\(\int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx\) [235]

Optimal result

Integrand size = 22, antiderivative size = 350 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {6 d (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \csc (a+b x)}{b}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4} \]

[Out]

-2*I*(d*x+c)^3*arctan(exp(I*(b*x+a)))/b-6*d*(d*x+c)^2*arctanh(exp(I*(b*x+a)))/b^2-(d*x+c)^3*csc(b*x+a)/b+6*I*d
^2*(d*x+c)*polylog(2,-exp(I*(b*x+a)))/b^3+3*I*d*(d*x+c)^2*polylog(2,-I*exp(I*(b*x+a)))/b^2-3*I*d*(d*x+c)^2*pol
ylog(2,I*exp(I*(b*x+a)))/b^2-6*I*d^2*(d*x+c)*polylog(2,exp(I*(b*x+a)))/b^3-6*d^3*polylog(3,-exp(I*(b*x+a)))/b^
4-6*d^2*(d*x+c)*polylog(3,-I*exp(I*(b*x+a)))/b^3+6*d^2*(d*x+c)*polylog(3,I*exp(I*(b*x+a)))/b^3+6*d^3*polylog(3
,exp(I*(b*x+a)))/b^4-6*I*d^3*polylog(4,-I*exp(I*(b*x+a)))/b^4+6*I*d^3*polylog(4,I*exp(I*(b*x+a)))/b^4

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2701, 327, 213, 4505, 6873, 12, 6874, 6408, 4266, 2611, 6744, 2320, 6724, 4268} \[ \int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {6 d (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \csc (a+b x)}{b} \]

[In]

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

[Out]

((-2*I)*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b - (6*d*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^2 - ((c + d*x)^3
*Csc[a + b*x])/b + ((6*I)*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^3 + ((3*I)*d*(c + d*x)^2*PolyLog[2, (-
I)*E^(I*(a + b*x))])/b^2 - ((3*I)*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - ((6*I)*d^2*(c + d*x)*Poly
Log[2, E^(I*(a + b*x))])/b^3 - (6*d^3*PolyLog[3, -E^(I*(a + b*x))])/b^4 - (6*d^2*(c + d*x)*PolyLog[3, (-I)*E^(
I*(a + b*x))])/b^3 + (6*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, E^(I*(a + b*x))])
/b^4 - ((6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 + ((6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4

Rule 12

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

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 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 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 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 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 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 6873

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

Rule 6874

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

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(739\) vs. \(2(350)=700\).

Time = 6.39 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.11 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {(c+d x)^3 \csc (a)}{b}+\frac {3 d \left ((c+d x)^2 \log \left (1-e^{i (a+b x)}\right )-(c+d x)^2 \log \left (1+e^{i (a+b x)}\right )+\frac {2 i d \left (b (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+i d \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )\right )}{b^2}+\frac {2 d \left (-i b (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+d \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )\right )}{b^2}\right )}{b^2}+\frac {-2 i b^3 c^3 \arctan \left (e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1-i e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1-i e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1+i e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1+i e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4}+\frac {\sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (-c^3 \sin \left (\frac {b x}{2}\right )-3 c^2 d x \sin \left (\frac {b x}{2}\right )-3 c d^2 x^2 \sin \left (\frac {b x}{2}\right )-d^3 x^3 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c^3 \sin \left (\frac {b x}{2}\right )+3 c^2 d x \sin \left (\frac {b x}{2}\right )+3 c d^2 x^2 \sin \left (\frac {b x}{2}\right )+d^3 x^3 \sin \left (\frac {b x}{2}\right )\right )}{2 b} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1157 vs. \(2 (313 ) = 626\).

Time = 1.92 (sec) , antiderivative size = 1158, normalized size of antiderivative = 3.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(1158\)

[In]

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

[Out]

