3.285 \(\int x^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx\)
Optimal. Leaf size=387 \[ \frac{9 i x^2 \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x^2 \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac{6 i x \text{PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac{6 i x \text{PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac{9 x \text{PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac{9 x \text{PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac{3 i \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac{3 i \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac{6 \text{PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac{6 \text{PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}-\frac{9 i \text{PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac{6 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{3 x^2 \csc (a+b x)}{2 b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b} \]
[Out]
((6*I)*x^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*x*ArcTanh[E^(I*(a + b*x))])/b^3 - (3*x^3*ArcTanh[E^(I*(a + b*x))]
)/b - (3*x^2*Csc[a + b*x])/(2*b^2) + ((3*I)*PolyLog[2, -E^(I*(a + b*x))])/b^4 + (((9*I)/2)*x^2*PolyLog[2, -E^(
I*(a + b*x))])/b^2 - ((6*I)*x*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((6*I)*x*PolyLog[2, I*E^(I*(a + b*x))])/
b^3 - ((3*I)*PolyLog[2, E^(I*(a + b*x))])/b^4 - (((9*I)/2)*x^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (9*x*PolyLog
[3, -E^(I*(a + b*x))])/b^3 + (6*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*PolyLog[3, I*E^(I*(a + b*x))])/b^4
+ (9*x*PolyLog[3, E^(I*(a + b*x))])/b^3 - ((9*I)*PolyLog[4, -E^(I*(a + b*x))])/b^4 + ((9*I)*PolyLog[4, E^(I*(a
+ b*x))])/b^4 + (3*x^3*Sec[a + b*x])/(2*b) - (x^3*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)
________________________________________________________________________________________
Rubi [A] time = 0.960072, antiderivative size = 387, normalized size of antiderivative = 1.,
number of steps used = 40, number of rules used = 18, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used =
{2622, 288, 321, 207, 4420, 14, 6273, 12, 4183, 2531, 6609, 2282, 6589, 6742, 4181, 2621, 2279,
2391} \[ \frac{9 i x^2 \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x^2 \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac{6 i x \text{PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac{6 i x \text{PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac{9 x \text{PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac{9 x \text{PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac{3 i \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac{3 i \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac{6 \text{PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac{6 \text{PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}-\frac{9 i \text{PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac{6 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{3 x^2 \csc (a+b x)}{2 b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[x^3*Csc[a + b*x]^3*Sec[a + b*x]^2,x]
[Out]
((6*I)*x^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*x*ArcTanh[E^(I*(a + b*x))])/b^3 - (3*x^3*ArcTanh[E^(I*(a + b*x))]
)/b - (3*x^2*Csc[a + b*x])/(2*b^2) + ((3*I)*PolyLog[2, -E^(I*(a + b*x))])/b^4 + (((9*I)/2)*x^2*PolyLog[2, -E^(
I*(a + b*x))])/b^2 - ((6*I)*x*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((6*I)*x*PolyLog[2, I*E^(I*(a + b*x))])/
b^3 - ((3*I)*PolyLog[2, E^(I*(a + b*x))])/b^4 - (((9*I)/2)*x^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (9*x*PolyLog
[3, -E^(I*(a + b*x))])/b^3 + (6*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*PolyLog[3, I*E^(I*(a + b*x))])/b^4
+ (9*x*PolyLog[3, E^(I*(a + b*x))])/b^3 - ((9*I)*PolyLog[4, -E^(I*(a + b*x))])/b^4 + ((9*I)*PolyLog[4, E^(I*(a
+ b*x))])/b^4 + (3*x^3*Sec[a + b*x])/(2*b) - (x^3*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)
Rule 2622
