\(\int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [236]

Optimal result

Integrand size = 28, antiderivative size = 882 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4} \]

[Out]

-I*(f*x+e)^3/a/d+2*b*(f*x+e)^3*arctanh(exp(I*(d*x+c)))/a^2/d-(f*x+e)^3*cot(d*x+c)/a/d+3*f*(f*x+e)^2*ln(1-exp(2
*I*(d*x+c)))/a/d^2-3*I*b*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a^2/d^2-3*I*f^2*(f*x+e)*polylog(2,exp(2*I*(d*x
+c)))/a/d^3-6*I*b^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d^3/(a^2-b^2)^(1/2)+6*b*
f^2*(f*x+e)*polylog(3,-exp(I*(d*x+c)))/a^2/d^3-6*b*f^2*(f*x+e)*polylog(3,exp(I*(d*x+c)))/a^2/d^3+3/2*f^3*polyl
og(3,exp(2*I*(d*x+c)))/a/d^4+6*I*b*f^3*polylog(4,-exp(I*(d*x+c)))/a^2/d^4+3*I*b*f*(f*x+e)^2*polylog(2,exp(I*(d
*x+c)))/a^2/d^2+I*b^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d/(a^2-b^2)^(1/2)-6*I*b*f^3*p
olylog(4,exp(I*(d*x+c)))/a^2/d^4-3*b^2*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d^2/(
a^2-b^2)^(1/2)+3*b^2*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d^2/(a^2-b^2)^(1/2)-I*b
^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d/(a^2-b^2)^(1/2)+6*I*b^2*f^2*(f*x+e)*polylog(3,
I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d^3/(a^2-b^2)^(1/2)+6*b^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2
-b^2)^(1/2)))/a^2/d^4/(a^2-b^2)^(1/2)-6*b^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d^4/(a^2
-b^2)^(1/2)

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {4631, 4269, 3798, 2221, 2611, 2320, 6724, 4268, 6744, 3404, 2296} \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right ) f^3}{2 a d^4}+\frac {6 i b \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right ) f^3}{a^2 d^4}-\frac {6 i b \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right ) f^3}{a^2 d^4}+\frac {6 b^2 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) f^3}{a^2 \sqrt {a^2-b^2} d^4}-\frac {6 b^2 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) f^3}{a^2 \sqrt {a^2-b^2} d^4}-\frac {3 i (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) f^2}{a d^3}+\frac {6 b (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) f^2}{a^2 d^3}-\frac {6 b (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) f^2}{a^2 d^3}-\frac {6 i b^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) f^2}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) f^2}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right ) f}{a d^2}-\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) f}{a^2 d^2}+\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) f}{a^2 d^2}-\frac {3 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) f}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) f}{a^2 \sqrt {a^2-b^2} d^2}-\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d} \]

[In]

Int[((e + f*x)^3*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

((-I)*(e + f*x)^3)/(a*d) + (2*b*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a^2*d) - ((e + f*x)^3*Cot[c + d*x])/(a*
d) - (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (I*b^2
*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (3*f*(e + f*x)^2*
Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((3*I)*b*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + ((3*I
)*b*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)
))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(
a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^
3) + (6*b*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x)
)])/(a^2*d^3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^
2 - b^2]*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^
2 - b^2]*d^3) + (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) + ((6*I)*b*f^3*PolyLog[4, -E^(I*(c + d*x))])
/(a^2*d^4) - ((6*I)*b*f^3*PolyLog[4, E^(I*(c + d*x))])/(a^2*d^4) + (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))
/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^4) - (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2
- b^2])])/(a^2*Sqrt[a^2 - b^2]*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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

