\(\int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx\) [279]

Optimal result

Integrand size = 24, antiderivative size = 601 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {6 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b} \]

[Out]

-9*d^2*(d*x+c)*polylog(3,-exp(I*(b*x+a)))/b^3+9*d^2*(d*x+c)*polylog(3,exp(I*(b*x+a)))/b^3-3*I*d^3*polylog(2,ex
p(I*(b*x+a)))/b^4+3/2*(d*x+c)^3*sec(b*x+a)/b+6*d^3*polylog(3,-I*exp(I*(b*x+a)))/b^4-6*d^3*polylog(3,I*exp(I*(b
*x+a)))/b^4+6*I*d^3*x^2*arctan(exp(I*(b*x+a)))/b^2+6*I*c*d^2*polylog(2,I*exp(I*(b*x+a)))/b^3+6*I*d^3*x*polylog
(2,I*exp(I*(b*x+a)))/b^3+9*I*d^3*polylog(4,exp(I*(b*x+a)))/b^4-6*d^3*x*arctanh(exp(I*(b*x+a)))/b^3-3*c*d^2*arc
tanh(cos(b*x+a))/b^3-3*c^2*d*arctanh(sin(b*x+a))/b^2-3/2*c^2*d*csc(b*x+a)/b^2-3/2*d^3*x^2*csc(b*x+a)/b^2-1/2*(
d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)/b-9*I*d^3*polylog(4,-exp(I*(b*x+a)))/b^4-3*(d*x+c)^3*arctanh(exp(I*(b*x+a)))/
b+3*I*d^3*polylog(2,-exp(I*(b*x+a)))/b^4+12*I*c*d^2*x*arctan(exp(I*(b*x+a)))/b^2+9/2*I*d*(d*x+c)^2*polylog(2,-
exp(I*(b*x+a)))/b^2-3*c*d^2*x*csc(b*x+a)/b^2-6*I*c*d^2*polylog(2,-I*exp(I*(b*x+a)))/b^3-6*I*d^3*x*polylog(2,-I
*exp(I*(b*x+a)))/b^3-9/2*I*d*(d*x+c)^2*polylog(2,exp(I*(b*x+a)))/b^2

Rubi [A] (verified)

Time = 2.61 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.00, number of steps used = 64, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2702, 294, 327, 213, 4505, 6820, 12, 6874, 6408, 4268, 2611, 6744, 2320, 6724, 4218, 464, 212, 4266, 2317, 2438, 2701, 6406, 3855, 14} \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {6 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b} \]

[In]

Int[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

((12*I)*c*d^2*x*ArcTan[E^(I*(a + b*x))])/b^2 + ((6*I)*d^3*x^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*d^3*x*ArcTanh[
E^(I*(a + b*x))])/b^3 - (3*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b - (3*c*d^2*ArcTanh[Cos[a + b*x]])/b^3 - (3*
c^2*d*ArcTanh[Sin[a + b*x]])/b^2 - (3*c^2*d*Csc[a + b*x])/(2*b^2) - (3*c*d^2*x*Csc[a + b*x])/b^2 - (3*d^3*x^2*
Csc[a + b*x])/(2*b^2) + ((3*I)*d^3*PolyLog[2, -E^(I*(a + b*x))])/b^4 + (((9*I)/2)*d*(c + d*x)^2*PolyLog[2, -E^
(I*(a + b*x))])/b^2 - ((6*I)*c*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 - ((6*I)*d^3*x*PolyLog[2, (-I)*E^(I*(
a + b*x))])/b^3 + ((6*I)*c*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + ((6*I)*d^3*x*PolyLog[2, I*E^(I*(a + b*x))]
)/b^3 - ((3*I)*d^3*PolyLog[2, E^(I*(a + b*x))])/b^4 - (((9*I)/2)*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^
2 - (9*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*d^3
*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + (9*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - ((9*I)*d^3*PolyLog[4
, -E^(I*(a + b*x))])/b^4 + ((9*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4 + (3*(c + d*x)^3*Sec[a + b*x])/(2*b) -
((c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

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

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4218

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p
)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*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 4505

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Modul
e[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u
, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 6406

Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 - u^2)), x], x] /; I
nverseFunctionFreeQ[u, x]

Rule 6408

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan
h[u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 - u^2)), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

