\(\int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [204]

Optimal result

Integrand size = 28, antiderivative size = 327 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 i (e+f x)^2}{a d}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3} \]

[Out]

-2*I*(f*x+e)^2/a/d+2*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d-(f*x+e)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d-(f*x+e)^2*c
ot(d*x+c)/a/d+4*f*(f*x+e)*ln(1-I*exp(I*(d*x+c)))/a/d^2+2*f*(f*x+e)*ln(1-exp(2*I*(d*x+c)))/a/d^2-2*I*f*(f*x+e)*
polylog(2,-exp(I*(d*x+c)))/a/d^2-4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3+2*I*f*(f*x+e)*polylog(2,exp(I*(d*x+
c)))/a/d^2-I*f^2*polylog(2,exp(2*I*(d*x+c)))/a/d^3+2*f^2*polylog(3,-exp(I*(d*x+c)))/a/d^3-2*f^2*polylog(3,exp(
I*(d*x+c)))/a/d^3

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {4631, 4269, 3798, 2221, 2317, 2438, 4268, 2611, 2320, 6724, 3399} \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {2 i (e+f x)^2}{a d} \]

[In]

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

[Out]

((-2*I)*(e + f*x)^2)/(a*d) + (2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) - ((e + f*x)^2*Cot[c/2 + Pi/4 + (d
*x)/2])/(a*d) - ((e + f*x)^2*Cot[c + d*x])/(a*d) + (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + (2*f*(
e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((
4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) + ((2*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (I
*f^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (2*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) - (2*f^2*PolyLog[
3, E^(I*(c + d*x))])/(a*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 2317

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

Rule 2320

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

Rule 2438

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

Rule 2611

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

Rule 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3798

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

Rule 4268

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

Rule 4269

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

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(709\) vs. \(2(327)=654\).

Time = 8.04 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.17 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i d^2 e (d e-2 f) x-i d^2 e (d e+2 f) x-\frac {2 i d^2 (e+f x)^2}{-1+e^{2 i c}}-2 d (d e-f) f x \log \left (1-e^{-i (c+d x)}\right )-d^2 f^2 x^2 \log \left (1-e^{-i (c+d x)}\right )+2 d f (d e+f) x \log \left (1+e^{-i (c+d x)}\right )+d^2 f^2 x^2 \log \left (1+e^{-i (c+d x)}\right )-d e (d e-2 f) \log \left (1-e^{i (c+d x)}\right )+d e (d e+2 f) \log \left (1+e^{i (c+d x)}\right )+2 i f (d e+f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+2 i d f^2 x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-2 i (d e-f) f \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-2 i d f^2 x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+2 f^2 \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )-2 f^2 \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )}{a d^3}-\frac {4 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^2 (\cos (c)-i \sin (c))}{2 f}-\frac {(e+f x) \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}\right )}{a d (\cos (c)+i (1+\sin (c)))}+\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {2 \left (e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

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

[Out]

(I*d^2*e*(d*e - 2*f)*x - I*d^2*e*(d*e + 2*f)*x - ((2*I)*d^2*(e + f*x)^2)/(-1 + E^((2*I)*c)) - 2*d*(d*e - f)*f*
x*Log[1 - E^((-I)*(c + d*x))] - d^2*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(d*e + f)*x*Log[1 + E^((-I)*(c
+ d*x))] + d^2*f^2*x^2*Log[1 + E^((-I)*(c + d*x))] - d*e*(d*e - 2*f)*Log[1 - E^(I*(c + d*x))] + d*e*(d*e + 2*
f)*Log[1 + E^(I*(c + d*x))] + (2*I)*f*(d*e + f)*PolyLog[2, -E^((-I)*(c + d*x))] + (2*I)*d*f^2*x*PolyLog[2, -E^
((-I)*(c + d*x))] - (2*I)*(d*e - f)*f*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*d*f^2*x*PolyLog[2, E^((-I)*(c + d
*x))] + 2*f^2*PolyLog[3, -E^((-I)*(c + d*x))] - 2*f^2*PolyLog[3, E^((-I)*(c + d*x))])/(a*d^3) - (4*f*(Cos[c] +
I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I
*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(a*d*(
Cos[c] + I*(1 + Sin[c]))) + (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*Si
n[(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) + (2*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(a*d*(Cos[c/2] + Sin[c/
2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (296 ) = 592\).