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 288
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 321
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 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 4420
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 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 6273
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 4183
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 2531
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 6609
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 2282
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 6589
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 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rule 4181
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 2621
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 2279
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 2391
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]
Rubi steps
\begin{align*} \int x^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx &=-\frac{3 x^3 \tanh ^{-1}(\cos (a+b x))}{2 b}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-3 \int x^2 \left (-\frac{3 \tanh ^{-1}(\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 x^3 \tanh ^{-1}(\cos (a+b x))}{2 b}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-3 \int \left (-\frac{3 x^2 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{x^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x)}{2 b}\right ) \, dx\\ &=-\frac{3 x^3 \tanh ^{-1}(\cos (a+b x))}{2 b}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{3 \int x^2 \left (-3+\csc ^2(a+b x)\right ) \sec (a+b x) \, dx}{2 b}+\frac{9 \int x^2 \tanh ^{-1}(\cos (a+b x)) \, dx}{2 b}\\ &=\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{3 \int b x^3 \csc (a+b x) \, dx}{2 b}+\frac{3 \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 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{3}{2} \int x^3 \csc (a+b x) \, dx+\frac{3 \int x^2 \csc ^2(a+b x) \sec (a+b x) \, dx}{2 b}-\frac{9 \int x^2 \sec (a+b x) \, dx}{2 b}\\ &=\frac{9 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{3 x^2 \tanh ^{-1}(\sin (a+b x))}{2 b^2}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{9 \int x \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac{9 \int x \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac{3 \int x \left (\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\csc (a+b x)}{b}\right ) \, dx}{b}-\frac{9 \int x^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{2 b}+\frac{9 \int x^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{2 b}\\ &=\frac{9 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{3 x^2 \tanh ^{-1}(\sin (a+b x))}{2 b^2}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{9 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{(9 i) \int \text{Li}_2\left (-i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac{(9 i) \int \text{Li}_2\left (i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac{(9 i) \int x \text{Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac{(9 i) \int x \text{Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^2}-\frac{3 \int \left (\frac{x \tanh ^{-1}(\sin (a+b x))}{b}-\frac{x \csc (a+b x)}{b}\right ) \, dx}{b}\\ &=\frac{9 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{3 x^2 \tanh ^{-1}(\sin (a+b x))}{2 b^2}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{9 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{9 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac{9 \int \text{Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac{9 \int \text{Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^3}-\frac{3 \int x \tanh ^{-1}(\sin (a+b x)) \, dx}{b^2}+\frac{3 \int x \csc (a+b x) \, dx}{b^2}\\ &=\frac{9 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{9 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{9 \text{Li}_3\left (-i e^{i (a+b