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]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\frac {3 (c+d x)^3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-(3 d) \int (c+d x)^2 \left (-\frac {3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 \sec (a+b x)}{2 b}-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b}\right ) \, dx \\ & = -\frac {3 (c+d x)^3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-(3 d) \int \frac {(c+d x)^2 \left (-3 \text {arctanh}(\cos (a+b x))-\left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)\right )}{2 b} \, dx \\ & = -\frac {3 (c+d x)^3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \left (-3 \text {arctanh}(\cos (a+b x))-\left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)\right ) \, dx}{2 b} \\ & = -\frac {3 (c+d x)^3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {(3 d) \int \left (-3 (c+d x)^2 \text {arctanh}(\cos (a+b x))-(c+d x)^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)\right ) \, dx}{2 b} \\ & = -\frac {3 (c+d x)^3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {(3 d) \int (c+d x)^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x) \, dx}{2 b}+\frac {(9 d) \int (c+d x)^2 \text {arctanh}(\cos (a+b x)) \, dx}{2 b} \\ & = \frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {3 \int b (-c-d x)^3 \csc (a+b x) \, dx}{2 b}+\frac {(3 d) \int \left (c^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)+2 c d x \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)+d^2 x^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)\right ) \, dx}{2 b} \\ & = \frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {3}{2} \int (-c-d x)^3 \csc (a+b x) \, dx+\frac {\left (3 c^2 d\right ) \int \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x) \, dx}{2 b}+\frac {\left (3 c d^2\right ) \int x \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x) \, dx}{b}+\frac {\left (3 d^3\right ) \int x^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x) \, dx}{2 b} \\ & = -\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {(9 d) \int (-c-d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{2 b}+\frac {(9 d) \int (-c-d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{2 b}+\frac {\left (3 c^2 d\right ) \text {Subst}\left (\int \frac {1-3 x^2}{x^2 \left (1-x^2\right )} \, dx,x,\sin (a+b x)\right )}{2 b^2}+\frac {\left (3 c d^2\right ) \int \left (-3 x \sec (a+b x)+x \csc ^2(a+b x) \sec (a+b x)\right ) \, dx}{b}+\frac {\left (3 d^3\right ) \int \left (-3 x^2 \sec (a+b x)+x^2 \csc ^2(a+b x) \sec (a+b x)\right ) \, dx}{2 b} \\ & = -\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {\left (3 c^2 d\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{b^2}+\frac {\left (9 i d^2\right ) \int (-c-d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (9 i d^2\right ) \int (-c-d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 c d^2\right ) \int x \csc ^2(a+b x) \sec (a+b x) \, dx}{b}-\frac {\left (9 c d^2\right ) \int x \sec (a+b x) \, dx}{b}+\frac {\left (3 d^3\right ) \int x^2 \csc ^2(a+b x) \sec (a+b x) \, dx}{2 b}-\frac {\left (9 d^3\right ) \int x^2 \sec (a+b x) \, dx}{2 b} \\ & = \frac {18 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {9 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}+\frac {3 c d^2 x \text {arctanh}(\sin (a+b x))}{b^2}+\frac {3 d^3 x^2 \text {arctanh}(\sin (a+b x))}{2 b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {\left (9 c d^2\right ) \int \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (9 c d^2\right ) \int \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (3 c d^2\right ) \int \left (\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}\right ) \, dx}{b}+\frac {\left (9 d^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (9 d^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (9 d^3\right ) \int x \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (9 d^3\right ) \int x \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (3 d^3\right ) \int x \left (\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}\right ) \, dx}{b} \\ & = \frac {18 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {9 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}+\frac {3 c d^2 x \text {arctanh}(\sin (a+b x))}{b^2}+\frac {3 d^3 x^2 \text {arctanh}(\sin (a+b x))}{2 b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {9 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {\left (9 i c d^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\frac {\left (9 i c d^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (3 c d^2\right ) \int \text {arctanh}(\sin (a+b x)) \, dx}{b^2}+\frac {\left (3 c d^2\right ) \int \csc (a+b x) \, dx}{b^2}-\frac {\left (9 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (9 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (9 i d^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (9 i d^3\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (3 d^3\right ) \int \left (\frac {x \text {arctanh}(\sin (a+b x))}{b}-\frac {x \csc (a+b x)}{b}\right ) \, dx}{b} \\ & = \frac {18 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {9 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}+\frac {3 d^3 x^2 \text {arctanh}(\sin (a+b x))}{2 b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {9 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {9 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {\left (3 c d^2\right ) \int b x \sec (a+b x) \, dx}{b^2}+\frac {\left (9 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac {\left (9 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac {\left (3 d^3\right ) \int x \text {arctanh}(\sin (a+b x)) \, dx}{b^2}+\frac {\left (3 d^3\right ) \int x \csc (a+b x) \, dx}{b^2} \\ & = \frac {18 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {9 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {9 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {9 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {9 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {\left (3 c d^2\right ) \int x \sec (a+b x) \, dx}{b}-\frac {\left (3 d^3\right ) \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (3 d^3\right ) \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (3 d^3\right ) \int b x^2 \sec (a+b x) \, dx}{2 b^2} \\ & = \frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {9 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {9 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {9 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {9 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {\left (3 c d^2\right ) \int \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 c d^2\right ) \int \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (3 d^3\right ) \int x^2 \sec (a+b x) \, dx}{2 b} \\ & = \frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {9 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {9 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {9 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {9 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {\left (3 i c d^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (3 i c d^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (3 d^3\right ) \int x \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 d^3\right ) \int x \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = \frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {6 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {9 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {\left (3 i d^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (3 i d^3\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^3} \\ & = \frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {6 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {9 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {9 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4} \\ & = \frac {12 i c d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 i d^3 x^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {3 c d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {3 c^2 d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 c^2 d \csc (a+b x)}{2 b^2}-\frac {3 c d^2 x \csc (a+b x)}{b^2}-\frac {3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {6 i c d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 x \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i c d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^3 x \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac {(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.19 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.51 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=-\frac {3 d \left (-2 i b^2 c^2 \arctan \left (e^{i (a+b x)}\right )+2 b^2 c d x \log \left (1-i e^{i (a+b x)}\right )+b^2 d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )-2 b^2 c d x \log \left (1+i e^{i (a+b x)}\right )-b^2 d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )+2 i b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-2 i b d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )\right )}{b^4}+\frac {3 \left (b^3 c^3 \log \left (1-e^{i (a+b x)}\right )+2 b c d^2 \log \left (1-e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1-e^{i (a+b x)}\right )+2 b d^3 x \log \left (1-e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1-e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1-e^{i (a+b x)}\right )-b^3 c^3 \log \left (1+e^{i (a+b x)}\right )-2 b c d^2 \log \left (1+e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1+e^{i (a+b x)}\right )-2 b d^3 x \log \left (1+e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1+e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1+e^{i (a+b x)}\right )+i d \left (2 d^2+3 b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )-i d \left (2 d^2+3 b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )\right )}{2 b^4}-\frac {\csc ^2(a+b x) \sec (a+b x) \left (-b c^3-3 b c^2 d x-3 b c d^2 x^2-b d^3 x^3+3 b c^3 \cos (2 a+2 b x)+9 b c^2 d x \cos (2 a+2 b x)+9 b c d^2 x^2 \cos (2 a+2 b x)+3 b d^3 x^3 \cos (2 a+2 b x)+3 c^2 d \sin (2 a+2 b x)+6 c d^2 x \sin (2 a+2 b x)+3 d^3 x^2 \sin (2 a+2 b x)\right )}{4 b^2} \]

