3.4.41 \(\int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [341]

Optimal. Leaf size=1144 \[ -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \cos (c+d x)}{a^2 b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x)^3 \cos (c+d x)}{a^2 b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a^2 b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f (e+f x)^2 \sin (c+d x)}{a^2 b d^2} \]

[Out]

6*(a^2-b^2)^(3/2)*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^2/d^4-6*(a^2-b^2)^(3/2)*f^3*poly
log(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^2/d^4+6*b*f^2*(f*x+e)*polylog(3,-exp(I*(d*x+c)))/a^2/d^3-1
/4*(f*x+e)^4/a/f+2*b*(f*x+e)^3*arctanh(exp(I*(d*x+c)))/a^2/d+6*b*f^2*(f*x+e)*cos(d*x+c)/a^2/d^3-(a^2-b^2)*(f*x
+e)^3*cos(d*x+c)/a^2/b/d-6*(a^2-b^2)*f^3*sin(d*x+c)/a^2/b/d^4-b*(f*x+e)^3*cos(d*x+c)/a^2/d-6*b*f^3*sin(d*x+c)/
a^2/d^4-1/4*(a^2-b^2)*(f*x+e)^4/a/b^2/f+I*(a^2-b^2)^(3/2)*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)
))/a^2/b^2/d-3*(a^2-b^2)^(3/2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^2/d^2+3*(a^
2-b^2)^(3/2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^2/d^2-I*(a^2-b^2)^(3/2)*(f*x+
e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^2/d+6*I*b*f^3*polylog(4,-exp(I*(d*x+c)))/a^2/d^4-3*I*b
*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a^2/d^2+3*f*(f*x+e)^2*ln(1-exp(2*I*(d*x+c)))/a/d^2+3*I*b*f*(f*x+e)^2*p
olylog(2,exp(I*(d*x+c)))/a^2/d^2-I*(f*x+e)^3/a/d-(f*x+e)^3*cot(d*x+c)/a/d+3/2*f^3*polylog(3,exp(2*I*(d*x+c)))/
a/d^4-6*I*(a^2-b^2)^(3/2)*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^2/d^3+6*I*(a^2-b
^2)^(3/2)*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^2/d^3+3*b*f*(f*x+e)^2*sin(d*x+c)
/a^2/d^2-3*I*f^2*(f*x+e)*polylog(2,exp(2*I*(d*x+c)))/a/d^3-6*b*f^2*(f*x+e)*polylog(3,exp(I*(d*x+c)))/a^2/d^3-6
*I*b*f^3*polylog(4,exp(I*(d*x+c)))/a^2/d^4+6*(a^2-b^2)*f^2*(f*x+e)*cos(d*x+c)/a^2/b/d^3+3*(a^2-b^2)*f*(f*x+e)^
2*sin(d*x+c)/a^2/b/d^2

________________________________________________________________________________________

Rubi [A]
time = 1.67, antiderivative size = 1144, normalized size of antiderivative = 1.00, number of steps used = 66, number of rules used = 20, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4639, 4493, 3392, 32, 3391, 3801, 3798, 2221, 2611, 2320, 6724, 4490, 3377, 2717, 2713, 4268, 6744, 4621, 3404, 2296} \begin {gather*} -\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {(e+f x)^4}{4 a f}+\frac {2 b \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^3}{a^2 d}-\frac {b \cos (c+d x) (e+f x)^3}{a^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^3}{a^2 b d}-\frac {\cot (c+d x) (e+f x)^3}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b^2 d}-\frac {i (e+f x)^3}{a d}+\frac {3 f \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 i b f \text {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)^2}{a^2 d^2}+\frac {3 i b f \text {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)^2}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b^2 d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b^2 d^2}+\frac {3 b f \sin (c+d x) (e+f x)^2}{a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)^2}{a^2 b d^2}+\frac {6 b f^2 \cos (c+d x) (e+f x)}{a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 \cos (c+d x) (e+f x)}{a^2 b d^3}-\frac {3 i f^2 \text {PolyLog}\left (2,e^{2 i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 b f^2 \text {PolyLog}\left (3,-e^{i (c+d x)}\right ) (e+f x)}{a^2 d^3}-\frac {6 b f^2 \text {PolyLog}\left (3,e^{i (c+d x)}\right ) (e+f x)}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^2 d^3}+\frac {3 f^3 \text {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a^2 b d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