x)}\right )}{b^4}-\frac{9 \text{Li}_3\left (i e^{i (a+b x)}\right )}{b^4}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac{(9 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac{(9 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac{3 \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac{3 \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^3}+\frac{3 \int b x^2 \sec (a+b x) \, dx}{2 b^2}\\ &=\frac{9 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{9 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{9 \text{Li}_3\left (-i e^{i (a+b x)}\right )}{b^4}-\frac{9 \text{Li}_3\left (i e^{i (a+b x)}\right )}{b^4}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac{9 i \text{Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac{3 \int x^2 \sec (a+b x) \, dx}{2 b}\\ &=\frac{6 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 i \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{9 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{9 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{3 i \text{Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{9 \text{Li}_3\left (-i e^{i (a+b x)}\right )}{b^4}-\frac{9 \text{Li}_3\left (i e^{i (a+b x)}\right )}{b^4}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac{9 i \text{Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac{3 \int x \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac{3 \int x \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}\\ &=\frac{6 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 i \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{6 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{6 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{3 i \text{Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{9 \text{Li}_3\left (-i e^{i (a+b x)}\right )}{b^4}-\frac{9 \text{Li}_3\left (i e^{i (a+b x)}\right )}{b^4}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac{9 i \text{Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac{(3 i) \int \text{Li}_2\left (-i e^{i (a+b x)}\right ) \, dx}{b^3}+\frac{(3 i) \int \text{Li}_2\left (i e^{i (a+b x)}\right ) \, dx}{b^3}\\ &=\frac{6 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 i \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{6 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{6 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{3 i \text{Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{9 \text{Li}_3\left (-i e^{i (a+b x)}\right )}{b^4}-\frac{9 \text{Li}_3\left (i e^{i (a+b x)}\right )}{b^4}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac{9 i \text{Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}\\ &=\frac{6 i x^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 x \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac{3 x^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 i \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i x^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac{6 i x \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^3}+\frac{6 i x \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^3}-\frac{3 i \text{Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac{9 i x^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac{9 x \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{6 \text{Li}_3\left (-i e^{i (a+b x)}\right )}{b^4}-\frac{6 \text{Li}_3\left (i e^{i (a+b x)}\right )}{b^4}+\frac{9 x \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac{9 i \text{Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac{9 i \text{Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 7.19335, size = 672, normalized size = 1.74 \[ \frac{3 \left (i \left (3 b^2 x^2+2\right ) \text{PolyLog}(2,-\cos (a+b x)-i \sin (a+b x))-i \left (3 b^2 x^2+2\right ) \text{PolyLog}(2,\cos (a+b x)+i \sin (a+b x))-6 b x \text{PolyLog}(3,-\cos (a+b x)-i \sin (a+b x))+6 b x \text{PolyLog}(3,\cos (a+b x)+i \sin (a+b x))-6 i \text{PolyLog}(4,-\cos (a+b x)-i \sin (a+b x))+6 i \text{PolyLog}(4,\cos (a+b x)+i \sin (a+b x))+b^3 x^3 \log (-i \sin (a+b x)-\cos (a+b x)+1)-b^3 x^3 \log (i \sin (a+b x)+\cos (a+b x)+1)+2 b x \log (-i \sin (a+b x)-\cos (a+b x)+1)-2 b x \log (i \sin (a+b x)+\cos (a+b x)+1)\right )}{2 b^4}+\frac{6 \left (i b x \text{PolyLog}(2,-\sin (a+b x)+i \cos (a+b x))-i b x \text{PolyLog}(2,\sin (a+b x)-i \cos (a+b x))-\text{PolyLog}(3,-\sin (a+b x)+i \cos (a+b x))+\text{PolyLog}(3,\sin (a+b x)-i \cos (a+b x))+i b^2 x^2 \tan ^{-1}(\cos (a+b x)+i \sin (a+b x))\right )}{b^4}+\frac{3 x^2 \csc \left (\frac{a}{2}\right ) \sin \left (\frac{b x}{2}\right ) \csc \left (\frac{a}{2}+\frac{b x}{2}\right )}{4 b^2}-\frac{3 x^2 \sec \left (\frac{a}{2}\right ) \sin \left (\frac{b x}{2}\right ) \sec \left (\frac{a}{2}+\frac{b x}{2}\right )}{4 b^2}+\frac{x^2 \csc (a) \sec (a) (2 b x \sin (a)-3 \cos (a))}{2 b^2}-\frac{x^3 \csc ^2\left (\frac{a}{2}+\frac{b x}{2}\right )}{8 b}+\frac{x^3 \sec ^2\left (\frac{a}{2}+\frac{b x}{2}\right )}{8 b}+\frac{x^3 \sin \left (\frac{b x}{2}\right )}{b \left (\cos \left (\frac{a}{2}\right )-\sin \left (\frac{a}{2}\right )\right ) \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )-\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}-\frac{x^3 \sin \left (\frac{b x}{2}\right )}{b \left (\sin \left (\frac{a}{2}\right )+\cos \left (\frac{a}{2}\right )\right ) \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )+\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x^3*Csc[a + b*x]^3*Sec[a + b*x]^2,x]
[Out]
-(x^3*Csc[a/2 + (b*x)/2]^2)/(8*b) + (6*(I*b^2*x^2*ArcTan[Cos[a + b*x] + I*Sin[a + b*x]] + I*b*x*PolyLog[2, I*C
os[a + b*x] - Sin[a + b*x]] - I*b*x*PolyLog[2, (-I)*Cos[a + b*x] + Sin[a + b*x]] - PolyLog[3, I*Cos[a + b*x] -
Sin[a + b*x]] + PolyLog[3, (-I)*Cos[a + b*x] + Sin[a + b*x]]))/b^4 + (3*(2*b*x*Log[1 - Cos[a + b*x] - I*Sin[a
+ b*x]] + b^3*x^3*Log[1 - Cos[a + b*x] - I*Sin[a + b*x]] - 2*b*x*Log[1 + Cos[a + b*x] + I*Sin[a + b*x]] - b^3
*x^3*Log[1 + Cos[a + b*x] + I*Sin[a + b*x]] + I*(2 + 3*b^2*x^2)*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] - I
*(2 + 3*b^2*x^2)*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] - 6*b*x*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]]
+ 6*b*x*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - (6*I)*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]] + (6*I)*P
olyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]))/(2*b^4) + (x^3*Sec[a/2 + (b*x)/2]^2)/(8*b) + (x^2*Csc[a]*Sec[a]*(-3
*Cos[a] + 2*b*x*Sin[a]))/(2*b^2) + (3*x^2*Csc[a/2]*Csc[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2) - (3*x^2*Sec[a/2]*
Sec[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2) + (x^3*Sin[(b*x)/2])/(b*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - S
in[a/2 + (b*x)/2])) - (x^3*Sin[(b*x)/2])/(b*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]))
________________________________________________________________________________________
Maple [F] time = 1.401, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( \csc \left ( bx+a \right ) \right ) ^{3} \left ( \sec \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*csc(b*x+a)^3*sec(b*x+a)^2,x)
[Out]
int(x^3*csc(b*x+a)^3*sec(b*x+a)^2,x)
________________________________________________________________________________________
Maxima [B] time = 3.74018, size = 5385, normalized size = 13.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="maxima")
[Out]
-1/4*(a^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)) - 4*((12*(b*x + a)^2 - 24*(b*x + a)*a + 12*a^2 + 12*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*cos(6*b*x +
6*a) - 12*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*cos(4*b*x + 4*a) - 12*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*cos(2
*b*x + 2*a) - (-12*I*(b*x + a)^2 + 24*I*(b*x + a)*a - 12*I*a^2)*sin(6*b*x + 6*a) - (12*I*(b*x + a)^2 - 24*I*(b
*x + a)*a + 12*I*a^2)*sin(4*b*x + 4*a) - (12*I*(b*x + a)^2 - 24*I*(b*x + a)*a + 12*I*a^2)*sin(2*b*x + 2*a))*ar