[In]

Integrate[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

(-3*d*((-2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 - I
*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + (2*I)*b*
d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] - 2*d^2*PolyL
og[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))]))/b^4 + (3*(b^3*c^3*Log[1 - E^(I*(a + b*x))]
+ 2*b*c*d^2*Log[1 - E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - E^(I*(a + b*x))] + 2*b*d^3*x*Log[1 - E^(I*(a + b
*x))] + 3*b^3*c*d^2*x^2*Log[1 - E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 - E^(I*(a + b*x))] - b^3*c^3*Log[1 + E^(I
*(a + b*x))] - 2*b*c*d^2*Log[1 + E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + E^(I*(a + b*x))] - 2*b*d^3*x*Log[1 +
E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 + E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + E^(I*(a + b*x))] + I*d*(2*d
^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] - I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*
x))] - 6*b*c*d^2*PolyLog[3, -E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, -E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3,
E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, -E^(I*(a + b*x))] + (6*I)*d^3*
PolyLog[4, E^(I*(a + b*x))]))/(2*b^4) - (Csc[a + b*x]^2*Sec[a + b*x]*(-(b*c^3) - 3*b*c^2*d*x - 3*b*c*d^2*x^2 -
b*d^3*x^3 + 3*b*c^3*Cos[2*a + 2*b*x] + 9*b*c^2*d*x*Cos[2*a + 2*b*x] + 9*b*c*d^2*x^2*Cos[2*a + 2*b*x] + 3*b*d^
3*x^3*Cos[2*a + 2*b*x] + 3*c^2*d*Sin[2*a + 2*b*x] + 6*c*d^2*x*Sin[2*a + 2*b*x] + 3*d^3*x^2*Sin[2*a + 2*b*x]))/
(4*b^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1612 vs. \(2 (535 ) = 1070\).

Time = 2.44 (sec) , antiderivative size = 1613, normalized size of antiderivative = 2.68

method	result	size
risch	\(\text {Expression too large to display}\)	\(1613\)

[In]

