\(\int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [237]

Optimal result

Integrand size = 28, antiderivative size = 639 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3} \]

[Out]

-I*(f*x+e)^2/a/d+2*b*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a^2/d-(f*x+e)^2*cot(d*x+c)/a/d+2*f*(f*x+e)*ln(1-exp(2*I
*(d*x+c)))/a/d^2-2*I*b*f*(f*x+e)*polylog(2,-exp(I*(d*x+c)))/a^2/d^2+2*I*b*f*(f*x+e)*polylog(2,exp(I*(d*x+c)))/
a^2/d^2-I*f^2*polylog(2,exp(2*I*(d*x+c)))/a/d^3+2*b*f^2*polylog(3,-exp(I*(d*x+c)))/a^2/d^3-2*b*f^2*polylog(3,e
xp(I*(d*x+c)))/a^2/d^3-I*b^2*(f*x+e)^2*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)+I*b^
2*(f*x+e)^2*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)-2*b^2*f*(f*x+e)*polylog(2,I*b*e
xp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d^2/(a^2-b^2)^(1/2)+2*b^2*f*(f*x+e)*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)-2*I*b^2*f^2*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*I*b^2*f^2*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)

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4631, 4269, 3798, 2221, 2317, 2438, 4268, 2611, 2320, 6724, 3404, 2296} \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^3 \sqrt {a^2-b^2}}+\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 d^3 \sqrt {a^2-b^2}}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^2 \sqrt {a^2-b^2}}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 d^2 \sqrt {a^2-b^2}}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i (e+f x)^2}{a d} \]

[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 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 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 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]

Rubi steps \begin{align*} \text {integral}& = \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 \text {arctanh}\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 \text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \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)^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 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (2 i b f^2\right ) \int \operatorname {PolyLog}\left (2,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 \text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,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 ) \text {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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \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 {2 b^2 f (e+f x) \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 {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}+\frac {\left (2 b^2 f^2\right ) \int \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 (2 b^2 f^2\right ) \int \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 {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \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 {2 b^2 f (e+f x) \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 {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {\left (2 i b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 \text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \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 {2 b^2 f (e+f x) \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 {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 i b^2 f^2 \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 {2 i b^2 f^2 \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} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.91 (sec) , antiderivative size = 868, normalized size of antiderivative = 1.36 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 \left (i d^2 e (b d e-2 a f) x-i d^2 e (b d e+2 a f) x-\frac {2 i a d^2 (e+f x)^2}{-1+e^{2 i c}}-2 d f (b d e-a f) x \log \left (1-e^{-i (c+d x)}\right )-b d^2 f^2 x^2 \log \left (1-e^{-i (c+d x)}\right )+2 d f (b d e+a f) x \log \left (1+e^{-i (c+d x)}\right )+b d^2 f^2 x^2 \log \left (1+e^{-i (c+d x)}\right )-d e (b d e-2 a f) \log \left (1-e^{i (c+d x)}\right )+d e (b d e+2 a f) \log \left (1+e^{i (c+d x)}\right )+2 i f (b d e+a f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+2 i b d f^2 x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+2 i f (-b d e+a f) \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-2 i b d f^2 x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+2 b f^2 \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )-2 b f^2 \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )\right )+\frac {2 i b^2 \left (-2 \sqrt {a^2-b^2} d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+2 \sqrt {a^2-b^2} d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-i \left (d^2 \left (2 \sqrt {-a^2+b^2} e^2 \arctan \left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+\sqrt {a^2-b^2} f x (2 e+f x) \left (\log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-\log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )\right )\right )+2 \sqrt {a^2-b^2} f^2 \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-2 \sqrt {a^2-b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )\right )\right )}{\sqrt {-\left (a^2-b^2\right )^2}}+a d^2 (e+f x)^2 \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+a d^2 (e+f x)^2 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{2 a^2 d^3} \]

[In]

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

[Out]

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

Maple [F]

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

[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)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2972 vs. \(2 (556) = 1112\).

Time = 0.54 (sec) , antiderivative size = 2972, normalized size of antiderivative = 4.65 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(2*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*s
in(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) - 2*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) + 2*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) - 2*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) - 2*(a^2*b - b^
3)*f^2*polylog(3, cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) - 2*(a^2*b - b^3)*f^2*polylog(3, cos(d*x + c) -
I*sin(d*x + c))*sin(d*x + c) + 2*(a^2*b - b^3)*f^2*polylog(3, -cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 2
*(a^2*b - b^3)*f^2*polylog(3, -cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 2*(I*b^3*d*f^2*x + I*b^3*d*e*f)*s
qrt(-(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) + 2*(-I*b^3*d*f^2*x - I*b^3*d*e*f)*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*co
s(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) + 2*(-I*b^3*d*f^2*x - I*b^3*d*e*f)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*c
os(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 2*(I*b^3*d*f^2*x + I*b^3*d*e
*f)*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))*sqr
t(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + (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) + (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) - (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)*lo
g(-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^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) - (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(-(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^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(-(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^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(-(-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^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(
-(-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) + 2*(I*(a^2*b - b^3)*d*f^2*x + I*(a^2*b - b^3)*d*e*f - I*(a^3 - a*b^2)*f^2)*dilog(cos(d*x + c) + I*si
n(d*x + c))*sin(d*x + c) + 2*(-I*(a^2*b - b^3)*d*f^2*x - I*(a^2*b - b^3)*d*e*f + I*(a^3 - a*b^2)*f^2)*dilog(co
s(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 2*(I*(a^2*b - b^3)*d*f^2*x + I*(a^2*b - b^3)*d*e*f + I*(a^3 - a*b^
2)*f^2)*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 2*(-I*(a^2*b - b^3)*d*f^2*x - I*(a^2*b - b^3)*d*e
*f - I*(a^3 - a*b^2)*f^2)*dilog(-cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + ((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) + ((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)
- ((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) - ((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) - ((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)*s
in(d*x + c) - ((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^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))

Sympy [F]

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

[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)

Maxima [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}