Time = 0.45 (sec) , antiderivative size = 984, normalized size of antiderivative = 3.01

method	result	size
risch	\(\text {Expression too large to display}\)	\(984\)

[In]

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

[Out]

-1/d^3/a*c^2*f^2*ln(exp(I*(d*x+c))-1)+1/d/a*f^2*ln(exp(I*(d*x+c))+1)*x^2-1/d/a*f^2*ln(1-exp(I*(d*x+c)))*x^2-2*
(-2*x^2*f^2+I*exp(I*(d*x+c))*f^2*x^2-4*f*e*x+2*I*exp(I*(d*x+c))*e*f*x-2*e^2+I*exp(I*(d*x+c))*e^2+f^2*x^2*exp(2
*I*(d*x+c))+2*e*f*x*exp(2*I*(d*x+c))+e^2*exp(2*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)/(exp(I*(d*x+c))+I)/d/a+2/a/d^2
*e*f*ln(exp(I*(d*x+c))-1)+2/a/d^2*e*f*ln(exp(I*(d*x+c))+1)+2/a/d^2*f^2*ln(1-exp(I*(d*x+c)))*x+2/a/d^2*f^2*ln(e
xp(I*(d*x+c))+1)*x+2/a/d^3*f^2*ln(1-exp(I*(d*x+c)))*c-2/a/d^3*c*f^2*ln(exp(I*(d*x+c))-1)-2*I/a/d^2*e*f*polylog
(2,-exp(I*(d*x+c)))+4*I/a/d^3*f^2*c*arctan(exp(I*(d*x+c)))+2*I/a/d^2*f^2*polylog(2,exp(I*(d*x+c)))*x-2*I/a/d^2
*f^2*polylog(2,-exp(I*(d*x+c)))*x-8*I/a/d^2*f^2*c*x-4*I/a/d^2*e*f*arctan(exp(I*(d*x+c)))+2*I/a/d^2*e*f*polylog
(2,exp(I*(d*x+c)))+2/a/d^2*f*e*ln(1+exp(2*I*(d*x+c)))-8/a/d^2*f*e*ln(exp(I*(d*x+c)))+4/a/d^2*f^2*ln(1-I*exp(I*
(d*x+c)))*x+4/a/d^3*f^2*ln(1-I*exp(I*(d*x+c)))*c-2/a/d^3*f^2*c*ln(1+exp(2*I*(d*x+c)))+8/a/d^3*f^2*c*ln(exp(I*(
d*x+c)))-1/d/a*e^2*ln(exp(I*(d*x+c))-1)+1/d/a*e^2*ln(exp(I*(d*x+c))+1)+1/d^3/a*f^2*ln(1-exp(I*(d*x+c)))*c^2-4*
I/a/d^3*f^2*c^2-2*I/a/d^3*f^2*polylog(2,-exp(I*(d*x+c)))-4*I/a/d*f^2*x^2+2/d/a*e*f*ln(exp(I*(d*x+c))+1)*x-2/d/
a*e*f*ln(1-exp(I*(d*x+c)))*x-2/d^2/a*e*f*ln(1-exp(I*(d*x+c)))*c+2/d^2/a*c*e*f*ln(exp(I*(d*x+c))-1)-2*I*f^2*pol
ylog(2,exp(I*(d*x+c)))/a/d^3-4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3+2*f^2*polylog(3,-exp(I*(d*x+c)))/a/d^3-
2*f^2*polylog(3,exp(I*(d*x+c)))/a/d^3

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. 2539 vs. \(2 (285) = 570\).

