\(\int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx\) [266]

Optimal result

Integrand size = 22, antiderivative size = 469 \[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {24 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}-\frac {24 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec (a+b x)}{b} \]

[Out]

24*I*d^4*polylog(4,-I*exp(I*(b*x+a)))/b^5-2*(d*x+c)^4*arctanh(exp(I*(b*x+a)))/b-24*I*d^4*polylog(4,I*exp(I*(b*
x+a)))/b^5-24*I*d^3*(d*x+c)*polylog(4,-exp(I*(b*x+a)))/b^4+24*I*d^3*(d*x+c)*polylog(4,exp(I*(b*x+a)))/b^4+4*I*
d*(d*x+c)^3*polylog(2,-exp(I*(b*x+a)))/b^2-12*d^2*(d*x+c)^2*polylog(3,-exp(I*(b*x+a)))/b^3+24*d^3*(d*x+c)*poly
log(3,-I*exp(I*(b*x+a)))/b^4-24*d^3*(d*x+c)*polylog(3,I*exp(I*(b*x+a)))/b^4+12*d^2*(d*x+c)^2*polylog(3,exp(I*(
b*x+a)))/b^3-4*I*d*(d*x+c)^3*polylog(2,exp(I*(b*x+a)))/b^2+8*I*d*(d*x+c)^3*arctan(exp(I*(b*x+a)))/b^2-12*I*d^2
*(d*x+c)^2*polylog(2,-I*exp(I*(b*x+a)))/b^3+12*I*d^2*(d*x+c)^2*polylog(2,I*exp(I*(b*x+a)))/b^3+24*d^4*polylog(
5,-exp(I*(b*x+a)))/b^5-24*d^4*polylog(5,exp(I*(b*x+a)))/b^5+(d*x+c)^4*sec(b*x+a)/b

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2702, 327, 213, 4505, 6873, 12, 6874, 6408, 4268, 2611, 6744, 2320, 6724, 4266} \[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {24 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}-\frac {24 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}-\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {(c+d x)^4 \sec (a+b x)}{b} \]

[In]

Int[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^2,x]

[Out]

((8*I)*d*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b^2 - (2*(c + d*x)^4*ArcTanh[E^(I*(a + b*x))])/b + ((4*I)*d*(c +
d*x)^3*PolyLog[2, -E^(I*(a + b*x))])/b^2 - ((12*I)*d^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((
12*I)*d^2*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - ((4*I)*d*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))])/b
^2 - (12*d^2*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (24*d^3*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))
])/b^4 - (24*d^3*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + (12*d^2*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))
])/b^3 - ((24*I)*d^3*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))])/b^4 + ((24*I)*d^4*PolyLog[4, (-I)*E^(I*(a + b*x))
])/b^5 - ((24*I)*d^4*PolyLog[4, I*E^(I*(a + b*x))])/b^5 + ((24*I)*d^3*(c + d*x)*PolyLog[4, E^(I*(a + b*x))])/b
^4 + (24*d^4*PolyLog[5, -E^(I*(a + b*x))])/b^5 - (24*d^4*PolyLog[5, E^(I*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a
+ b*x])/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 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 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 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)^4 \text {arctanh}(\cos (a+b x))}{b}+\frac {(c+d x)^4 \sec (a+b x)}{b}-(4 d) \int (c+d x)^3 \left (-\frac {\text {arctanh}(\cos (a+b x))}{b}+\frac {\sec (a+b x)}{b}\right ) \, dx \\ & = -\frac {(c+d x)^4 \text {arctanh}(\cos (a+b x))}{b}+\frac {(c+d x)^4 \sec (a+b x)}{b}-(4 d) \int \frac {(c+d x)^3 (-\text {arctanh}(\cos (a+b x))+\sec (a+b x))}{b} \, dx \\ & = -\frac {(c+d x)^4 \text {arctanh}(\cos (a+b x))}{b}+\frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {(4 d) \int (c+d x)^3 (-\text {arctanh}(\cos (a+b x))+\sec (a+b x)) \, dx}{b} \\ & = -\frac {(c+d x)^4 \text {arctanh}(\cos (a+b x))}{b}+\frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {(4 d) \int \left (-(c+d x)^3 \text {arctanh}(\cos (a+b x))+(c+d x)^3 \sec (a+b x)\right ) \, dx}{b} \\ & = -\frac {(c+d x)^4 \text {arctanh}(\cos (a+b x))}{b}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {(4 d) \int (c+d x)^3 \text {arctanh}(\cos (a+b x)) \, dx}{b}-\frac {(4 d) \int (c+d x)^3 \sec (a+b x) \, dx}{b} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {\int b (c+d x)^4 \csc (a+b x) \, dx}{b}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (24 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (24 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^3}+\int (c+d x)^4 \csc (a+b x) \, dx \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {(4 d) \int (c+d x)^3 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac {(4 d) \int (c+d x)^3 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}-\frac {\left (24 d^4\right ) \int \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right ) \, dx}{b^4}+\frac {\left (24 d^4\right ) \int \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {\left (12 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (12 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (24 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}-\frac {\left (24 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (24 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (24 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right ) \, dx}{b^3} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (24 i d^4\right ) \int \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right ) \, dx}{b^4}-\frac {\left (24 i d^4\right ) \int \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (24 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}-\frac {\left (24 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {24 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}-\frac {24 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.67 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.48 \[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {b^4 (c+d x)^4 \log \left (1-e^{i (a+b x)}\right )-b^4 (c+d x)^4 \log \left (1+e^{i (a+b x)}\right )-4 d \left (-2 i b^3 c^3 \arctan \left (e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1-i e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1-i e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1+i e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1+i e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )\right )+4 i d \left (b^3 (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )-6 d^2 \left (b (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )+i d \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )\right )\right )-4 i d \left (b^3 (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )-6 d^2 \left (b (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )+i d \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )\right )\right )+b^4 (c+d x)^4 \sec (a+b x)}{b^5} \]