int((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

6*I/b^4*a*d^3*dilog(1+I*exp(I*(b*x+a)))-9*I/b^2*c*d^2*polylog(2,exp(I*(b*x+a)))*x+9*I/b^2*c*d^2*polylog(2,-exp
(I*(b*x+a)))*x+3/b^3*d^3*ln(1-exp(I*(b*x+a)))*x-3/b^3*d^3*ln(exp(I*(b*x+a))+1)*x-3/b^3*c*d^2*ln(exp(I*(b*x+a))
+1)+3/b^3*c*d^2*ln(exp(I*(b*x+a))-1)-3/b^4*a*d^3*ln(exp(I*(b*x+a))-1)+3/b^4*d^3*ln(1-exp(I*(b*x+a)))*a-3*I*d^3
*polylog(2,exp(I*(b*x+a)))/b^4-9*I*d^3*polylog(4,-exp(I*(b*x+a)))/b^4+6/b^2*d^2*c*ln(1+I*exp(I*(b*x+a)))*x-6/b
^2*d^2*c*ln(1-I*exp(I*(b*x+a)))*x+6/b^3*d^2*c*ln(1+I*exp(I*(b*x+a)))*a-6/b^3*d^2*c*ln(1-I*exp(I*(b*x+a)))*a+6*
I/b^2*d*c^2*arctan(exp(I*(b*x+a)))+6*I/b^4*d^3*a^2*arctan(exp(I*(b*x+a)))+6*I*d^3*x*polylog(2,I*exp(I*(b*x+a))
)/b^3+9*I*d^3*polylog(4,exp(I*(b*x+a)))/b^4+9/2/b^3*c*d^2*a^2*ln(exp(I*(b*x+a))-1)-9/2/b^2*c^2*d*a*ln(exp(I*(b
*x+a))-1)+9/2/b^2*d*c^2*ln(1-exp(I*(b*x+a)))*a-9/2/b^3*c*d^2*ln(1-exp(I*(b*x+a)))*a^2+6*d^3*polylog(3,-I*exp(I
*(b*x+a)))/b^4-6*d^3*polylog(3,I*exp(I*(b*x+a)))/b^4-3/2/b*c^3*ln(exp(I*(b*x+a))+1)+3/2/b*c^3*ln(exp(I*(b*x+a)
)-1)+3/b^2*d^3*ln(1+I*exp(I*(b*x+a)))*x^2+3/b^4*d^3*a^2*ln(1-I*exp(I*(b*x+a)))-3/b^4*d^3*a^2*ln(1+I*exp(I*(b*x
+a)))-3/b^2*d^3*ln(1-I*exp(I*(b*x+a)))*x^2+9/2/b*d*c^2*ln(1-exp(I*(b*x+a)))*x-9/2/b*d*c^2*ln(exp(I*(b*x+a))+1)
*x+9/2/b*c*d^2*ln(1-exp(I*(b*x+a)))*x^2-9/2/b*c*d^2*ln(exp(I*(b*x+a))+1)*x^2-3/2/b^4*d^3*a^3*ln(exp(I*(b*x+a))
-1)+3/2/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^3+9/b^3*d^3*polylog(3,exp(I*(b*x+a)))*x-9/b^3*d^3*polylog(3,-exp(I*(b*x
+a)))*x+3/2/b*d^3*ln(1-exp(I*(b*x+a)))*x^3-3/2/b*d^3*ln(exp(I*(b*x+a))+1)*x^3+9/b^3*c*d^2*polylog(3,exp(I*(b*x
+a)))-9/b^3*c*d^2*polylog(3,-exp(I*(b*x+a)))-12*I/b^3*d^2*c*a*arctan(exp(I*(b*x+a)))+1/b^2/(exp(2*I*(b*x+a))-1
)^2/(exp(2*I*(b*x+a))+1)*(3*d^3*x^3*b*exp(5*I*(b*x+a))+9*c*d^2*x^2*b*exp(5*I*(b*x+a))+9*c^2*d*x*b*exp(5*I*(b*x
+a))-2*d^3*x^3*b*exp(3*I*(b*x+a))+3*c^3*b*exp(5*I*(b*x+a))-6*c*d^2*x^2*b*exp(3*I*(b*x+a))-3*I*d^3*x^2*exp(5*I*
(b*x+a))-6*c^2*d*x*b*exp(3*I*(b*x+a))+3*d^3*x^3*b*exp(I*(b*x+a))+3*I*d^3*x^2*exp(I*(b*x+a))-2*c^3*b*exp(3*I*(b
*x+a))+9*c*d^2*x^2*b*exp(I*(b*x+a))-3*I*c^2*d*exp(5*I*(b*x+a))+9*c^2*d*x*b*exp(I*(b*x+a))+3*c^3*b*exp(I*(b*x+a
))+3*I*c^2*d*exp(I*(b*x+a))+6*I*c*d^2*x*exp(I*(b*x+a))-6*I*c*d^2*x*exp(5*I*(b*x+a)))+3*I*d^3*polylog(2,-exp(I*
(b*x+a)))/b^4-9/2*I/b^2*d^3*polylog(2,exp(I*(b*x+a)))*x^2+9/2*I/b^2*d^3*polylog(2,-exp(I*(b*x+a)))*x^2+9/2*I/b
^2*d*c^2*polylog(2,-exp(I*(b*x+a)))-9/2*I/b^2*d*c^2*polylog(2,exp(I*(b*x+a)))-6*I*d^3*x*polylog(2,-I*exp(I*(b*
x+a)))/b^3-6*I/b^4*a*d^3*dilog(1-I*exp(I*(b*x+a)))-6*I/b^3*c*d^2*dilog(1+I*exp(I*(b*x+a)))+6*I/b^3*c*d^2*dilog
(1-I*exp(I*(b*x+a)))-6*I/b^4*d^3*polylog(2,-I*exp(I*(b*x+a)))*a+6*I/b^4*d^3*polylog(2,I*exp(I*(b*x+a)))*a

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. 3173 vs. \(2 (509) = 1018\).