((-I)*(e + f*x)^3)/(a*d) - (e + f*x)^4/(4*a*f) - ((a^2 - b^2)*(e + f*x)^4)/(4*a*b^2*f) + (2*b*(e + f*x)^3*ArcT
anh[E^(I*(c + d*x))])/(a^2*d) + (6*b*f^2*(e + f*x)*Cos[c + d*x])/(a^2*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*Cos[
c + d*x])/(a^2*b*d^3) - (b*(e + f*x)^3*Cos[c + d*x])/(a^2*d) - ((a^2 - b^2)*(e + f*x)^3*Cos[c + d*x])/(a^2*b*d
) - ((e + f*x)^3*Cot[c + d*x])/(a*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqr
t[a^2 - b^2])])/(a^2*b^2*d) + (I*(a^2 - b^2)^(3/2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b
^2])])/(a^2*b^2*d) + (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((3*I)*b*f*(e + f*x)^2*PolyLog[2
, -E^(I*(c + d*x))])/(a^2*d^2) + ((3*I)*b*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (3*(a^2 - b^2
)^(3/2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) + (3*(a^2 - b^2)^
(3/2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) - ((3*I)*f^2*(e + f
*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a^2*d^3) - (6
*b*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a^2*d^3) - ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog[3, (I
*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^3) + ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog[3,
(I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^3) + (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4
) + ((6*I)*b*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a^2*d^4) - ((6*I)*b*f^3*PolyLog[4, E^(I*(c + d*x))])/(a^2*d^4)
+ (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^4) - (6*(a^2 -
b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^4) - (6*b*f^3*Sin[c + d*x]
)/(a^2*d^4) - (6*(a^2 - b^2)*f^3*Sin[c + d*x])/(a^2*b*d^4) + (3*b*f*(e + f*x)^2*Sin[c + d*x])/(a^2*d^2) + (3*(
a^2 - b^2)*f*(e + f*x)^2*Sin[c + d*x])/(a^2*b*d^2)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 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 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 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 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 3392

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

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 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[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 4490

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

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {\int (e+f x)^3 \cos ^2(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos ^3(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int (e+f x)^3 \, dx}{2 a}-\frac {\int (e+f x)^3 \, dx}{a}+\frac {\int (e+f x)^3 \cos ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos (c+d x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {(3 f) \int (e+f x)^2 \cot (c+d x) \, dx}{a d}+\frac {\left (3 f^2\right ) \int (e+f x) \cos ^2(c+d x) \, dx}{2 a d^2}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {3 (e+f x)^4}{8 a f}+\frac {3 f^3 \cos ^2(c+d x)}{8 a d^4}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}+\frac {\int (e+f x)^3 \, dx}{2 a}-\frac {b \int (e+f x)^3 \csc (c+d x) \, dx}{a^2}+\frac {b \int (e+f x)^3 \sin (c+d x) \, dx}{a^2}-\frac {\left (a \left (1-\frac {b^2}{a^2}\right )\right ) \int (e+f x)^3 \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x)^3 \sin (c+d x) \, dx}{b}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {(e+f x)^3}{a+b \sin (c+d x)} \, dx}{b^2}-\frac {(6 i f) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1-e^{2 i (c+d x)}} \, dx}{a d}+\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2}-\frac {\left (3 f^2\right ) \int (e+f x) \cos ^2(c+d x) \, dx}{2 a d^2}\\ &=\frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {\left (2 \left (a^2-b^2\right )^2\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2 b^2}+\frac {(3 b f) \int (e+f x)^2 \cos (c+d x) \, dx}{a^2 d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \cos (c+d x) \, dx}{b d}-\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}+\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}+\frac {\left (6 i b f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{a^2 d^2}-\frac {\left (6 \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{b d^2}+\frac {\left (3 i f^3\right ) \int \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (3 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d}-\frac {\left (3 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {\left (6 b f^3\right ) \int \cos (c+d x) \, dx}{a^2 d^3}-\frac {\left (6 b f^3\right ) \int \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \text {Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a^2 d^3}-\frac {\left (6 \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int \cos (c+d x) \, dx}{b d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^2}-\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^2}+\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^4}-\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^4}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^3}-\frac {\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^4}-\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^4}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4632\) vs. \(2(1144)=2288\).
time = 40.58, size = 4632, normalized size = 4.05 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^3*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