[In]

Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^2,x]

[Out]

(b^4*(c + d*x)^4*Log[1 - E^(I*(a + b*x))] - b^4*(c + d*x)^4*Log[1 + E^(I*(a + b*x))] - 4*d*((-2*I)*b^3*c^3*Arc
Tan[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))]) + (4
*I)*d*(b^3*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))] - 6
*d^2*(b*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))] + I*d*PolyLog[5, -E^(I*(a + b*x))])) - (4*I)*d*(b^3*(c + d*x)^3
*PolyLog[2, E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))] - 6*d^2*(b*(c + d*x)*PolyLo
g[4, E^(I*(a + b*x))] + I*d*PolyLog[5, E^(I*(a + b*x))])) + b^4*(c + d*x)^4*Sec[a + b*x])/b^5

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1865 vs. \(2 (422 ) = 844\).

Time = 2.05 (sec) , antiderivative size = 1866, normalized size of antiderivative = 3.98

method	result	size
risch	\(\text {Expression too large to display}\)	\(1866\)

[In]

int((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/b^5*a^4*d^4*ln(exp(I*(b*x+a))-1)+12/b^3*c^2*d^2*polylog(3,exp(I*(b*x+a)))-24/b^4*d^4*polylog(3,I*exp(I*(b*x+
a)))*x+24/b^4*d^4*polylog(3,-I*exp(I*(b*x+a)))*x+12/b^3*d^4*polylog(3,exp(I*(b*x+a)))*x^2-12/b^3*d^4*polylog(3
,-exp(I*(b*x+a)))*x^2+4/b^5*a^3*d^4*ln(1+I*exp(I*(b*x+a)))+4/b^2*d^4*ln(1+I*exp(I*(b*x+a)))*x^3-4/b^2*d^4*ln(1
-I*exp(I*(b*x+a)))*x^3-24/b^4*c*d^3*polylog(3,I*exp(I*(b*x+a)))+24/b^4*c*d^3*polylog(3,-I*exp(I*(b*x+a)))-4/b^
5*a^3*d^4*ln(1-I*exp(I*(b*x+a)))+1/b*d^4*ln(1-exp(I*(b*x+a)))*x^4-1/b^5*d^4*ln(1-exp(I*(b*x+a)))*a^4-1/b*d^4*l
n(exp(I*(b*x+a))+1)*x^4-24*I*d^4*polylog(4,I*exp(I*(b*x+a)))/b^5+12/b^3*c^2*d^2*ln(1+I*exp(I*(b*x+a)))*a-12/b^
3*c^2*d^2*ln(1-I*exp(I*(b*x+a)))*a-4/b*c^3*d*ln(exp(I*(b*x+a))+1)*x+4/b*c^3*d*ln(1-exp(I*(b*x+a)))*x-24/b^3*d^
3*c*polylog(3,-exp(I*(b*x+a)))*x+12/b^4*c*d^3*ln(1-I*exp(I*(b*x+a)))*a^2+4/b*d^3*c*ln(1-exp(I*(b*x+a)))*x^3+4/
b^4*d^3*c*ln(1-exp(I*(b*x+a)))*a^3-4/b*d^3*c*ln(exp(I*(b*x+a))+1)*x^3+24/b^3*d^3*c*polylog(3,exp(I*(b*x+a)))*x
-12/b^4*c*d^3*ln(1+I*exp(I*(b*x+a)))*a^2+6/b^3*a^2*c^2*d^2*ln(exp(I*(b*x+a))-1)+6/b*c^2*d^2*ln(1-exp(I*(b*x+a)
))*x^2-4/b^4*a^3*c*d^3*ln(exp(I*(b*x+a))-1)+24*d^4*polylog(5,-exp(I*(b*x+a)))/b^5-24*d^4*polylog(5,exp(I*(b*x+
a)))/b^5-12/b^3*c^2*d^2*polylog(3,-exp(I*(b*x+a)))-1/b*c^4*ln(exp(I*(b*x+a))+1)+1/b*c^4*ln(exp(I*(b*x+a))-1)+2
