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

Optimal. Leaf size=882 \[ -\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4} \]

[Out]

-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)+2*b*(f*x+e)^3*a
rctanh(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-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*polylog(4,exp(I*(d*x+c)))/a^2/d^4+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*polylog(3,exp(2*I*(d*x+c)))/a/d^4-3*I*f^2*(f*x+e)*poly
log(2,exp(2*I*(d*x+c)))/a/d^3+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)+3*I*b*f*(f*x+e)^2*polylog(2,exp(I*(d*x+c)))/a^2/d^2-I*(f*x+e)^3/a/d-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)+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*I*b*f^3*polylog(4,-exp(I*(d*x+c)))/a^2/d^4+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]
time = 0.97, antiderivative size = 882, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {4631, 4269, 3798, 2221, 2611, 2320, 6724, 4268, 6744, 3404, 2296} \begin {gather*} \frac {3 \text {PolyLog}\left (3,e^{2 i (c+d x)}\right ) f^3}{2 a d^4}+\frac {6 i b \text {PolyLog}\left (4,-e^{i (c+d x)}\right ) f^3}{a^2 d^4}-\frac {6 i b \text {PolyLog}\left (4,e^{i (c+d x)}\right ) f^3}{a^2 d^4}+\frac {6 b^2 \text {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 \text {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) \text {PolyLog}\left (2,e^{2 i (c+d x)}\right ) f^2}{a d^3}+\frac {6 b (e+f x) \text {PolyLog}\left (3,-e^{i (c+d x)}\right ) f^2}{a^2 d^3}-\frac {6 b (e+f x) \text {PolyLog}\left (3,e^{i (c+d x)}\right ) f^2}{a^2 d^3}-\frac {6 i b^2 (e+f x) \text {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) \text {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 \text {PolyLog}\left (2,-e^{i (c+d x)}\right ) f}{a^2 d^2}+\frac {3 i b (e+f x)^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right ) f}{a^2 d^2}-\frac {3 b^2 (e+f x)^2 \text {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 \text {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 \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (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) \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \text {Li}_2\left (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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 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 {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (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 \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 b f^3\right ) \int \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \text {Li}_3\left (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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (\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 \text {Li}_2\left (\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) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (\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) \text {Li}_2\left (\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 {\text {Li}_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 {\text {Li}_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 {\text {Li}_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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (\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 \text {Li}_2\left (\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) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (\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) \text {Li}_3\left (\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 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}+\frac {\left (6 i b^2 f^3\right ) \int \text {Li}_3\left (\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 \text {Li}_3\left (\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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (\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 \text {Li}_2\left (\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) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (\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) \text {Li}_3\left (\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 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}+\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\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 {\text {Li}_3\left (\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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (\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 \text {Li}_2\left (\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) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (\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) \text {Li}_3\left (\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 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \text {Li}_4\left (\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 \text {Li}_4\left (\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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2452\) vs. \(2(882)=1764\).
time = 41.17, size = 2452, normalized size = 2.78 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((b*e^3*Log[Tan[(c + d*x)/2]])/(a^2*d)) - (3*b*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))] - PolyLog[2, E^(I*(c + d*x))])))/(a^2*d^2
) - (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)) + (6*b*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^2*d^3) - (b*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*Si
n[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
]]))/(a^2*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)) + (I*b^2*((3*I)*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/
(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + (3*I)*Sqrt[a^2 - b^2
]*d^3*e*f^2*x^2*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Si
n[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + I*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2
*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + 3*Sqrt[a
^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^
2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) - 3*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (
b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*
(Cos[c] + I*Sin[c]) + (6*I)*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*
Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) + (6*I)*Sqrt[a^2 - b^2]*d*
f^3*x*PolyLog[3, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])
^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) - 6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*
x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) + 6*Sqrt[a^2 - b
^2]*f^3*PolyLog[4, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[
c])^2] + a*Sin[c])]*(Cos[c] + I*Sin[c]) + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*(Cos[2*c + d*x] + I*Sin[2*c
+ d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + 3*Sq
rt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)
*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*(Cos[2*c
+ d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c
] + Sin[c]) + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sq
rt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog
[3, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[
c])]*((-I)*Cos[c] + Sin[c]) - (2*I)*d^3*e^3*ArcTan[(b*Cos[c + d*x] + I*(a + b*Sin[c + d*x]))/Sqrt[a^2 - b^2]]*
Sqrt[(-a^2 + b^2)*(Cos[2*c] + I*Sin[2*c])]))/(a^2*Sqrt[a^2 - b^2]*d^4*Sqrt[(-a^2 + b^2)*(Cos[2*c] + I*Sin[2*c]
)]) + (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) - (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 [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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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. 4553 vs. \(2 (786) = 1572\).
time = 0.77, size = 4553, normalized size = 5.16 \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)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(-6*I*b^3*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*
b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) + 6*I*b^3*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(I*a*
cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) +
6*I*b^3*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin
(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) - 6*I*b^3*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-I*a*cos(
d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) - 6*I*
(a^2*b - b^3)*f^3*polylog(4, cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 6*I*(a^2*b - b^3)*f^3*polylog(4, co
s(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - 6*I*(a^2*b - b^3)*f^3*polylog(4, -cos(d*x + c) + I*sin(d*x + c))*s
in(d*x + c) + 6*I*(a^2*b - b^3)*f^3*polylog(4, -cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 3*(I*b^3*d^2*f^3
*x^2 + 2*I*b^3*d^2*f^2*x*e + I*b^3*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x + c) - a*sin(d*x + c)
+ (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 3*(-I*b^3*d^2*f^3*x^2
- 2*I*b^3*d^2*f^2*x*e - I*b^3*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x + c) - a*sin(d*x + c) - (b*
cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 3*(-I*b^3*d^2*f^3*x^2 - 2*I
*b^3*d^2*f^2*x*e - I*b^3*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(
d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 3*(I*b^3*d^2*f^3*x^2 + 2*I*b^3*
d^2*f^2*x*e + I*b^3*d^2*f*e^2)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x +
c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) - (b^3*c^3*f^3 - 3*b^3*c^2*d*f^2*e + 3
*b^3*c*d^2*f*e^2 - b^3*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(
a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) - (b^3*c^3*f^3 - 3*b^3*c^2*d*f^2*e + 3*b^3*c*d^2*f*e^2 - b^3*d^3*e^3)*sq
rt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a)*sin(d*x +
c) + (b^3*c^3*f^3 - 3*b^3*c^2*d*f^2*e + 3*b^3*c*d^2*f*e^2 - b^3*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)*log(-2*b*cos(
d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) + (b^3*c^3*f^3 - 3*b^3*c^2*d*
f^2*e + 3*b^3*c*d^2*f*e^2 - b^3*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2
*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a)*sin(d*x + c) - (b^3*d^3*f^3*x^3 + b^3*c^3*f^3 + 3*(b^3*d^3*f*x + b^3*c*d^2*
f)*e^2 + 3*(b^3*d^3*f^2*x^2 - b^3*c^2*d*f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c)
+ (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + (b^3*d^3*f^3*x^3 + b^3*c^
3*f^3 + 3*(b^3*d^3*f*x + b^3*c*d^2*f)*e^2 + 3*(b^3*d^3*f^2*x^2 - b^3*c^2*d*f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(
-(I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d
*x + c) - (b^3*d^3*f^3*x^3 + b^3*c^3*f^3 + 3*(b^3*d^3*f*x + b^3*c*d^2*f)*e^2 + 3*(b^3*d^3*f^2*x^2 - b^3*c^2*d*
f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(-(-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*
sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + (b^3*d^3*f^3*x^3 + b^3*c^3*f^3 + 3*(b^3*d^3*f*x + b^3*c*d^2*f)*e
^2 + 3*(b^3*d^3*f^2*x^2 - b^3*c^2*d*f^2)*e)*sqrt(-(a^2 - b^2)/b^2)*log(-(-I*a*cos(d*x + c) - a*sin(d*x + c) -
(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + 6*(b^3*d*f^3*x + b^3*d*f^2*e
)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*
sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) - 6*(b^3*d*f^3*x + b^3*d*f^2*e)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(I*
a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c)
+ 6*(b^3*d*f^3*x + b^3*d*f^2*e)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(-I*a*cos(d*x + c) + a*sin(d*x + c) + (b*co
s(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) - 6*(b^3*d*f^3*x + b^3*d*f^2*e)*sqrt(-(
a^2 - b^2)/b^2)*polylog(3, -(-I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a
^2 - b^2)/b^2))/b)*sin(d*x + c) + 3*(I*(a^2*b - b^3)*d^2*f^3*x^2 - 2*I*(a^3 - a*b^2)*d*f^3*x + I*(a^2*b - b^3)
*d^2*f*e^2 + 2*I*((a^2*b - b^3)*d^2*f^2*x - (a^3 - a*b^2)*d*f^2)*e)*dilog(cos(d*x + c) + I*sin(d*x + c))*sin(d
*x + c) + 3*(-I*(a^2*b - b^3)*d^2*f^3*x^2 + 2*I*(a^3 - a*b^2)*d*f^3*x - I*(a^2*b - b^3)*d^2*f*e^2 - 2*I*((a^2*
b - b^3)*d^2*f^2*x - (a^3 - a*b^2)*d*f^2)*e)*dilog(cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 3*(I*(a^2*b -
b^3)*d^2*f^3*x^2 + 2*I*(a^3 - a*b^2)*d*f^3*x + I*(a^2*b - b^3)*d^2*f*e^2 + 2*I*((a^2*b - b^3)*d^2*f^2*x + (a^
3 - a*b^2)*d*f^2)*e)*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 3*(-I*(a^2*b - b^3)*d^2*f^3*x^2 - 2*
I*(a^3 - a*b^2)*d*f^3*x - I*(a^2*b - b^3)*d^2*f...

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)^2/(a+b*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)^2*(a + b*sin(c + d*x))),x)

[Out]

\text{Hanged}

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