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

Optimal. Leaf size=639 \[ -\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 d^2 \sqrt{a^2-b^2}}+\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 d^2 \sqrt{a^2-b^2}}-\frac{2 i b^2 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 d^3 \sqrt{a^2-b^2}}+\frac{2 i b^2 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 d^3 \sqrt{a^2-b^2}}-\frac{2 i b f (e+f x) \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac{2 i b f (e+f x) \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac{2 b f^2 \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac{2 b f^2 \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac{i f^2 \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac{i b^2 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 d \sqrt{a^2-b^2}}+\frac{i b^2 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 d \sqrt{a^2-b^2}}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac{2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac{(e+f x)^2 \cot (c+d x)}{a d}-\frac{i (e+f x)^2}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.20559, antiderivative size = 639, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4535, 4184, 3717, 2190, 2279, 2391, 4183, 2531, 2282, 6589, 3323, 2264} \[ -\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 d^2 \sqrt{a^2-b^2}}+\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 d^2 \sqrt{a^2-b^2}}-\frac{2 i b^2 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 d^3 \sqrt{a^2-b^2}}+\frac{2 i b^2 f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 d^3 \sqrt{a^2-b^2}}-\frac{2 i b f (e+f x) \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac{2 i b f (e+f x) \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac{2 b f^2 \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac{2 b f^2 \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac{i f^2 \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac{i b^2 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 d \sqrt{a^2-b^2}}+\frac{i b^2 (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 d \sqrt{a^2-b^2}}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac{2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac{(e+f x)^2 \cot (c+d x)}{a d}-\frac{i (e+f x)^2}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4535

Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csc[c + d*x]^(n - 1))/(a +
b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4184

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

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

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

Rule 2391

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

Rule 4183

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

Rule 2531

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

Rule 2282

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

Rule 6589

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

Rule 3323

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 2264

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]

Rubi steps

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

Mathematica [A]  time = 11.9213, size = 911, normalized size = 1.43 \[ \frac{i \left (-2 \sqrt{a^2-b^2} d f (e+f x) \text{PolyLog}\left (2,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )+2 \sqrt{a^2-b^2} d f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right )-i \left (\left (2 \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right ) e^2+\sqrt{a^2-b^2} f x (2 e+f x) \left (\log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )-\log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right )\right )\right ) d^2+2 \sqrt{a^2-b^2} f^2 \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )-2 \sqrt{a^2-b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right )\right )\right ) b^2}{a^2 \sqrt{-\left (a^2-b^2\right )^2} d^3}+\frac{-b d^2 x^2 \log \left (1-e^{-i (c+d x)}\right ) f^2+b d^2 x^2 \log \left (1+e^{-i (c+d x)}\right ) f^2+2 b \left (i d x \text{PolyLog}\left (2,-e^{-i (c+d x)}\right )+\text{PolyLog}\left (3,-e^{-i (c+d x)}\right )\right ) f^2-2 i b \left (d x \text{PolyLog}\left (2,e^{-i (c+d x)}\right )-i \text{PolyLog}\left (3,e^{-i (c+d x)}\right )\right ) f^2-2 d (b d e-a f) x \log \left (1-e^{-i (c+d x)}\right ) f+2 d (b d e+a f) x \log \left (1+e^{-i (c+d x)}\right ) f+2 i (b d e+a f) \text{PolyLog}\left (2,-e^{-i (c+d x)}\right ) f+2 i (a f-b d e) \text{PolyLog}\left (2,e^{-i (c+d x)}\right ) f-\frac{2 i a d^2 (e+f x)^2}{-1+e^{2 i c}}+i d e (b d e-2 a f) \left (d x+i \log \left (1-e^{i (c+d x)}\right )\right )+d e (b d e+2 a f) \left (\log \left (1+e^{i (c+d x)}\right )-i d x\right )}{a^2 d^3}+\frac{\csc \left (\frac{c}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\sin \left (\frac{d x}{2}\right ) e^2+2 f x \sin \left (\frac{d x}{2}\right ) e+f^2 x^2 \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 (\sin \left (\frac{d x}{2}\right ) e^2+2 f x \sin \left (\frac{d x}{2}\right ) e+f^2 x^2 \sin \left (\frac{d x}{2}\right )\right )}{2 a d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((-2*I)*a*d^2*(e + f*x)^2)/(-1 + E^((2*I)*c)) - 2*d*f*(b*d*e - a*f)*x*Log[1 - E^((-I)*(c + d*x))] - b*d^2*f^2
*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(b*d*e + a*f)*x*Log[1 + E^((-I)*(c + d*x))] + b*d^2*f^2*x^2*Log[1 + E
^((-I)*(c + d*x))] + I*d*e*(b*d*e - 2*a*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d*e*(b*d*e + 2*a*f)*((-I)*d*x
+ Log[1 + E^(I*(c + d*x))]) + (2*I)*f*(b*d*e + a*f)*PolyLog[2, -E^((-I)*(c + d*x))] + (2*I)*f*(-(b*d*e) + a*f)
*PolyLog[2, E^((-I)*(c + d*x))] + 2*b*f^2*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*
x))]) - (2*I)*b*f^2*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/(a^2*d^3) + (I*b^
2*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 -
b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e
^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d
*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^
2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(
I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(a^2*Sqrt[-(a^2 - b^2)^2]*d^3) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^2
*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(e^2*Sin[
(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d)

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Maple [F]  time = 2.214, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 4.9288, size = 6947, normalized size = 10.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out