*exp(I*(b*x+a))*(d^4*x^4+4*c*d^3*x^3+6*c^2*d^2*x^2+4*c^3*d*x+c^4)/b/(exp(2*I*(b*x+a))+1)+24*I*d^4*polylog(4,-I
*exp(I*(b*x+a)))/b^5-24*I/b^3*c^2*d^2*a*arctan(exp(I*(b*x+a)))+24*I/b^3*c*d^3*polylog(2,I*exp(I*(b*x+a)))*x+24
*I/b^4*c*d^3*polylog(2,I*exp(I*(b*x+a)))*a-12*I/b^2*c^2*d^2*polylog(2,exp(I*(b*x+a)))*x+24*I/b^4*a^2*d^3*c*arc
tan(exp(I*(b*x+a)))+12*I/b^2*c^2*d^2*polylog(2,-exp(I*(b*x+a)))*x-24*I/b^3*c*d^3*polylog(2,-I*exp(I*(b*x+a)))*
x-24*I/b^4*c*d^3*polylog(2,-I*exp(I*(b*x+a)))*a+24*I/b^4*a*c*d^3*dilog(1+I*exp(I*(b*x+a)))-24*I/b^4*a*c*d^3*di
log(1-I*exp(I*(b*x+a)))-12*I/b^2*d^3*c*polylog(2,exp(I*(b*x+a)))*x^2+12*I/b^2*d^3*c*polylog(2,-exp(I*(b*x+a)))
*x^2-6/b*c^2*d^2*ln(exp(I*(b*x+a))+1)*x^2-4/b^2*a*c^3*d*ln(exp(I*(b*x+a))-1)+12/b^2*c^2*d^2*ln(1+I*exp(I*(b*x+
a)))*x-12/b^2*c^2*d^2*ln(1-I*exp(I*(b*x+a)))*x+4/b^2*c^3*d*ln(1-exp(I*(b*x+a)))*a-12/b^2*c*d^3*ln(1-I*exp(I*(b
*x+a)))*x^2+12/b^2*c*d^3*ln(1+I*exp(I*(b*x+a)))*x^2-6/b^3*c^2*d^2*ln(1-exp(I*(b*x+a)))*a^2+24*I/b^4*d^4*polylo
g(4,exp(I*(b*x+a)))*x-12*I/b^5*a^2*d^4*polylog(2,I*exp(I*(b*x+a)))-8*I/b^5*d^4*a^3*arctan(exp(I*(b*x+a)))+24*I
/b^4*d^3*c*polylog(4,exp(I*(b*x+a)))+12*I/b^5*a^2*d^4*polylog(2,-I*exp(I*(b*x+a)))-24*I/b^4*d^4*polylog(4,-exp
(I*(b*x+a)))*x-12*I/b^5*a^2*d^4*dilog(1+I*exp(I*(b*x+a)))+12*I/b^5*a^2*d^4*dilog(1-I*exp(I*(b*x+a)))-12*I/b^3*
d^4*polylog(2,-I*exp(I*(b*x+a)))*x^2+12*I/b^3*d^4*polylog(2,I*exp(I*(b*x+a)))*x^2-12*I/b^3*c^2*d^2*dilog(1+I*e
xp(I*(b*x+a)))+12*I/b^3*c^2*d^2*dilog(1-I*exp(I*(b*x+a)))-4*I/b^2*c^3*d*polylog(2,exp(I*(b*x+a)))+8*I/b^2*c^3*
d*arctan(exp(I*(b*x+a)))-24*I/b^4*d^3*c*polylog(4,-exp(I*(b*x+a)))-4*I/b^2*d^4*polylog(2,exp(I*(b*x+a)))*x^3+4
*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)))*x^3+4*I/b^2*c^3*d*polylog(2,-exp(I*(b*x+a)))

Fricas [F(-2)]

Exception generated. \[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^2,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   Too many variables

Sympy [F]

\[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\int \left (c + d x\right )^{4} \csc {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**4*csc(b*x+a)*sec(b*x+a)**2,x)

[Out]

Integral((c + d*x)**4*csc(a + b*x)*sec(a + b*x)**2, 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. 5709 vs. \(2 (403) = 806\).

