3*d^3*f*x + 2*d^2*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^
2*x + 2*d*f^2)*e)*cos(d*x + c)^2 + (3*d^3*f*x + 2*d^2*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^2*x + 2*d*f^2)*e)*sin(
d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + 1) - 12*(c^2*f^3 - 2*c*d*f^2*e - (c^2*f^3 - 2*c*d*f^2*e + d^2*f*
e^2)*cos(d*x + c)^3 + d^2*f*e^2 - (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*cos(d*x + c)^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 - (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*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 + 2*d^2*f^3*x^2 + 2*d*f^3*x - (d^
3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*d*f^3*x + d^3*e^3 + (3*d^3*f*x + 2*d^2*f)*e^2 + (3*d^3*f^2*x^2 + 4*d^2*f^2*x + 2
*d*f^2)*e)*cos(d*x + c)^3 + d^3*e^3 - (d^3*f^3*...

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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