3.4.29 \(\int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\) [329]

Optimal. Leaf size=763 \[ -\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {6 f^3 \cos (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {6 i \left (a^2-b^2\right ) f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {6 i \left (a^2-b^2\right ) f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {3 i f^3 \text {Li}_4\left (e^{2 i (c+d x)}\right )}{4 a d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {(e+f x)^3 \sin (c+d x)}{b d} \]

[Out]

-3*I*(a^2-b^2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a/b^2/d^2-3/2*I*f*(f*x+e)^2*polyl
og(2,exp(2*I*(d*x+c)))/a/d^2+6*f^3*cos(d*x+c)/b/d^4-3*f*(f*x+e)^2*cos(d*x+c)/b/d^2+(f*x+e)^3*ln(1-exp(2*I*(d*x
+c)))/a/d+(a^2-b^2)*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a/b^2/d+(a^2-b^2)*(f*x+e)^3*ln(1-I*
b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a/b^2/d+6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2
)))/a/b^2/d^4+3/4*I*f^3*polylog(4,exp(2*I*(d*x+c)))/a/d^4+6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a
^2-b^2)^(1/2)))/a/b^2/d^4+3/2*f^2*(f*x+e)*polylog(3,exp(2*I*(d*x+c)))/a/d^3+6*(a^2-b^2)*f^2*(f*x+e)*polylog(3,
I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a/b^2/d^3+6*(a^2-b^2)*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2
-b^2)^(1/2)))/a/b^2/d^3-1/4*I*(f*x+e)^4/a/f-3*I*(a^2-b^2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2
)^(1/2)))/a/b^2/d^2-1/4*I*(a^2-b^2)*(f*x+e)^4/a/b^2/f+6*f^2*(f*x+e)*sin(d*x+c)/b/d^3-(f*x+e)^3*sin(d*x+c)/b/d

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 763, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 17, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4639, 4493, 4489, 3392, 32, 2715, 8, 3798, 2221, 2611, 6744, 2320, 6724, 4621, 3377, 2718, 4615} \begin {gather*} \frac {6 i f^3 \left (a^2-b^2\right ) \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {6 i f^3 \left (a^2-b^2\right ) \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a b^2 d^4}+\frac {6 f^2 \left (a^2-b^2\right ) (e+f x) \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 {6 f^2 \left (a^2-b^2\right ) (e+f x) \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 {3 i f \left (a^2-b^2\right ) (e+f x)^2 \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 {3 i f \left (a^2-b^2\right ) (e+f x)^2 \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 {3 i f^3 \text {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a d^4}+\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}-\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \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)^3 \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)^4}{4 a b^2 f}+\frac {(e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {i (e+f x)^4}{4 a f}+\frac {6 f^3 \cos (c+d x)}{b d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}-\frac {(e+f x)^3 \sin (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 2715

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

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 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 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 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 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]

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 (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cos ^2(c+d x) \cot (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^3 \cot (c+d x) \, dx}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \, dx}{b}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {(e+f x)^3 \cos (c+d x)}{a+b \sin (c+d x)} \, dx\\ &=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {(e+f x)^3 \sin (c+d x)}{b d}-\frac {(2 i) \int \frac {e^{2 i (c+d x)} (e+f x)^3}{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)^3}{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)^3}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx+\frac {(3 f) \int (e+f x)^2 \sin (c+d x) \, dx}{b d}\\ &=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \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)^3 \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)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \sin (c+d x)}{b d}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d}+\frac {\left (3 \left (-\frac {a}{b}+\frac {b}{a}\right ) f\right ) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d}+\frac {\left (3 \left (-\frac {a}{b}+\frac {b}{a}\right ) f\right ) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d}+\frac {\left (6 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{b d^2}\\ &=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \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)^3 \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)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {(e+f x)^3 \sin (c+d x)}{b d}+\frac {\left (3 i f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 i \left (a^2-b^2\right ) f^2\right ) \int (e+f x) \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 (6 i \left (a^2-b^2\right ) f^2\right ) \int (e+f x) \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 (6 f^3\right ) \int \sin (c+d x) \, dx}{b d^3}\\ &=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {6 f^3 \cos (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \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)^3 \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)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2-b^2\right ) 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 b^2 d^3}+\frac {6 \left (a^2-b^2\right ) 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 b^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {(e+f x)^3 \sin (c+d x)}{b d}-\frac {\left (3 f^3\right ) \int \text {Li}_3\left (e^{2 i (c+d x)}\right ) \, dx}{2 a d^3}-\frac {\left (6 \left (a^2-b^2\right ) f^3\right ) \int \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a b^2 d^3}-\frac {\left (6 \left (a^2-b^2\right ) f^3\right ) \int \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a b^2 d^3}\\ &=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {6 f^3 \cos (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \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)^3 \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)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2-b^2\right ) 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 b^2 d^3}+\frac {6 \left (a^2-b^2\right ) 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 b^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {(e+f x)^3 \sin (c+d x)}{b d}+\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{4 a d^4}+\frac {\left (6 i \left (a^2-b^2\right ) 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 b^2 d^4}+\frac {\left (6 i \left (a^2-b^2\right ) 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 b^2 d^4}\\ &=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {6 f^3 \cos (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \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)^3 \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)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i \left (a^2-b^2\right ) 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 b^2 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2-b^2\right ) 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 b^2 d^3}+\frac {6 \left (a^2-b^2\right ) 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 b^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {6 i \left (a^2-b^2\right ) f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {6 i \left (a^2-b^2\right ) f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {3 i f^3 \text {Li}_4\left (e^{2 i (c+d x)}\right )}{4 a d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {(e+f x)^3 \sin (c+d x)}{b d}\\ \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. \(3808\) vs. \(2(763)=1526\).
time = 8.56, size = 3808, normalized size = 4.99 \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])/(a + b*Sin[c + d*x]),x]

