\(\int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [338]

Optimal result

Integrand size = 34, antiderivative size = 616 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4639, 4493, 3377, 2717, 4495, 4268, 2317, 2438, 4489, 3391, 3798, 2221, 2611, 2320, 6724, 4621, 4615} \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 f^2 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {2 f^2 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^3}+\frac {2 i f \left (a^2-b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {2 i f \left (a^2-b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b d}+\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {i b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a^2 d^2}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i b (e+f x)^3}{3 a^2 f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}-\frac {(e+f x)^2 \csc (c+d x)}{a d} \]

[In]

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

[Out]

((I/3)*b*(e + f*x)^3)/(a^2*f) + ((I/3)*(a^2 - b^2)*(e + f*x)^3)/(a^2*b*f) - (4*f*(e + f*x)*ArcTanh[E^(I*(c + d
*x))])/(a*d^2) - ((e + f*x)^2*Csc[c + d*x])/(a*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^2*b*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^2*b*d) - (b*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + ((2*I)*f^2*PolyLog[2, -E^(I*(c + d*x))])/
(a*d^3) - ((2*I)*f^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^3) + ((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^2*b*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^2*b*d^2) + (I*b*f*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*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^2*b*d^3) - (2*(a^2 - b^2)*f^2*PolyL
og[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^3) - (b*f^2*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a
^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 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 2717

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

Rule 3377

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 3391

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 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 4489

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 4493

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 4495

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[
(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4615

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]

Rule 4621

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 4639

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 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 \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = -\frac {\int (e+f x)^2 \cos (c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \cos ^2(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {(e+f x)^2 \sin (c+d x)}{a d}+\frac {\int (e+f x)^2 \cos (c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x)^2 \cos (c+d x)}{a+b \sin (c+d x)} \, dx+\frac {(2 f) \int (e+f x) \csc (c+d x) \, dx}{a d}+\frac {(2 f) \int (e+f x) \sin (c+d x) \, dx}{a d} \\ & = \frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{3 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1-e^{2 i (c+d x)}} \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\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 (1-\frac {b^2}{a^2}\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}{a d}+\frac {\left (2 f^2\right ) \int \cos (c+d x) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d^2} \\ & = \frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{3 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (2 \left (1-\frac {b^2}{a^2}\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 (1-\frac {b^2}{a^2}\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 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}-\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}-\frac {\left (2 f^2\right ) \int \cos (c+d x) \, dx}{a d^2} \\ & = \frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{3 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {2 i \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a^2 d^2}-\frac {\left (i b f^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (2 i \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d^2}-\frac {\left (2 i \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d^2} \\ & = \frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{3 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {2 i \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a^2 d^2}-\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a^2 d^3}-\frac {\left (2 \left (1-\frac {b^2}{a^2}\right ) 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 )}{b d^3}-\frac {\left (2 \left (1-\frac {b^2}{a^2}\right ) 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 )}{b d^3} \\ & = \frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{3 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {2 i \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (1-\frac {b^2}{a^2}\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {2 \left (1-\frac {b^2}{a^2}\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 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. \(1752\) vs. \(2(616)=1232\).

Time = 9.80 (sec) , antiderivative size = 1752, normalized size of antiderivative = 2.84 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {-\frac {3 i e (b d e-2 a f) x}{d}-\frac {3 i e (b d e+2 a f) x}{d}-\frac {2 i b (e+f x)^3}{\left (-1+e^{2 i c}\right ) f}+\frac {6 f (b d e-a f) x \log \left (1-e^{-i (c+d x)}\right )}{d^2}+\frac {3 b f^2 x^2 \log \left (1-e^{-i (c+d x)}\right )}{d}+\frac {6 f (b d e+a f) x \log \left (1+e^{-i (c+d x)}\right )}{d^2}+\frac {3 b f^2 x^2 \log \left (1+e^{-i (c+d x)}\right )}{d}+\frac {3 e (b d e-2 a f) \log \left (1-e^{i (c+d x)}\right )}{d^2}+\frac {3 e (b d e+2 a f) \log \left (1+e^{i (c+d x)}\right )}{d^2}+\frac {6 i f (b d e+a f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )}{d^3}+\frac {6 i b f^2 x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )}{d^2}+\frac {6 i f (b d e-a f) \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )}{d^3}+\frac {6 i b f^2 x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )}{d^2}+\frac {6 b f^2 \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )}{d^3}+\frac {6 b f^2 \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )}{d^3}}{3 a^2}+\frac {\left (a^2-b^2\right ) \left (6 i d^3 e^2 e^{2 i c} x+6 i d^3 e e^{2 i c} f x^2+2 i d^3 e^{2 i c} f^2 x^3+3 d^2 e^2 \log \left (b-2 i a e^{i (c+d x)}-b e^{2 i (c+d x)}\right )-3 d^2 e^2 e^{2 i c} \log \left (b-2 i a e^{i (c+d x)}-b e^{2 i (c+d x)}\right )+6 d^2 e f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 d^2 e e^{2 i c} f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+3 d^2 f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-3 d^2 e^{2 i c} f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 d^2 e f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 d^2 e e^{2 i c} f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+3 d^2 f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-3 d^2 e^{2 i c} f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 i d \left (-1+e^{2 i c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (2 c+d x)}}{a e^{i c}+i \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 i d \left (-1+e^{2 i c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (2 c+d x)}}{a e^{i c}+i \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 e^{2 i c} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (2 c+d x)}}{a e^{i c}+i \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 e^{2 i c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )\right )}{3 a^2 b d^3 \left (-1+e^{2 i c}\right )}+\frac {\left (-3 b e^2-6 b e f x-3 b f^2 x^2-3 a d e^2 x \cos (c)-3 a d e f x^2 \cos (c)-a d f^2 x^3 \cos (c)\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{6 a b 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 {\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} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^2*cos(d*x+c)*cot(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. 2529 vs. \(2 (549) = 1098\).

Time = 0.50 (sec) , antiderivative size = 2529, normalized size of antiderivative = 4.11 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((f*x+e)^2*cos(d*x+c)*cot(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}