-6*I*d^3*polylog(4,-I*exp(I*(b*x+a)))/b^4+6*I*d^3*polylog(4,I*exp(I*(b*x+a)))/b^4+6*I/b^2*a*c^2*d*arctan(exp(I
*(b*x+a)))+6*I/b^2*d^2*c*polylog(2,-I*exp(I*(b*x+a)))*x-6*I/b^2*d^2*c*polylog(2,I*exp(I*(b*x+a)))*x-6*I/b^3*a^
2*c*d^2*arctan(exp(I*(b*x+a)))-3/b^2*d^3*ln(exp(I*(b*x+a))+1)*x^2+3/b^2*d^3*ln(1-exp(I*(b*x+a)))*x^2-3/b^2*c^2
*d*ln(exp(I*(b*x+a))+1)+3/b^2*c^2*d*ln(exp(I*(b*x+a))-1)-1/b*d^3*ln(1+I*exp(I*(b*x+a)))*x^3-6/b^3*d^2*c*polylo
g(3,-I*exp(I*(b*x+a)))+1/b*d^3*ln(1-I*exp(I*(b*x+a)))*x^3+3/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^2+6/b^3*d^2*c*polyl
og(3,I*exp(I*(b*x+a)))-1/b^4*a^3*d^3*ln(1+I*exp(I*(b*x+a)))+1/b^4*a^3*d^3*ln(1-I*exp(I*(b*x+a)))-6/b^3*d^3*pol
ylog(3,-I*exp(I*(b*x+a)))*x+3/b^4*d^3*a^2*ln(exp(I*(b*x+a))-1)+6/b^3*d^3*polylog(3,I*exp(I*(b*x+a)))*x-2*I/b*c
^3*arctan(exp(I*(b*x+a)))-6*d^3*polylog(3,-exp(I*(b*x+a)))/b^4+6*d^3*polylog(3,exp(I*(b*x+a)))/b^4+6/b^3*d^3*l
n(1-exp(I*(b*x+a)))*a*x-3/b*c^2*d*ln(1+I*exp(I*(b*x+a)))*x-3/b^3*a^2*c*d^2*ln(1-I*exp(I*(b*x+a)))+3/b^2*c^2*d*
ln(1-I*exp(I*(b*x+a)))*a-3/b^2*c^2*d*ln(1+I*exp(I*(b*x+a)))*a-2*I*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)*exp(I*(b
*x+a))/b/(exp(2*I*(b*x+a))-1)-6/b^2*c*d^2*ln(exp(I*(b*x+a))+1)*x+3/b*c^2*d*ln(1-I*exp(I*(b*x+a)))*x-6*I/b^4*d^
3*polylog(2,exp(I*(b*x+a)))*a-6*I/b^4*a*d^3*dilog(exp(I*(b*x+a))+1)+6*I/b^3*d^3*polylog(2,-exp(I*(b*x+a)))*x+6
*I/b^4*d^3*polylog(2,-exp(I*(b*x+a)))*a-6*I/b^3*d^3*polylog(2,exp(I*(b*x+a)))*x-3*I/b^2*c^2*d*polylog(2,I*exp(
I*(b*x+a)))+2*I/b^4*a^3*d^3*arctan(exp(I*(b*x+a)))+6*I/b^3*c*d^2*dilog(exp(I*(b*x+a))+1)+3*I/b^2*d^3*polylog(2
,-I*exp(I*(b*x+a)))*x^2-3*I/b^2*d^3*polylog(2,I*exp(I*(b*x+a)))*x^2-6*I/b^4*d^3*a*dilog(exp(I*(b*x+a)))+6*I/b^
3*c*d^2*dilog(exp(I*(b*x+a)))+3*I/b^2*c^2*d*polylog(2,-I*exp(I*(b*x+a)))-3/b*d^2*c*ln(1+I*exp(I*(b*x+a)))*x^2+
3/b*d^2*c*ln(1-I*exp(I*(b*x+a)))*x^2+3/b^3*a^2*c*d^2*ln(1+I*exp(I*(b*x+a)))-6/b^3*c*d^2*a*ln(exp(I*(b*x+a))-1)

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. 1753 vs. \(2 (296) = 592\).

Time = 0.36 (sec) , antiderivative size = 1753, normalized size of antiderivative = 5.01 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

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. 3257 vs. \(2 (296) = 592\).

Time = 0.99 (sec) , antiderivative size = 3257, normalized size of antiderivative = 9.31 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(c^3*(2/sin(b*x + a) - log(sin(b*x + a) + 1) + log(sin(b*x + a) - 1)) - 3*a*c^2*d*(2/sin(b*x + a) - log(s
in(b*x + a) + 1) + log(sin(b*x + a) - 1))/b + 3*a^2*c*d^2*(2/sin(b*x + a) - log(sin(b*x + a) + 1) + log(sin(b*
x + a) - 1))/b^2 - a^3*d^3*(2/sin(b*x + a) - log(sin(b*x + a) + 1) + log(sin(b*x + a) - 1))/b^3 - 2*(2*((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) - ((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))*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 - 2*I*a*b*c*d^2 + I*a^2*d^3)*(b*x + a))
*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 1) + 2*((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) - ((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))*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 - 2*I*a*b*c*d^2 + I*a^2*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*x + a),
-sin(b*x + a) + 1) + 6*(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(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(2*b*x + 2*a)
)*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3 - (b^2*c^2*d - 2*a*b*c*d^2 +
a^2*d^3)*cos(2*b*x + 2*a) + (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*a^2*d^3)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a),
cos(b*x + a) - 1) + 6*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) - ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^
3)*(b*x + a))*cos(2*b*x + 2*a) - (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arc
tan2(sin(b*x + a), -cos(b*x + a) + 1) - 4*((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))*cos(b*x + a) + 6*(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(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(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 6*(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(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(2*b*x + 2*a))*dilog(-I*e^(I*b*x + I*a)) - 12*(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(2*b*x + 2*a) + (-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*sin(2*b*x + 2*a))*dilo
g(-e^(I*b*x + I*a)) + 12*(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(2*b*x + 2*a)
- (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) - 3*(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(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(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^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(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(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + (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) + (-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))*cos(2*b*x + 2*
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 - 2*a*b*c*d^2 + a^2*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) + (-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) + (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))*cos(2*b*x + 2*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 - 2*a*b*c*d^2 + a^2*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) + 12*(d^3*cos(2*b*x + 2*a) + I*d^3*sin(2*b*x +
2*a) - d^3)*polylog(4, I*e^(I*b*x + I*a)) - 12*(d^3*cos(2*b*x + 2*a) + I*d^3*sin(2*b*x + 2*a) - d^3)*polylog(
4, -I*e^(I*b*x + I*a)) - 12*(-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(2*b*x + 2*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, I*e^(I*b*x + I*a)) - 12*(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(2*b*x + 2*a) + (b*c*d^2 + (b
*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, -I*e^(I*b*x + I*a)) - 12*(-I*d^3*cos(2*b*x + 2*a) + d^3*sin(
2*b*x + 2*a) + I*d^3)*polylog(3, -e^(I*b*x + I*a)) - 12*(I*d^3*cos(2*b*x + 2*a) - d^3*sin(2*b*x + 2*a) - I*d^3
)*polylog(3, e^(I*b*x + I*a)) - 4*(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))*sin(b*x + a))/(-2*I*b^3*cos(2*b*x + 2*a) + 2*b^3*sin(2*b*x + 2*a) + 2*I*
b^3))/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}