[Out]

-1/4*(e*f^2*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
))]))/(a*d^3*E^(I*c)) - (E^(I*c)*f^3*Csc[c]*(x^4 + (-1 + E^((-2*I)*c))*x^4 + ((-1 + E^((2*I)*c))*(2*d^4*x^4 +
(4*I)*d^3*x^3*Log[1 - E^((2*I)*(c + d*x))] + 6*d^2*x^2*PolyLog[2, E^((2*I)*(c + d*x))] + (6*I)*d*x*PolyLog[3,
E^((2*I)*(c + d*x))] - 3*PolyLog[4, E^((2*I)*(c + d*x))]))/(2*d^4*E^((2*I)*c))))/(4*a) + ((a^2 - b^2)*((-4*I)*
d^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 - (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*
f^3*x^4 - (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (2*I)*d^3*e^3*E^((2*I)*
c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*
(-1 + E^((2*I)*(c + d*x)))^2] + d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*
x)))^2] - 6*d^3*e^2*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^3*
e^2*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^3*e*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^3*e*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)])] - 2*d^3*f^3*x^3*Log[1 + (b*E
^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*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^3*e^2*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^3*e^2*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^3*e*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^3*e*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)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2
*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*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^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*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^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E
^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*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*d*e*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*d*f^3*x*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*d*E^((2*I)*c)*f^3*x*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*d*e*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*d*e*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)]))] - 1
2*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*E^((2*I)*
c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*f^3*Poly
Log[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*E^((2*I)*c)*f^3*PolyLo
g[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*f^3*PolyLog[4, -((b*E^(I
*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*
(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(2*a*b^2*d^4*(-1 + E^((2*I)*c))) + (e^3*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)) + Csc[c]*(Cos[c
+ d*x]/(8*b^2*d^4) - ((I/8)*Sin[c + d*x])/(b^2*d^4))*(4*a*d^4*e^3*x*Cos[d*x] + 6*a*d^4*e^2*f*x^2*Cos[d*x] + 4
*a*d^4*e*f^2*x^3*Cos[d*x] + a*d^4*f^3*x^4*Cos[d*x] + 4*a*d^4*e^3*x*Cos[2*c + d*x] + 6*a*d^4*e^2*f*x^2*Cos[2*c
+ d*x] + 4*a*d^4*e*f^2*x^3*Cos[2*c + d*x] + a*d^4*f^3*x^4*Cos[2*c + d*x] - 2*b*d^3*e^3*Cos[c + 2*d*x] - (6*I)*
b*d^2*e^2*f*Cos[c + 2*d*x] + 12*b*d*e*f^2*Cos[c + 2*d*x] + (12*I)*b*f^3*Cos[c + 2*d*x] - 6*b*d^3*e^2*f*x*Cos[c
+ 2*d*x] - (12*I)*b*d^2*e*f^2*x*Cos[c + 2*d*x] + 12*b*d*f^3*x*Cos[c + 2*d*x] - 6*b*d^3*e*f^2*x^2*Cos[c + 2*d*
x] - (6*I)*b*d^2*f^3*x^2*Cos[c + 2*d*x] - 2*b*d^3*f^3*x^3*Cos[c + 2*d*x] + 2*b*d^3*e^3*Cos[3*c + 2*d*x] + (6*I
)*b*d^2*e^2*f*Cos[3*c + 2*d*x] - 12*b*d*e*f^2*Cos[3*c + 2*d*x] - (12*I)*b*f^3*Cos[3*c + 2*d*x] + 6*b*d^3*e^2*f
*x*Cos[3*c + 2*d*x] + (12*I)*b*d^2*e*f^2*x*Cos[3*c + 2*d*x] - 12*b*d*f^3*x*Cos[3*c + 2*d*x] + 6*b*d^3*e*f^2*x^
2*Cos[3*c + 2*d*x] + (6*I)*b*d^2*f^3*x^2*Cos[3*c + 2*d*x] + 2*b*d^3*f^3*x^3*Cos[3*c + 2*d*x] - (4*I)*b*d^3*e^3
*Sin[c] - 12*b*d^2*e^2*f*Sin[c] + (24*I)*b*d*e*f^2*Sin[c] + 24*b*f^3*Sin[c] - (12*I)*b*d^3*e^2*f*x*Sin[c] - 24
*b*d^2*e*f^2*x*Sin[c] + (24*I)*b*d*f^3*x*Sin[c]...

