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

Optimal. Leaf size=566 \[ -\frac{2 i f \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^2}-\frac{2 i f \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^2}+\frac{2 f^2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^3}+\frac{2 f^2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^3}-\frac{i f (e+f x) \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}+\frac{\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d}+\frac{\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d}-\frac{i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}+\frac{(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac{i (e+f x)^3}{3 a f}-\frac{2 f (e+f x) \cos (c+d x)}{b d^2}+\frac{2 f^2 \sin (c+d x)}{b d^3}-\frac{(e+f x)^2 \sin (c+d x)}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.12232, antiderivative size = 566, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.382, Rules used = {4543, 4408, 4404, 3310, 3717, 2190, 2531, 2282, 6589, 4525, 3296, 2637, 4519} \[ -\frac{2 i f \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^2}-\frac{2 i f \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^2}+\frac{2 f^2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^3}+\frac{2 f^2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^3}-\frac{i f (e+f x) \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}+\frac{\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d}+\frac{\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d}-\frac{i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}+\frac{(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac{i (e+f x)^3}{3 a f}-\frac{2 f (e+f x) \cos (c+d x)}{b d^2}+\frac{2 f^2 \sin (c+d x)}{b d^3}-\frac{(e+f x)^2 \sin (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4543

Int[(Cos[(c_.) + (d_.)*(x_)]^(p_.)*Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin
[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Dist[b/a
, Int[((e + f*x)^m*Cos[c + d*x]^(p + 1)*Cot[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] && IGtQ[p, 0]

Rule 4408

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4404

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c +
d*x)^m*Sin[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

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 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 4525

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

Rule 3296

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rubi steps

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

Mathematica [B]  time = 9.36273, size = 1834, normalized size = 3.24 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(E^(I*c)*f^2*Csc[c]*((2*d^3*x^3)/E^((2*I)*c) + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 - E^((-I)*(c + d*x))] +
(3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 + E^((-I)*(c + d*x))] - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, -E^((-I)
*(c + d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, E^((-I)
*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(6*a*d^3) + ((a^2 - b^2)*((-12*I)*d^3*e^2*E^((
2*I)*c)*x - (12*I)*d^3*e*E^((2*I)*c)*f*x^2 - (4*I)*d^3*E^((2*I)*c)*f^2*x^3 - (6*I)*d^2*e^2*ArcTan[(2*a*E^(I*(c
+ d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (6*I)*d^2*e^2*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^
((2*I)*(c + d*x))))] - 3*d^2*e^2*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 3*d^2*e^2
*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 12*d^2*e*f*x*Log[1 + (b*E^(I*
(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c
+ d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^
(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c
) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 +
b^2)*E^((2*I)*c)])] + 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*
E^((2*I)*c)])] - 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] +
6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12
*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2
*I)*c)])] - (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a
^2 + b^2)*E^((2*I)*c)]))] - 12*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I
)*c)])] + 12*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])
] - 12*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*E^((2*I)*c
)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(6*a*b^2*d^3*(-1 +
E^((2*I)*c))) + (a*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Cos[c]*Csc[c/2]*Sec[c/2])/(6*b^2) - (Cos[d*x]*(2*d*e*f*Cos[c
] + 2*d*f^2*x*Cos[c] + d^2*e^2*Sin[c] - 2*f^2*Sin[c] + 2*d^2*e*f*x*Sin[c] + d^2*f^2*x^2*Sin[c]))/(b*d^3) + (e^
2*Csc[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d*(Cos[c]^2 + Sin[c]^2)) - ((d^2*
e^2*Cos[c] - 2*f^2*Cos[c] + 2*d^2*e*f*x*Cos[c] + d^2*f^2*x^2*Cos[c] - 2*d*e*f*Sin[c] - 2*d*f^2*x*Sin[c])*Sin[d
*x])/(b*d^3) - (e*f*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^2*Sqrt[Sec[c]^2*(Cos[c]^2 + Sin[c]^2)])

________________________________________________________________________________________

Maple [F]  time = 3.634, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\cot \left ( dx+c \right ) }{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*cos(d*x+c)^2*cot(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