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

Rule 2611

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

Rule 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 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 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*} \text {integral}& = \frac {\int (e+f x)^3 \csc ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {b \int (e+f x)^3 \csc (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3}{a+b \sin (c+d x)} \, dx}{a^2}+\frac {(3 f) \int (e+f x)^2 \cot (c+d x) \, dx}{a d} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {\left (2 b^2\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2}-\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 b f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a^2 d} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{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 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {\left (2 i b^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 \sqrt {a^2-b^2}}+\frac {\left (2 i b^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 \sqrt {a^2-b^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 (6 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a^2 d^2} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}+\frac {\left (3 i b^2 f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 \sqrt {a^2-b^2} d}-\frac {\left (3 i b^2 f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 \sqrt {a^2-b^2} d}+\frac {\left (3 i f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx}{a^2 d^3} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 \sqrt {a^2-b^2} d^2}-\frac {\left (6 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 \sqrt {a^2-b^2} d^2}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^4}-\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^4} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {\left (6 i b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 \sqrt {a^2-b^2} d^3}-\frac {\left (6 i b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 \sqrt {a^2-b^2} d^3} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 \sqrt {a^2-b^2} d^4}-\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 \sqrt {a^2-b^2} d^4} \\ & = -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 9.07 (sec) , antiderivative size = 1735, normalized size of antiderivative = 1.97 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i d^3 e^2 (b d e-3 a f) x-i d^3 e^2 (b d e+3 a f) x-\frac {2 i a d^3 (e+f x)^3}{-1+e^{2 i c}}-3 d^2 e f (b d e-2 a f) x \log \left (1-e^{-i (c+d x)}\right )-3 d^2 f^2 (b d e-a f) x^2 \log \left (1-e^{-i (c+d x)}\right )-b d^3 f^3 x^3 \log \left (1-e^{-i (c+d x)}\right )+3 d^2 e f (b d e+2 a f) x \log \left (1+e^{-i (c+d x)}\right )+3 d^2 f^2 (b d e+a f) x^2 \log \left (1+e^{-i (c+d x)}\right )+b d^3 f^3 x^3 \log \left (1+e^{-i (c+d x)}\right )-d^2 e^2 (b d e-3 a f) \log \left (1-e^{i (c+d x)}\right )+d^2 e^2 (b d e+3 a f) \log \left (1+e^{i (c+d x)}\right )+3 i d e f (b d e+2 a f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+6 i d f^2 (b d e+a f) x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+3 i b d^2 f^3 x^2 \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-3 i d e f (b d e-2 a f) \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-6 i d f^2 (b d e-a f) x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-3 i b d^2 f^3 x^2 \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+6 f^2 (b d e+a f) \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )+6 b d f^3 x \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )+6 f^2 (-b d e+a f) \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )-6 b d f^3 x \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )-6 i b f^3 \operatorname {PolyLog}\left (4,-e^{-i (c+d x)}\right )+6 i b f^3 \operatorname {PolyLog}\left (4,e^{-i (c+d x)}\right )}{a^2 d^4}+\frac {b^2 \left (2 \sqrt {-a^2+b^2} d^3 e^3 \arctan \left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+\sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-\sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-3 i \sqrt {a^2-b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+3 i \sqrt {a^2-b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+6 \sqrt {a^2-b^2} d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+6 \sqrt {a^2-b^2} d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-6 \sqrt {a^2-b^2} d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-6 \sqrt {a^2-b^2} d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+6 i \sqrt {a^2-b^2} f^3 \operatorname {PolyLog}\left (4,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-6 i \sqrt {a^2-b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )\right )}{a^2 \sqrt {-\left (a^2-b^2\right )^2} d^4}+\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \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 )}{2 a d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \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 )}{2 a d} \]

[In]

Integrate[((e + f*x)^3*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

(I*d^3*e^2*(b*d*e - 3*a*f)*x - I*d^3*e^2*(b*d*e + 3*a*f)*x - ((2*I)*a*d^3*(e + f*x)^3)/(-1 + E^((2*I)*c)) - 3*
d^2*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c + d*x))] - 3*d^2*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))
] - b*d^3*f^3*x^3*Log[1 - E^((-I)*(c + d*x))] + 3*d^2*e*f*(b*d*e + 2*a*f)*x*Log[1 + E^((-I)*(c + d*x))] + 3*d^
2*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I)*(c + d*x))] + b*d^3*f^3*x^3*Log[1 + E^((-I)*(c + d*x))] - d^2*e^2*(b*d
*e - 3*a*f)*Log[1 - E^(I*(c + d*x))] + d^2*e^2*(b*d*e + 3*a*f)*Log[1 + E^(I*(c + d*x))] + (3*I)*d*e*f*(b*d*e +
2*a*f)*PolyLog[2, -E^((-I)*(c + d*x))] + (6*I)*d*f^2*(b*d*e + a*f)*x*PolyLog[2, -E^((-I)*(c + d*x))] + (3*I)*
b*d^2*f^3*x^2*PolyLog[2, -E^((-I)*(c + d*x))] - (3*I)*d*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))] - (
6*I)*d*f^2*(b*d*e - a*f)*x*PolyLog[2, E^((-I)*(c + d*x))] - (3*I)*b*d^2*f^3*x^2*PolyLog[2, E^((-I)*(c + d*x))]
+ 6*f^2*(b*d*e + a*f)*PolyLog[3, -E^((-I)*(c + d*x))] + 6*b*d*f^3*x*PolyLog[3, -E^((-I)*(c + d*x))] + 6*f^2*(
-(b*d*e) + a*f)*PolyLog[3, E^((-I)*(c + d*x))] - 6*b*d*f^3*x*PolyLog[3, E^((-I)*(c + d*x))] - (6*I)*b*f^3*Poly
Log[4, -E^((-I)*(c + d*x))] + (6*I)*b*f^3*PolyLog[4, E^((-I)*(c + d*x))])/(a^2*d^4) + (b^2*(2*Sqrt[-a^2 + b^2]
*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c
+ d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a +
Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*
Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^
2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c +
d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-
I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sq
rt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*
Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^
2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I
*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + S
qrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(a
^2*Sqrt[-(a^2 - b^2)^2]*d^4) + (Csc[c/2]*Csc[c/2 + (d*x)/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]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/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]))/(2*a*d)

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]

[In]

int((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)

[Out]

int((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   Too many variables

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

int((e + f*x)^3/(sin(c + d*x)^2*(a + b*sin(c + d*x))),x)

[Out]

\text{Hanged}