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Maple [F]
time = 0.80, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{a +b \sin \left (d x +c \right )}\, dx\]

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)/(a+b*sin(d*x+c)),x)

[Out]

int((f*x+e)^3*cos(d*x+c)^2*cot(d*x+c)/(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)/(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. 3448 vs. \(2 (702) = 1404\).
time = 0.76, size = 3448, normalized size = 4.52 \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)/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(6*I*b^2*f^3*polylog(4, cos(d*x + c) + I*sin(d*x + c)) - 6*I*b^2*f^3*polylog(4, cos(d*x + c) - I*sin(d*x +
c)) - 6*I*b^2*f^3*polylog(4, -cos(d*x + c) + I*sin(d*x + c)) + 6*I*b^2*f^3*polylog(4, -cos(d*x + c) - I*sin(d
*x + c)) - 6*I*(a^2 - b^2)*f^3*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) - 6*I*(a^2 - b^2)*f^3*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) + 6*I*(a^2 - b^2)*f^3*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) + 6*I*(a^2 - b^2)*f^3*polylo
g(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) - 6
*(a*b*d^2*f^3*x^2 + 2*a*b*d^2*f^2*x*e + a*b*d^2*f*e^2 - 2*a*b*f^3)*cos(d*x + c) - 3*(I*(a^2 - b^2)*d^2*f^3*x^2
+ 2*I*(a^2 - b^2)*d^2*f^2*x*e + I*(a^2 - b^2)*d^2*f*e^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) - 3*(I*(a^2 - b^2)*d^2*f^3*x^2 + 2*I*(a^2 - b^2)
*d^2*f^2*x*e + I*(a^2 - b^2)*d^2*f*e^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) - 3*(-I*(a^2 - b^2)*d^2*f^3*x^2 - 2*I*(a^2 - b^2)*d^2*f^2*x*e - I*
(a^2 - b^2)*d^2*f*e^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) - 3*(-I*(a^2 - b^2)*d^2*f^3*x^2 - 2*I*(a^2 - b^2)*d^2*f^2*x*e - I*(a^2 - b^2)*d^2*
f*e^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) - 3*(I*b^2*d^2*f^3*x^2 + 2*I*b^2*d^2*f^2*x*e + I*b^2*d^2*f*e^2)*dilog(cos(d*x + c) + I*sin(d*x + c
)) - 3*(-I*b^2*d^2*f^3*x^2 - 2*I*b^2*d^2*f^2*x*e - I*b^2*d^2*f*e^2)*dilog(cos(d*x + c) - I*sin(d*x + c)) - 3*(
-I*b^2*d^2*f^3*x^2 - 2*I*b^2*d^2*f^2*x*e - I*b^2*d^2*f*e^2)*dilog(-cos(d*x + c) + I*sin(d*x + c)) - 3*(I*b^2*d
^2*f^3*x^2 + 2*I*b^2*d^2*f^2*x*e + I*b^2*d^2*f*e^2)*dilog(-cos(d*x + c) - I*sin(d*x + c)) - ((a^2 - b^2)*c^3*f
^3 - 3*(a^2 - b^2)*c^2*d*f^2*e + 3*(a^2 - b^2)*c*d^2*f*e^2 - (a^2 - b^2)*d^3*e^3)*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)*c^3*f^3 - 3*(a^2 - b^2)*c^2*d*f^2*e + 3*(a^
2 - b^2)*c*d^2*f*e^2 - (a^2 - b^2)*d^3*e^3)*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)*c^3*f^3 - 3*(a^2 - b^2)*c^2*d*f^2*e + 3*(a^2 - b^2)*c*d^2*f*e^2 - (a^2 - b^2)*d^3