-((b*e^3*Log[Tan[(c + d*x)/2]])/(a^2*d)) - (3*b*e^2*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c +
d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a^2*d^2
) - (f^3*Csc[c]*(2*d^2*x^2*(2*d*E^((2*I)*c)*x + (3*I)*(-1 + E^((2*I)*c))*Log[1 - E^((2*I)*(c + d*x))]) + 6*d*(
-1 + E^((2*I)*c))*x*PolyLog[2, E^((2*I)*(c + d*x))] + (3*I)*(-1 + E^((2*I)*c))*PolyLog[3, E^((2*I)*(c + d*x))]
))/(4*a*d^4*E^(I*c)) + (6*b*e*f^2*(d^2*x^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - I*d*x*PolyLog[2, -Cos[c +
d*x] - I*Sin[c + d*x]] + I*d*x*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + PolyLog[3, -Cos[c + d*x] - I*Sin[c
+ d*x]] - PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]]))/(a^2*d^3) - (b*f^3*(-2*d^3*x^3*ArcTanh[Cos[c + d*x] + I*
Sin[c + d*x]] + (3*I)*d^2*x^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] - (3*I)*d^2*x^2*PolyLog[2, Cos[c + d*
x] + I*Sin[c + d*x]] - 6*d*x*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] + 6*d*x*PolyLog[3, Cos[c + d*x] + I*Si
n[c + d*x]] - (6*I)*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]] + (6*I)*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x
]]))/(a^2*d^4) + (3*e^2*f*Csc[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d^2*(Cos[
c]^2 + Sin[c]^2)) + (I*(a^2 - b^2)^(3/2)*((3*I)*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*(Cos[2*c + d*x] + I*Sin
[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + (3*I)*
Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*
(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[c]) + I*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*(Cos[2*c +
d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c])]*(Cos[c] + I*Sin[
c]) + 3*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + S
qrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) - 3*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^
2*PolyLog[2, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2]
+ a*Sin[c])]*(Cos[c] + I*Sin[c]) + (6*I)*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*(Cos[2*c + d*x] + I*Sin[2*c
+ d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) + (6*I)*Sqrt
[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[
c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) - 6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*(Cos[2*c + d*x] +