Time = 0.38 (sec) , antiderivative size = 2539, normalized size of antiderivative = 7.76 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(2*d^2*f^2*x^2 + 4*d^2*e*f*x + 2*d^2*e^2 - 4*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c)^2 - 2*(d^
2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c) - 2*(-I*d*f^2*x - I*d*e*f + (I*d*f^2*x + I*d*e*f - I*f^2)*cos(
d*x + c)^2 + I*f^2 + (-I*d*f^2*x - I*d*e*f + I*f^2 + (-I*d*f^2*x - I*d*e*f + I*f^2)*cos(d*x + c))*sin(d*x + c)
)*dilog(cos(d*x + c) + I*sin(d*x + c)) - 2*(I*d*f^2*x + I*d*e*f + (-I*d*f^2*x - I*d*e*f + I*f^2)*cos(d*x + c)^
2 - I*f^2 + (I*d*f^2*x + I*d*e*f - I*f^2 + (I*d*f^2*x + I*d*e*f - I*f^2)*cos(d*x + c))*sin(d*x + c))*dilog(cos
(d*x + c) - I*sin(d*x + c)) - 4*(-I*f^2*cos(d*x + c)^2 + I*f^2 + (I*f^2*cos(d*x + c) + I*f^2)*sin(d*x + c))*di
log(I*cos(d*x + c) - sin(d*x + c)) - 4*(I*f^2*cos(d*x + c)^2 - I*f^2 + (-I*f^2*cos(d*x + c) - I*f^2)*sin(d*x +
c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) - 2*(-I*d*f^2*x - I*d*e*f + (I*d*f^2*x + I*d*e*f + I*f^2)*cos(d*x +
c)^2 - I*f^2 + (-I*d*f^2*x - I*d*e*f - I*f^2 + (-I*d*f^2*x - I*d*e*f - I*f^2)*cos(d*x + c))*sin(d*x + c))*dil
og(-cos(d*x + c) + I*sin(d*x + c)) - 2*(I*d*f^2*x + I*d*e*f + (-I*d*f^2*x - I*d*e*f - I*f^2)*cos(d*x + c)^2 +
I*f^2 + (I*d*f^2*x + I*d*e*f + I*f^2 + (I*d*f^2*x + I*d*e*f + I*f^2)*cos(d*x + c))*sin(d*x + c))*dilog(-cos(d*
x + c) - I*sin(d*x + c)) + (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f - (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*(d^2*e*f +
d*f^2)*x)*cos(d*x + c)^2 + 2*(d^2*e*f + d*f^2)*x + (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*(d^2*e*f + d*f^2)*x +
(d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*(d^2*e*f + d*f^2)*x)*cos(d*x + c))*sin(d*x + c))*log(cos(d*x + c) + I*sin
(d*x + c) + 1) + 4*(d*e*f - c*f^2 - (d*e*f - c*f^2)*cos(d*x + c)^2 + (d*e*f - c*f^2 + (d*e*f - c*f^2)*cos(d*x
+ c))*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) + (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f - (d^2*f^2*x^2 +
d^2*e^2 + 2*d*e*f + 2*(d^2*e*f + d*f^2)*x)*cos(d*x + c)^2 + 2*(d^2*e*f + d*f^2)*x + (d^2*f^2*x^2 + d^2*e^2 +
2*d*e*f + 2*(d^2*e*f + d*f^2)*x + (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*(d^2*e*f + d*f^2)*x)*cos(d*x + c))*sin(
d*x + c))*log(cos(d*x + c) - I*sin(d*x + c) + 1) + 4*(d*f^2*x + c*f^2 - (d*f^2*x + c*f^2)*cos(d*x + c)^2 + (d*
f^2*x + c*f^2 + (d*f^2*x + c*f^2)*cos(d*x + c))*sin(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) + 4*(d*f^
2*x + c*f^2 - (d*f^2*x + c*f^2)*cos(d*x + c)^2 + (d*f^2*x + c*f^2 + (d*f^2*x + c*f^2)*cos(d*x + c))*sin(d*x +