*e^3)*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)*c^3*f^3
- 3*(a^2 - b^2)*c^2*d*f^2*e + 3*(a^2 - b^2)*c*d^2*f*e^2 - (a^2 - b^2)*d^3*e^3)*log(-2*b*cos(d*x + c) - 2*I*b*s
in(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) + ((a^2 - b^2)*d^3*f^3*x^3 + (a^2 - b^2)*c^3*f^3 + 3*((a^2 -
b^2)*d^3*f*x + (a^2 - b^2)*c*d^2*f)*e^2 + 3*((a^2 - b^2)*d^3*f^2*x^2 - (a^2 - b^2)*c^2*d*f^2)*e)*log(-(I*a*co
s(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) + ((a^2 - b^2
)*d^3*f^3*x^3 + (a^2 - b^2)*c^3*f^3 + 3*((a^2 - b^2)*d^3*f*x + (a^2 - b^2)*c*d^2*f)*e^2 + 3*((a^2 - b^2)*d^3*f
^2*x^2 - (a^2 - b^2)*c^2*d*f^2)*e)*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) + ((a^2 - b^2)*d^3*f^3*x^3 + (a^2 - b^2)*c^3*f^3 + 3*((a^2 - b^2)*d^3*f*x +
(a^2 - b^2)*c*d^2*f)*e^2 + 3*((a^2 - b^2)*d^3*f^2*x^2 - (a^2 - b^2)*c^2*d*f^2)*e)*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) + ((a^2 - b^2)*d^3*f^3*x^3 +
(a^2 - b^2)*c^3*f^3 + 3*((a^2 - b^2)*d^3*f*x + (a^2 - b^2)*c*d^2*f)*e^2 + 3*((a^2 - b^2)*d^3*f^2*x^2 - (a^2 -
b^2)*c^2*d*f^2)*e)*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) + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*f^2*x^2*e + 3*b^2*d^3*f*x*e^2 + b^2*d^3*e^3)*log(cos(d*x +
c) + I*sin(d*x + c) + 1) + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*f^2*x^2*e + 3*b^2*d^3*f*x*e^2 + b^2*d^3*e^3)*log(cos(d
*x + c) - I*sin(d*x + c) + 1) - (b^2*c^3*f^3 - 3*b^2*c^2*d*f^2*e + 3*b^2*c*d^2*f*e^2 - b^2*d^3*e^3)*log(-1/2*c
os(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) - (b^2*c^3*f^3 - 3*b^2*c^2*d*f^2*e + 3*b^2*c*d^2*f*e^2 - b^2*d^3*e^3)*
log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) + (b^2*d^3*f^3*x^3 + b^2*c^3*f^3 + 3*(b^2*d^3*f*x + b^2*c*d^
2*f)*e^2 + 3*(b^2*d^3*f^2*x^2 - b^2*c^2*d*f^2)*e)*log(-cos(d*x + c) + I*sin(d*x + c) + 1) + (b^2*d^3*f^3*x^3 +
b^2*c^3*f^3 + 3*(b^2*d^3*f*x + b^2*c*d^2*f)*e^2 + 3*(b^2*d^3*f^2*x^2 - b^2*c^2*d*f^2)*e)*log(-cos(d*x + c) -
I*sin(d*x + c) + 1) + 6*((a^2 - b^2)*d*f^3*x + (a^2 - b^2)*d*f^2*e)*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) + 6*((a^2 - b^2)*d*f^3*x + (a^2 - b^2)*d
*f^2*e)*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) + 6*((a^2 - b^2)*d*f^3*x + (a^2 - b^2...

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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 {\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)/(a+b*sin(d*x+c)),x)

[Out]

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

[Out]

\text{Hanged}

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