I*Sin[2*c + d*x]))/(I*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] - a*Sin[c]))]*(Cos[c] + I*Sin[c]) +
6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(C
os[c] + I*Sin[c])^2] + a*Sin[c])]*(Cos[c] + I*Sin[c]) + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*(Cos[2*c + d*
x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] +
Sin[c]) + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqr
t[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1
- (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c]
)]*((-I)*Cos[c] + Sin[c]) + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)
*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) + 6*Sqrt[a^2 - b^2]*d
*f^3*x*PolyLog[3, (b*(Cos[2*c + d*x] + I*Sin[2*c + d*x]))/((-I)*a*Cos[c] + Sqrt[(-a^2 + b^2)*(Cos[c] + I*Sin[c
])^2] + a*Sin[c])]*((-I)*Cos[c] + Sin[c]) - (2*I)*d^3*e^3*ArcTan[(b*Cos[c + d*x] + I*(a + b*Sin[c + d*x]))/Sqr
t[a^2 - b^2]]*Sqrt[(-a^2 + b^2)*(Cos[2*c] + I*Sin[2*c])]))/(a^2*b^2*d^4*Sqrt[(-a^2 + b^2)*(Cos[2*c] + I*Sin[2*
c])]) + Csc[c]*Csc[c + d*x]*(Cos[c + d*x]/(16*a*b^2*d^4) - ((I/16)*Sin[c + d*x])/(a*b^2*d^4))*((8*I)*b^2*d^3*e
^3*Cos[c] + (24*I)*b^2*d^3*e^2*f*x*Cos[c] + (24*I)*b^2*d^3*e*f^2*x^2*Cos[c] + (8*I)*b^2*d^3*f^3*x^3*Cos[c] - 2
*a*b*d^3*e^3*Cos[d*x] + (18*I)*a*b*d^2*e^2*f*Cos[d*x] + 12*a*b*d*e*f^2*Cos[d*x] - (36*I)*a*b*f^3*Cos[d*x] - 6*
a*b*d^3*e^2*f*x*Cos[d*x] + (36*I)*a*b*d^2*e*f^2*x*Cos[d*x] + 12*a*b*d*f^3*x*Cos[d*x] - 6*a*b*d^3*e*f^2*x^2*Cos
[d*x] + (18*I)*a*b*d^2*f^3*x^2*Cos[d*x] - 2*a*b*d^3*f^3*x^3*Cos[d*x] + 2*a*b*d^3*e^3*Cos[2*c + d*x] - (18*I)*a
*b*d^2*e^2*f*Cos[2*c + d*x] - 12*a*b*d*e*f^2*Cos[2*c + d*x] + (36*I)*a*b*f^3*Cos[2*c + d*x] + 6*a*b*d^3*e^2*f*
x*Cos[2*c + d*x] - (36*I)*a*b*d^2*e*f^2*x*Cos[2*c + d*x] - 12*a*b*d*f^3*x*Cos[2*c + d*x] + 6*a*b*d^3*e*f^2*x^2
*Cos[2*c + d*x] - (18*I)*a*b*d^2*f^3*x^2*Cos[2*c + d*x] + 2*a*b*d^3*f^3*x^3*Cos[2*c + d*x] - (8*I)*b^2*d^3*e^3
*Cos[c + 2*d*x] - 4*a^2*d^4*e^3*x*Cos[c + 2*d*x] - (24*I)*b^2*d^3*e^2*f*x*Cos[c + 2*d*x] - 6*a^2*d^4*e^2*f*x^2
*Cos[c + 2*d*x] - (24*I)*b^2*d^3*e*f^2*x^2*Cos[c + 2*d*x] - 4*a^2*d^4*e*f^2*x^3*Cos[c + 2*d*x] - (8*I)*b^2*d^3
*f^3*x^3*Cos[c + 2*d*x] - a^2*d^4*f^3*x^4*Cos[c...