Time = 1.77 (sec) , antiderivative size = 5709, normalized size of antiderivative = 12.17 \[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^2,x, algorithm="maxima")

[Out]

1/2*(c^4*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1)) - 4*a*c^3*d*(2/cos(b*x + a) - log(co
s(b*x + a) + 1) + log(cos(b*x + a) - 1))/b + 6*a^2*c^2*d^2*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b
*x + a) - 1))/b^2 - 4*a^3*c*d^3*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b^3 + a^4*d^4
*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b^4 + 2*(8*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*
a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2
*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^
4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^3*c^3*d - 3*I*a*b^
2*c^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c
^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 1) + 8*(
b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*
(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d
^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x
+ 2*a) + (I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3*(I*b*c*d^3 - I
*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b
*x + a), -sin(b*x + a) + 1) - 2*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*
d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a) + ((b*x + a)^
4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d -
3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3
+ I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*
b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1
) - 2*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2
+ 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a) + ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4
)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b
*c*d^3 - a^3*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6
*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*
d^3 + I*a^3*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + 4*(-I*(b*x + a)^4*d^4
+ 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b
^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*cos(b*x + a) + 24*(b^2*c^2*d^2 - 2*a*b*
c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d
^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*(b*x + a)^
2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 24*(b^2*c^
2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 +
(b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3
- I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-I*e^(I*b*x + I
*a)) + 8*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x +
a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*
x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*
cos(2*b*x + 2*a) + (I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3*(I*b
*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dil
og(-e^(I*b*x + I*a)) - 8*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3
- a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a
^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*
d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I
*a^3*d^4 + 3*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*si
n(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) - (-I*(b*x + a)^4*d^4 - 4*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^3 - 6*(I*b^2*
c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^2 - 4*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 - I*a^
3*d^4)*(b*x + a) + (-I*(b*x + a)^4*d^4 - 4*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^3 - 6*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^
3 + I*a^2*d^4)*(b*x + a)^2 - 4*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 - I*a^3*d^4)*(b*x + a))*cos(
2*b*x + 2*a) + ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b
*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(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)^4*d^4 - 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 - 6
*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 - 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*
d^3 + I*a^3*d^4)*(b*x + a) + (I*(b*x + a)^4*d^4 - 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 - 6*(-I*b^2*c^2*d^2 + 2
*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 - 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*
x + a))*cos(2*b*x + 2*a) - ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 +
a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(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) + 4*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*
d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 +
I*a^2*d^4)*(b*x + a) + (I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3
*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a)
- (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2
+ 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(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) + 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I*a^3*d^4 + 3
*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a) + (-I*b^3*c^3*d
+ 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I*a^3*d^4 + 3*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2
+ 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 +
3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a
^2*d^4)*(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) + 48*(-I*d^4*co
s(2*b*x + 2*a) + d^4*sin(2*b*x + 2*a) - I*d^4)*polylog(5, -e^(I*b*x + I*a)) + 48*(I*d^4*cos(2*b*x + 2*a) - d^4
*sin(2*b*x + 2*a) + I*d^4)*polylog(5, e^(I*b*x + I*a)) - 48*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x + 2*a) + d
^4)*polylog(4, I*e^(I*b*x + I*a)) + 48*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x + 2*a) + d^4)*polylog(4, -I*e^(
I*b*x + I*a)) - 48*(b*c*d^3 + (b*x + a)*d^4 - a*d^4 + (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a) - (-I
*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, -e^(I*b*x + I*a)) + 48*(b*c*d^3 + (b*x + a)
*d^4 - a*d^4 + (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a) + (I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*si
n(2*b*x + 2*a))*polylog(4, e^(I*b*x + I*a)) + 48*(I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4 + (I*b*c*d^3 + I*(b*x
+ a)*d^4 - I*a*d^4)*cos(2*b*x + 2*a) - (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog(3, I*e^(I*b
*x + I*a)) + 48*(-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4 + (-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*cos(2*b*x +
2*a) + (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog(3, -I*e^(I*b*x + I*a)) + 24*(I*b^2*c^2*d^2
- 2*I*a*b*c*d^3 + I*(b*x + a)^2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a) + (I*b^2*c^2*d^2 - 2*I*a*
b*c*d^3 + I*(b*x + a)^2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (b^2*c^2*d^2 -
2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, -e^(I*b
*x + I*a)) + 24*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*
x + a) + (-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))
*cos(2*b*x + 2*a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*si
n(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) + 4*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^
2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x +
a))*sin(b*x + a))/(-2*I*b^4*cos(2*b*x + 2*a) + 2*b^4*sin(2*b*x + 2*a) - 2*I*b^4))/b

Giac [F]

\[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \csc \left (b x + a\right ) \sec \left (b x + a\right )^{2} \,d x } \]

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*csc(b*x + a)*sec(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Hanged} \]

[In]

int((c + d*x)^4/(cos(a + b*x)^2*sin(a + b*x)),x)

[Out]

\text{Hanged}