Time = 0.43 (sec) , antiderivative size = 3173, normalized size of antiderivative = 5.28 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x + 4*b^3*c^3 - 6*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3
*c^2*d*x + b^3*c^3)*cos(b*x + a)^2 - 6*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos(b*x + a)*sin(b*x + a) + 3
*((3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 2*I*d^3)*cos(b*x + a)^3 + (-3*I*b^2*d^3*x^2 - 6*I*b^2*c
*d^2*x - 3*I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a))*dilog(cos(b*x + a) + I*sin(b*x + a)) + 3*((-3*I*b^2*d^3*x^2 -
6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a)^3 + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d
+ 2*I*d^3)*cos(b*x + a))*dilog(cos(b*x + a) - I*sin(b*x + a)) + 12*((-I*b*d^3*x - I*b*c*d^2)*cos(b*x + a)^3 +
(I*b*d^3*x + I*b*c*d^2)*cos(b*x + a))*dilog(I*cos(b*x + a) + sin(b*x + a)) + 12*((-I*b*d^3*x - I*b*c*d^2)*cos
(b*x + a)^3 + (I*b*d^3*x + I*b*c*d^2)*cos(b*x + a))*dilog(I*cos(b*x + a) - sin(b*x + a)) + 12*((I*b*d^3*x + I*
b*c*d^2)*cos(b*x + a)^3 + (-I*b*d^3*x - I*b*c*d^2)*cos(b*x + a))*dilog(-I*cos(b*x + a) + sin(b*x + a)) + 12*((
I*b*d^3*x + I*b*c*d^2)*cos(b*x + a)^3 + (-I*b*d^3*x - I*b*c*d^2)*cos(b*x + a))*dilog(-I*cos(b*x + a) - sin(b*x
+ a)) + 3*((3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 2*I*d^3)*cos(b*x + a)^3 + (-3*I*b^2*d^3*x^2 -
6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a))*dilog(-cos(b*x + a) + I*sin(b*x + a)) + 3*((-3*I*b^2
*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a)^3 + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*
I*b^2*c^2*d + 2*I*d^3)*cos(b*x + a))*dilog(-cos(b*x + a) - I*sin(b*x + a)) + 3*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2
+ b^3*c^3 + 2*b*c*d^2 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a)^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3
+ 2*b*c*d^2 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a))*log(cos(b*x + a) + I*sin(b*x + a) + 1) + 6*((b^2*c^2*d
- 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)^3 - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a))*log(cos(b*x + a) +
I*sin(b*x + a) + I) + 3*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 2*b*c*d^2 + (3*b^3*c^2*d + 2*b*d^3)*x)*co
s(b*x + a)^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 2*b*c*d^2 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a))
*log(cos(b*x + a) - I*sin(b*x + a) + 1) - 6*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)^3 - (b^2*c^2*d -
2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a))*log(cos(b*x + a) - I*sin(b*x + a) + I) + 6*((b^2*d^3*x^2 + 2*b^2*c*d^2*x
+ 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)^3 - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a))
*log(I*cos(b*x + a) + sin(b*x + a) + 1) - 6*((b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a
)^3 - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a))*log(I*cos(b*x + a) - sin(b*x + a) +
1) + 6*((b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)^3 - (b^2*d^3*x^2 + 2*b^2*c*d^2*x +
2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a))*log(-I*cos(b*x + a) + sin(b*x + a) + 1) - 6*((b^2*d^3*x^2 + 2*b^2*c*d^2*x
+ 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)^3 - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a))
*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - 3*((b^3*c^3 - 3*a*b^2*c^2*d + (3*a^2 + 2)*b*c*d^2 - (a^3 + 2*a)*d^3
)*cos(b*x + a)^3 - (b^3*c^3 - 3*a*b^2*c^2*d + (3*a^2 + 2)*b*c*d^2 - (a^3 + 2*a)*d^3)*cos(b*x + a))*log(-1/2*co
s(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - 3*((b^3*c^3 - 3*a*b^2*c^2*d + (3*a^2 + 2)*b*c*d^2 - (a^3 + 2*a)*d^3)*
cos(b*x + a)^3 - (b^3*c^3 - 3*a*b^2*c^2*d + (3*a^2 + 2)*b*c*d^2 - (a^3 + 2*a)*d^3)*cos(b*x + a))*log(-1/2*cos(
b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - 3*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^
3 + 2*a)*d^3 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a)^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*
a^2*b*c*d^2 + (a^3 + 2*a)*d^3 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a))*log(-cos(b*x + a) + I*sin(b*x + a) +
1) + 6*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)^3 - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a))
*log(-cos(b*x + a) + I*sin(b*x + a) + I) - 3*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 +
(a^3 + 2*a)*d^3 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a)^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2 + (a^3 + 2*a)*d^3 + (3*b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a))*log(-cos(b*x + a) - I*sin(b*x + a
) + 1) - 6*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)^3 - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x +
a))*log(-cos(b*x + a) - I*sin(b*x + a) + I) + 18*(-I*d^3*cos(b*x + a)^3 + I*d^3*cos(b*x + a))*polylog(4, cos(
b*x + a) + I*sin(b*x + a)) + 18*(I*d^3*cos(b*x + a)^3 - I*d^3*cos(b*x + a))*polylog(4, cos(b*x + a) - I*sin(b*
x + a)) + 18*(-I*d^3*cos(b*x + a)^3 + I*d^3*cos(b*x + a))*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) + 18*(I*d
^3*cos(b*x + a)^3 - I*d^3*cos(b*x + a))*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) - 18*((b*d^3*x + b*c*d^2)*c
os(b*x + a)^3 - (b*d^3*x + b*c*d^2)*cos(b*x + a))*polylog(3, cos(b*x + a) + I*sin(b*x + a)) - 18*((b*d^3*x + b
*c*d^2)*cos(b*x + a)^3 - (b*d^3*x + b*c*d^2)*cos(b*x + a))*polylog(3, cos(b*x + a) - I*sin(b*x + a)) - 12*(d^3
*cos(b*x + a)^3 - d^3*cos(b*x + a))*polylog(3, I*cos(b*x + a) + sin(b*x + a)) + 12*(d^3*cos(b*x + a)^3 - d^3*c
os(b*x + a))*polylog(3, I*cos(b*x + a) - sin(b*x + a)) - 12*(d^3*cos(b*x + a)^3 - d^3*cos(b*x + a))*polylog(3,
-I*cos(b*x + a) + sin(b*x + a)) + 12*(d^3*cos(b*x + a)^3 - d^3*cos(b*x + a))*polylog(3, -I*cos(b*x + a) - sin
(b*x + a)) + 18*((b*d^3*x + b*c*d^2)*cos(b*x + a)^3 - (b*d^3*x + b*c*d^2)*cos(b*x + a))*polylog(3, -cos(b*x +
a) + I*sin(b*x + a)) + 18*((b*d^3*x + b*c*d^2)*cos(b*x + a)^3 - (b*d^3*x + b*c*d^2)*cos(b*x + a))*polylog(3, -
cos(b*x + a) - I*sin(b*x + a)))/(b^4*cos(b*x + a)^3 - b^4*cos(b*x + a))