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*cos(d*x+c)^2*cot(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 3.79429, size = 5391, normalized size = 9.52 \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*cos(d*x+c)^2*cot(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(2*b^2*f^2*polylog(3, cos(d*x + c) + I*sin(d*x + c)) + 2*b^2*f^2*polylog(3, cos(d*x + c) - I*sin(d*x + c))
+ 2*b^2*f^2*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) + 2*b^2*f^2*polylog(3, -cos(d*x + c) - I*sin(d*x + c))
+ 2*(a^2 - b^2)*f^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) + 2*(a^2 - b^2)*f^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) + 2*(a^2 - b^2)*f^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) + 2*(a^2 - b^2)*f^2*polyl
og(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) - 4
*(a*b*d*f^2*x + a*b*d*e*f)*cos(d*x + c) + (2*I*(a^2 - b^2)*d*f^2*x + 2*I*(a^2 - b^2)*d*e*f)*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) +
(2*I*(a^2 - b^2)*d*f^2*x + 2*I*(a^2 - b^2)*d*e*f)*dilog(-1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*co
s(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (-2*I*(a^2 - b^2)*d*f^2*x - 2*I*(a^2 - b
^2)*d*e*f)*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) + (-2*I*(a^2 - b^2)*d*f^2*x - 2*I*(a^2 - b^2)*d*e*f)*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) + (-2*I*b^
2*d*f^2*x - 2*I*b^2*d*e*f)*dilog(cos(d*x + c) + I*sin(d*x + c)) + (2*I*b^2*d*f^2*x + 2*I*b^2*d*e*f)*dilog(cos(
d*x + c) - I*sin(d*x + c)) + (2*I*b^2*d*f^2*x + 2*I*b^2*d*e*f)*dilog(-cos(d*x + c) + I*sin(d*x + c)) + (-2*I*b
^2*d*f^2*x - 2*I*b^2*d*e*f)*dilog(-cos(d*x + c) - I*sin(d*x + c)) + ((a^2 - b^2)*d^2*e^2 - 2*(a^2 - b^2)*c*d*e
*f + (a^2 - b^2)*c^2*f^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) + ((
a^2 - b^2)*d^2*e^2 - 2*(a^2 - b^2)*c*d*e*f + (a^2 - b^2)*c^2*f^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) + ((a^2 - b^2)*d^2*e^2 - 2*(a^2 - b^2)*c*d*e*f + (a^2 - b^2)*c^2*f^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) + ((a^2 - b^2)*d^2*e^2 - 2*(a^2 -
b^2)*c*d*e*f + (a^2 - b^2)*c^2*f^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) + ((a^2 - b^2)*d^2*f^2*x^2 + 2*(a^2 - b^2)*d^2*e*f*x + 2*(a^2 - b^2)*c*d*e*f - (a^2 - b^2)*c^2*f^2)*lo
g(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) + ((a^2 - b^2)*d^2*f^2*x^2 + 2*(a^2 - b^2)*d^2*e*f*x + 2*(a^2 - b^2)*c*d*e*f - (a^2 - b^2)*c^2*f^2)*lo
g(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) + ((a^2 - b^2)*d^2*f^2*x^2 + 2*(a^2 - b^2)*d^2*e*f*x + 2*(a^2 - b^2)*c*d*e*f - (a^2 - b^2)*c^2*f^2)*lo
g(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) + ((a^2 - b^2)*d^2*f^2*x^2 + 2*(a^2 - b^2)*d^2*e*f*x + 2*(a^2 - b^2)*c*d*e*f - (a^2 - b^2)*c^2*f^2)*l
og(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) + (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(cos(d*x + c) + I*sin(d*x + c) + 1) + (b^2*d^
2*f^2*x^2 + 2*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(cos(d*x + c) - I*sin(d*x + c) + 1) + (b^2*d^2*e^2 - 2*b^2*c*d*e
*f + b^2*c^2*f^2)*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) + (b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f
^2)*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) + (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b
^2*c^2*f^2)*log(-cos(d*x + c) + I*sin(d*x + c) + 1) + (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2
*c^2*f^2)*log(-cos(d*x + c) - I*sin(d*x + c) + 1) - 2*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + a*b*d^2*e^2 - 2*a*b
*f^2)*sin(d*x + c))/(a*b^2*d^3)

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

[Out]

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

[Out]

Timed out