c))*log(-I*cos(d*x + c) + sin(d*x + c) + 1) - (d^2*e^2 - 2*(c + 1)*d*e*f + (c^2 + 2*c)*f^2 - (d^2*e^2 - 2*(c +
1)*d*e*f + (c^2 + 2*c)*f^2)*cos(d*x + c)^2 + (d^2*e^2 - 2*(c + 1)*d*e*f + (c^2 + 2*c)*f^2 + (d^2*e^2 - 2*(c +
1)*d*e*f + (c^2 + 2*c)*f^2)*cos(d*x + c))*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) - (
d^2*e^2 - 2*(c + 1)*d*e*f + (c^2 + 2*c)*f^2 - (d^2*e^2 - 2*(c + 1)*d*e*f + (c^2 + 2*c)*f^2)*cos(d*x + c)^2 + (
d^2*e^2 - 2*(c + 1)*d*e*f + (c^2 + 2*c)*f^2 + (d^2*e^2 - 2*(c + 1)*d*e*f + (c^2 + 2*c)*f^2)*cos(d*x + c))*sin(
d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) - (d^2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)*f^2 - (d^
2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)*f^2 + 2*(d^2*e*f - d*f^2)*x)*cos(d*x + c)^2 + 2*(d^2*e*f - d*f^2)*x + (d^2
*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)*f^2 + 2*(d^2*e*f - d*f^2)*x + (d^2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)*f^2 +
2*(d^2*e*f - d*f^2)*x)*cos(d*x + c))*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + 1) + 4*(d*e*f - c*f^2
- (d*e*f - c*f^2)*cos(d*x + c)^2 + (d*e*f - c*f^2 + (d*e*f - c*f^2)*cos(d*x + c))*sin(d*x + c))*log(-cos(d*x +
c) + I*sin(d*x + c) + I) - (d^2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)*f^2 - (d^2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c
)*f^2 + 2*(d^2*e*f - d*f^2)*x)*cos(d*x + c)^2 + 2*(d^2*e*f - d*f^2)*x + (d^2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)
*f^2 + 2*(d^2*e*f - d*f^2)*x + (d^2*f^2*x^2 + 2*c*d*e*f - (c^2 + 2*c)*f^2 + 2*(d^2*e*f - d*f^2)*x)*cos(d*x + c
))*sin(d*x + c))*log(-cos(d*x + c) - I*sin(d*x + c) + 1) + 2*(f^2*cos(d*x + c)^2 - f^2 - (f^2*cos(d*x + c) + f
^2)*sin(d*x + c))*polylog(3, cos(d*x + c) + I*sin(d*x + c)) + 2*(f^2*cos(d*x + c)^2 - f^2 - (f^2*cos(d*x + c)
+ f^2)*sin(d*x + c))*polylog(3, cos(d*x + c) - I*sin(d*x + c)) - 2*(f^2*cos(d*x + c)^2 - f^2 - (f^2*cos(d*x +
c) + f^2)*sin(d*x + c))*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) - 2*(f^2*cos(d*x + c)^2 - f^2 - (f^2*cos(d*
x + c) + f^2)*sin(d*x + c))*polylog(3, -cos(d*x + c) - I*sin(d*x + c)) - 2*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^
2 + 2*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c))*sin(d*x + c))/(a*d^3*cos(d*x + c)^2 - a*d^3 - (a*d^3
*cos(d*x + c) + a*d^3)*sin(d*x + c))

Sympy [F]

\[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{2} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

(Integral(e**2*csc(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f**2*x**2*csc(c + d*x)**2/(sin(c + d*x) + 1),
x) + Integral(2*e*f*x*csc(c + d*x)**2/(sin(c + d*x) + 1), x))/a

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}