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**3*csc(b*x+a)**3*sec(b*x+a)**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8032 vs. \(2 (509) = 1018\).

Time = 3.62 (sec) , antiderivative size = 8032, normalized size of antiderivative = 13.36 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="maxima")

[Out]

1/4*(c^3*(2*(3*cos(b*x + a)^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)) - 3*log(cos(b*x + a) + 1) + 3*log(cos(b*x +
a) - 1)) - 3*a*c^2*d*(2*(3*cos(b*x + a)^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)) - 3*log(cos(b*x + a) + 1) + 3*
log(cos(b*x + a) - 1))/b + 3*a^2*c*d^2*(2*(3*cos(b*x + a)^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)) - 3*log(cos(b
*x + a) + 1) + 3*log(cos(b*x + a) - 1))/b^2 - a^3*d^3*(2*(3*cos(b*x + a)^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)
) - 3*log(cos(b*x + a) + 1) + 3*log(cos(b*x + a) - 1))/b^3 + 4*(12*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3
+ a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2
- a*d^3)*(b*x + a))*cos(6*b*x + 6*a) - (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d
^3)*(b*x + a))*cos(4*b*x + 4*a) - (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(
b*x + a))*cos(2*b*x + 2*a) + (I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a
*d^3)*(b*x + a))*sin(6*b*x + 6*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*
d^2 + I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^2*d^3 - I*a^2*d^3 + 2
*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 1) + 12*(b^2*c^2*d -
2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a
)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(6*b*x + 6*a) - (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d
^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 +
a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 +
I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(6*b*x + 6*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)
^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*
(b*x + a)^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), -sin
(b*x + a) + 1) - 6*((b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6
*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a) + ((b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x +
a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(6*b*x + 6*a) - ((b*x + a)^3*d^3 + 2*b*c*d^
2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(4
*b*x + 4*a) - ((b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*
c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^3*d^3 - 2*I*b*c*d^2 + 2*I*a*d^3 + 3*(-I*b
*c*d^2 + I*a*d^3)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 + (-3*I*a^2 - 2*I)*d^3)*(b*x + a))*sin(6*b*x +
6*a) - (I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d -
6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a))*sin(4*b*x + 4*a) - (I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^
3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a))*sin
(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 12*(b*c*d^2 - a*d^3 + (b*c*d^2 - a*d^3)*cos(6*b*x + 6
*a) - (b*c*d^2 - a*d^3)*cos(4*b*x + 4*a) - (b*c*d^2 - a*d^3)*cos(2*b*x + 2*a) + (I*b*c*d^2 - I*a*d^3)*sin(6*b*
x + 6*a) + (-I*b*c*d^2 + I*a*d^3)*sin(4*b*x + 4*a) + (-I*b*c*d^2 + I*a*d^3)*sin(2*b*x + 2*a))*arctan2(sin(b*x
+ a), cos(b*x + a) - 1) - 6*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 +
(3*a^2 + 2)*d^3)*(b*x + a) + ((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 +
(3*a^2 + 2)*d^3)*(b*x + a))*cos(6*b*x + 6*a) - ((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^
2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - ((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x
+ a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^3*d^3 + 3*
(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 + (-3*I*a^2 - 2*I)*d^3)*(b*x + a))*sin(6*
b*x + 6*a) - (I*(b*x + a)^3*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*
a^2 + 2*I)*d^3)*(b*x + a))*sin(4*b*x + 4*a) - (I*(b*x + a)^3*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*
b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x +
a) + 1) + 12*(-I*(b*x + a)^3*d^3 - b^2*c^2*d + 2*a*b*c*d^2 - a^2*d^3 + (-3*I*b*c*d^2 + (3*I*a - 1)*d^3)*(b*x
+ a)^2 + (-3*I*b^2*c^2*d + 2*(3*I*a - 1)*b*c*d^2 + (-3*I*a^2 + 2*a)*d^3)*(b*x + a))*cos(5*b*x + 5*a) + 8*(I*(b