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Maple [F]
time = 0.62, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \left (\cot ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*cos(d*x+c)^2*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)

[Out]

int((f*x+e)^3*cos(d*x+c)^2*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)^2*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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4695 vs. \(2 (1052) = 2104\).
time = 0.96, size = 4695, normalized size = 4.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)^2*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(12*a^2*b*d^2*f^3*x^2 + 24*a^2*b*d^2*f^2*x*e - 12*I*b^3*f^3*polylog(4, cos(d*x + c) + I*sin(d*x + c))*sin(
d*x + c) + 12*I*b^3*f^3*polylog(4, cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - 12*I*b^3*f^3*polylog(4, -cos(
d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 12*I*b^3*f^3*polylog(4, -cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c)
+ 12*a^2*b*d^2*f*e^2 - 12*I*(a^2*b - b^3)*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(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) + 12*I*(a^2*b - b^3)*f^3*
sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sq
rt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) + 12*I*(a^2*b - b^3)*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-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) - 12*I*(
a^2*b - b^3)*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-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) - 24*a^2*b*f^3 - 6*(-I*(a^2*b - b^3)*d^2*f^3*x^2 - 2*I
*(a^2*b - b^3)*d^2*f^2*x*e - I*(a^2*b - b^3)*d^2*f*e^2)*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))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) - 6*(I*(a^2*b
- b^3)*d^2*f^3*x^2 + 2*I*(a^2*b - b^3)*d^2*f^2*x*e + I*(a^2*b - b^3)*d^2*f*e^2)*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))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin
(d*x + c) - 6*(I*(a^2*b - b^3)*d^2*f^3*x^2 + 2*I*(a^2*b - b^3)*d^2*f^2*x*e + I*(a^2*b - b^3)*d^2*f*e^2)*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))*sqrt(-(a^2 -
b^2)/b^2) - b)/b + 1)*sin(d*x + c) - 6*(-I*(a^2*b - b^3)*d^2*f^3*x^2 - 2*I*(a^2*b - b^3)*d^2*f^2*x*e - I*(a^2*
b - b^3)*d^2*f*e^2)*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*s
in(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) - 2*((a^2*b - b^3)*c^3*f^3 - 3*(a^2*b - b^3)*c^2*
d*f^2*e + 3*(a^2*b - b^3)*c*d^2*f*e^2 - (a^2*b - b^3)*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x + c) + 2
*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) - 2*((a^2*b - b^3)*c^3*f^3 - 3*(a^2*b - b
^3)*c^2*d*f^2*e + 3*(a^2*b - b^3)*c*d^2*f*e^2 - (a^2*b - b^3)*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x
+ c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a)*sin(d*x + c) + 2*((a^2*b - b^3)*c^3*f^3 - 3*(a
^2*b - b^3)*c^2*d*f^2*e + 3*(a^2*b - b^3)*c*d^2*f*e^2 - (a^2*b - b^3)*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)*log(-2*b
*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) + 2*((a^2*b - b^3)*c^3*f
^3 - 3*(a^2*b - b^3)*c^2*d*f^2*e + 3*(a^2*b - b^3)*c*d^2*f*e^2 - (a^2*b - b^3)*d^3*e^3)*sqrt(-(a^2 - b^2)/b^2)
*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a)*sin(d*x + c) - 2*((a^2*b - b
^3)*d^3*f^3*x^3 + (a^2*b - b^3)*c^3*f^3 + 3*((a^2*b - b^3)*d^3*f*x + (a^2*b - b^3)*c*d^2*f)*e^2 + 3*((a^2*b -
b^3)*d^3*f^2*x^2 - (a^2*b - b^3)*c^2*d*f^2)*e)*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*((a^2*b - b^3)*d^3*f^3*x
^3 + (a^2*b - b^3)*c^3*f^3 + 3*((a^2*b - b^3)*d^3*f*x + (a^2*b - b^3)*c*d^2*f)*e^2 + 3*((a^2*b - b^3)*d^3*f^2*
x^2 - (a^2*b - b^3)*c^2*d*f^2)*e)*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*((a^2*b - b^3)*d^3*f^3*x^3 + (a^2*b -
b^3)*c^3*f^3 + 3*((a^2*b - b^3)*d^3*f*x + (a^2*b - b^3)*c*d^2*f)*e^2 + 3*((a^2*b - b^3)*d^3*f^2*x^2 - (a^2*b
- b^3)*c^2*d*f^2)*e)*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*s
in(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + 2*((a^2*b - b^3)*d^3*f^3*x^3 + (a^2*b - b^3)*c^3*f^
3 + 3*((a^2*b - b^3)*d^3*f*x + (a^2*b - b^3)*c*d^2*f)*e^2 + 3*((a^2*b - b^3)*d^3*f^2*x^2 - (a^2*b - b^3)*c^2*d
*f^2)*e)*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) + 12*((a^2*b - b^3)*d*f^3*x + (a^2*b - b^3)*d*f^2*e)*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) - 12*((a^2*b - b^3)*d*f^3*x + (a^2*b - b^3)*d*f^2*e)*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) + 12*((a^2*b - b^3)*d*f^3*x + (a^2*b - b^3)*d*f^2*e)*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) - 12*((a^2*
b - b^3)*d*f^3*x + (a^2*b - b^3)*d*f^2*e)*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(-...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{3} \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*cos(d*x+c)**2*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)

[Out]

Integral((e + f*x)**3*cos(c + d*x)**2*cot(c + d*x)**2/(a + b*sin(c + d*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)^2*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^2*cot(c + d*x)^2*(e + f*x)^3)/(a + b*sin(c + d*x)),x)

[Out]

\text{Hanged}

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