*x + a)^3*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2*d^3)*(b*x + a))*c
os(3*b*x + 3*a) + 12*(-I*(b*x + a)^3*d^3 + b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3 + (-3*I*b*c*d^2 + (3*I*a + 1)*d^3
)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 2*(3*I*a + 1)*b*c*d^2 + (-3*I*a^2 - 2*a)*d^3)*(b*x + a))*cos(b*x + a) + 24*(
b*c*d^2 + (b*x + a)*d^3 - a*d^3 + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(6*b*x + 6*a) - (b*c*d^2 + (b*x + a)*d^
3 - a*d^3)*cos(4*b*x + 4*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(2*b*x + 2*a) + (I*b*c*d^2 + I*(b*x + a)*d^
3 - I*a*d^3)*sin(6*b*x + 6*a) + (-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*sin(4*b*x + 4*a) + (-I*b*c*d^2 - I*(b
*x + a)*d^3 + I*a*d^3)*sin(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 24*(b*c*d^2 + (b*x + a)*d^3 - a*d^3 + (b*c
*d^2 + (b*x + a)*d^3 - a*d^3)*cos(6*b*x + 6*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(4*b*x + 4*a) - (b*c*d^2
+ (b*x + a)*d^3 - a*d^3)*cos(2*b*x + 2*a) - (-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*sin(6*b*x + 6*a) - (I*b*
c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*sin(4*b*x + 4*a) - (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*sin(2*b*x + 2*a)
)*dilog(-I*e^(I*b*x + I*a)) + 6*(3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2
- a*d^3)*(b*x + a) + (3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2 - a*d^3)*(b
*x + a))*cos(6*b*x + 6*a) - (3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2 - a*
d^3)*(b*x + a))*cos(4*b*x + 4*a) - (3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d
^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + 3*I*(b*x + a)^2*d^3 + (3*I*a^2 + 2*
I)*d^3 + 6*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(6*b*x + 6*a) + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 - 3*I*(b*x + a)
^2*d^3 + (-3*I*a^2 - 2*I)*d^3 + 6*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-3*I*b^2*c^2*d + 6*I*a
*b*c*d^2 - 3*I*(b*x + a)^2*d^3 + (-3*I*a^2 - 2*I)*d^3 + 6*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*
dilog(-e^(I*b*x + I*a)) - 6*(3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2 - a*
d^3)*(b*x + a) + (3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x +
a))*cos(6*b*x + 6*a) - (3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2 - a*d^3)
*(b*x + a))*cos(4*b*x + 4*a) - (3*b^2*c^2*d - 6*a*b*c*d^2 + 3*(b*x + a)^2*d^3 + (3*a^2 + 2)*d^3 + 6*(b*c*d^2 -
a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 - 3*I*(b*x + a)^2*d^3 + (-3*I*a^2 - 2*I)
*d^3 + 6*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(6*b*x + 6*a) - (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + 3*I*(b*x + a)^2
*d^3 + (3*I*a^2 + 2*I)*d^3 + 6*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) - (3*I*b^2*c^2*d - 6*I*a*b*c*
d^2 + 3*I*(b*x + a)^2*d^3 + (3*I*a^2 + 2*I)*d^3 + 6*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e
^(I*b*x + I*a)) + 3*(I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*
b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a) + (I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*
(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a))*cos(6*b*x
+ 6*a) + (-I*(b*x + a)^3*d^3 - 2*I*b*c*d^2 + 2*I*a*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + (-3*I*b^2*c^2
*d + 6*I*a*b*c*d^2 + (-3*I*a^2 - 2*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (-I*(b*x + a)^3*d^3 - 2*I*b*c*d^2 + 2
*I*a*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 + (-3*I*a^2 - 2*I)*d^3)*(b*x
+ a))*cos(2*b*x + 2*a) - ((b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^
2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*sin(6*b*x + 6*a) + ((b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*
(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*sin(4*b*x + 4*a) + ((
b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 +
2)*d^3)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + 3*(-I*(b*x +
a)^3*d^3 - 2*I*b*c*d^2 + 2*I*a*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 +
(-3*I*a^2 - 2*I)*d^3)*(b*x + a) + (-I*(b*x + a)^3*d^3 - 2*I*b*c*d^2 + 2*I*a*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b
*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 + (-3*I*a^2 - 2*I)*d^3)*(b*x + a))*cos(6*b*x + 6*a) + (I*(b*x + a)
^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I
*a^2 + 2*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*(I*b*c*d^2 - I
*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a))*cos(2*b*x + 2*a) + ((b*
x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2
)*d^3)*(b*x + a))*sin(6*b*x + 6*a) - ((b*x + a)^3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2
+ (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*sin(4*b*x + 4*a) - ((b*x + a)^3*d^3 + 2*b*c*d^2 - 2
*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*sin(2*b*x
+ 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 6*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x +
a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a) + (I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 +
I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*cos(6*b*x + 6*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^
2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(
b*x + a)^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (b^2*c^2*d - 2*a*b*c*d^2 +
(b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(6*b*x + 6*a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x
+ a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)
^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*si
n(b*x + a) + 1) + 6*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(
b*x + a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))
*cos(6*b*x + 6*a) + (I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*
x + a))*cos(4*b*x + 4*a) + (I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d
^3)*(b*x + a))*cos(2*b*x + 2*a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(
b*x + a))*sin(6*b*x + 6*a) - (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x +
a))*sin(4*b*x + 4*a) - (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*
sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) - 36*(d^3*cos(6*b*x + 6*a) - d^3*c
os(4*b*x + 4*a) - d^3*cos(2*b*x + 2*a) + I*d^3*sin(6*b*x + 6*a) - I*d^3*sin(4*b*x + 4*a) - I*d^3*sin(2*b*x + 2
*a) + d^3)*polylog(4, -e^(I*b*x + I*a)) + 36*(d^3*cos(6*b*x + 6*a) - d^3*cos(4*b*x + 4*a) - d^3*cos(2*b*x + 2*
a) + I*d^3*sin(6*b*x + 6*a) - I*d^3*sin(4*b*x + 4*a) - I*d^3*sin(2*b*x + 2*a) + d^3)*polylog(4, e^(I*b*x + I*a
)) + 24*(I*d^3*cos(6*b*x + 6*a) - I*d^3*cos(4*b*x + 4*a) - I*d^3*cos(2*b*x + 2*a) - d^3*sin(6*b*x + 6*a) + d^3
*sin(4*b*x + 4*a) + d^3*sin(2*b*x + 2*a) + I*d^3)*polylog(3, I*e^(I*b*x + I*a)) + 24*(-I*d^3*cos(6*b*x + 6*a)
+ I*d^3*cos(4*b*x + 4*a) + I*d^3*cos(2*b*x + 2*a) + d^3*sin(6*b*x + 6*a) - d^3*sin(4*b*x + 4*a) - d^3*sin(2*b*
x + 2*a) - I*d^3)*polylog(3, -I*e^(I*b*x + I*a)) + 36*(I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3 + (I*b*c*d^2 + I*
(b*x + a)*d^3 - I*a*d^3)*cos(6*b*x + 6*a) + (-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*cos(4*b*x + 4*a) + (-I*b*
c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*cos(2*b*x + 2*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(6*b*x + 6*a) + (b*
c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(4*b*x + 4*a) + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(
3, -e^(I*b*x + I*a)) + 36*(-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3 + (-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*c
os(6*b*x + 6*a) + (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*cos(4*b*x + 4*a) + (I*b*c*d^2 + I*(b*x + a)*d^3 - I*
a*d^3)*cos(2*b*x + 2*a) + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(6*b*x + 6*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^
3)*sin(4*b*x + 4*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) + 12*((b
*x + a)^3*d^3 - I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*a^2*d^3 + (3*b*c*d^2 - (3*a + I)*d^3)*(b*x + a)^2 + (3*b^2*c^2
*d - 2*(3*a + I)*b*c*d^2 + (3*a^2 + 2*I*a)*d^3)*(b*x + a))*sin(5*b*x + 5*a) - 8*((b*x + a)^3*d^3 + 3*(b*c*d^2
- a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*sin(3*b*x + 3*a) + 12*((b*x + a)^3*d^3
+ I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2*d^3 + (3*b*c*d^2 - (3*a - I)*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 2*(3*a -
I)*b*c*d^2 + (3*a^2 - 2*I*a)*d^3)*(b*x + a))*sin(b*x + a))/(-4*I*b^3*cos(6*b*x + 6*a) + 4*I*b^3*cos(4*b*x + 4
*a) + 4*I*b^3*cos(2*b*x + 2*a) + 4*b^3*sin(6*b*x + 6*a) - 4*b^3*sin(4*b*x + 4*a) - 4*b^3*sin(2*b*x + 2*a) - 4*
I*b^3))/b

Giac [F]

\[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*csc(b*x + a)^3*sec(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}