ctan2(cos(b*x + a), sin(b*x + a) + 1) + (12*(b*x + a)^2 - 24*(b*x + a)*a + 12*a^2 + 12*((b*x + a)^2 - 2*(b*x +
a)*a + a^2)*cos(6*b*x + 6*a) - 12*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*cos(4*b*x + 4*a) - 12*((b*x + a)^2 - 2*
(b*x + a)*a + a^2)*cos(2*b*x + 2*a) - (-12*I*(b*x + a)^2 + 24*I*(b*x + a)*a - 12*I*a^2)*sin(6*b*x + 6*a) - (12
*I*(b*x + a)^2 - 24*I*(b*x + a)*a + 12*I*a^2)*sin(4*b*x + 4*a) - (12*I*(b*x + a)^2 - 24*I*(b*x + a)*a + 12*I*a
^2)*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), -sin(b*x + a) + 1) - (6*(b*x + a)^3 - 18*(b*x + a)^2*a + 6*(3*a^2
+ 2)*(b*x + a) + 6*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*cos(6*b*x + 6*a) - 6*((b*x +
a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*cos(4*b*x + 4*a) - 6*((b*x + a)^3 - 3*(b*x + a)^2*a + (3
*a^2 + 2)*(b*x + a) - 2*a)*cos(2*b*x + 2*a) + (6*I*(b*x + a)^3 - 18*I*(b*x + a)^2*a + (18*I*a^2 + 12*I)*(b*x +
a) - 12*I*a)*sin(6*b*x + 6*a) + (-6*I*(b*x + a)^3 + 18*I*(b*x + a)^2*a + (-18*I*a^2 - 12*I)*(b*x + a) + 12*I*
a)*sin(4*b*x + 4*a) + (-6*I*(b*x + a)^3 + 18*I*(b*x + a)^2*a + (-18*I*a^2 - 12*I)*(b*x + a) + 12*I*a)*sin(2*b*
x + 2*a) - 12*a)*arctan2(sin(b*x + a), cos(b*x + a) + 1) - (12*a*cos(6*b*x + 6*a) - 12*a*cos(4*b*x + 4*a) - 12
*a*cos(2*b*x + 2*a) + 12*I*a*sin(6*b*x + 6*a) - 12*I*a*sin(4*b*x + 4*a) - 12*I*a*sin(2*b*x + 2*a) + 12*a)*arct
an2(sin(b*x + a), cos(b*x + a) - 1) - (6*(b*x + a)^3 - 18*(b*x + a)^2*a + 6*(3*a^2 + 2)*(b*x + a) + 6*((b*x +
a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a))*cos(6*b*x + 6*a) - 6*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 +
2)*(b*x + a))*cos(4*b*x + 4*a) - 6*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a))*cos(2*b*x + 2*a) +
(6*I*(b*x + a)^3 - 18*I*(b*x + a)^2*a + (18*I*a^2 + 12*I)*(b*x + a))*sin(6*b*x + 6*a) + (-6*I*(b*x + a)^3 + 1
8*I*(b*x + a)^2*a + (-18*I*a^2 - 12*I)*(b*x + a))*sin(4*b*x + 4*a) + (-6*I*(b*x + a)^3 + 18*I*(b*x + a)^2*a +
(-18*I*a^2 - 12*I)*(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 -
12*(b*x + a)^2*(3*I*a - 1) + (36*I*a^2 - 24*a)*(b*x + a) + 12*a^2)*cos(5*b*x + 5*a) - (-8*I*(b*x + a)^3 + 24*
I*(b*x + a)^2*a - 24*I*(b*x + a)*a^2)*cos(3*b*x + 3*a) - (12*I*(b*x + a)^3 - 12*(b*x + a)^2*(3*I*a + 1) + (36*
I*a^2 + 24*a)*(b*x + a) - 12*a^2)*cos(b*x + a) + (24*b*x*cos(6*b*x + 6*a) - 24*b*x*cos(4*b*x + 4*a) - 24*b*x*c
os(2*b*x + 2*a) + 24*I*b*x*sin(6*b*x + 6*a) - 24*I*b*x*sin(4*b*x + 4*a) - 24*I*b*x*sin(2*b*x + 2*a) + 24*b*x)*
dilog(I*e^(I*b*x + I*a)) - (24*b*x*cos(6*b*x + 6*a) - 24*b*x*cos(4*b*x + 4*a) - 24*b*x*cos(2*b*x + 2*a) + 24*I
*b*x*sin(6*b*x + 6*a) - 24*I*b*x*sin(4*b*x + 4*a) - 24*I*b*x*sin(2*b*x + 2*a) + 24*b*x)*dilog(-I*e^(I*b*x + I*
a)) + (18*(b*x + a)^2 - 36*(b*x + a)*a + 18*a^2 + 6*(3*(b*x + a)^2 - 6*(b*x + a)*a + 3*a^2 + 2)*cos(6*b*x + 6*
a) - 6*(3*(b*x + a)^2 - 6*(b*x + a)*a + 3*a^2 + 2)*cos(4*b*x + 4*a) - 6*(3*(b*x + a)^2 - 6*(b*x + a)*a + 3*a^2
+ 2)*cos(2*b*x + 2*a) - (-18*I*(b*x + a)^2 + 36*I*(b*x + a)*a - 18*I*a^2 - 12*I)*sin(6*b*x + 6*a) - (18*I*(b*
x + a)^2 - 36*I*(b*x + a)*a + 18*I*a^2 + 12*I)*sin(4*b*x + 4*a) - (18*I*(b*x + a)^2 - 36*I*(b*x + a)*a + 18*I*
a^2 + 12*I)*sin(2*b*x + 2*a) + 12)*dilog(-e^(I*b*x + I*a)) - (18*(b*x + a)^2 - 36*(b*x + a)*a + 18*a^2 + 6*(3*
(b*x + a)^2 - 6*(b*x + a)*a + 3*a^2 + 2)*cos(6*b*x + 6*a) - 6*(3*(b*x + a)^2 - 6*(b*x + a)*a + 3*a^2 + 2)*cos(
4*b*x + 4*a) - 6*(3*(b*x + a)^2 - 6*(b*x + a)*a + 3*a^2 + 2)*cos(2*b*x + 2*a) + (18*I*(b*x + a)^2 - 36*I*(b*x
+ a)*a + 18*I*a^2 + 12*I)*sin(6*b*x + 6*a) + (-18*I*(b*x + a)^2 + 36*I*(b*x + a)*a - 18*I*a^2 - 12*I)*sin(4*b*
x + 4*a) + (-18*I*(b*x + a)^2 + 36*I*(b*x + a)*a - 18*I*a^2 - 12*I)*sin(2*b*x + 2*a) + 12)*dilog(e^(I*b*x + I*
a)) - (-3*I*(b*x + a)^3 + 9*I*(b*x + a)^2*a + (-9*I*a^2 - 6*I)*(b*x + a) + (-3*I*(b*x + a)^3 + 9*I*(b*x + a)^2
*a + (-9*I*a^2 - 6*I)*(b*x + a) + 6*I*a)*cos(6*b*x + 6*a) + (3*I*(b*x + a)^3 - 9*I*(b*x + a)^2*a + (9*I*a^2 +
6*I)*(b*x + a) - 6*I*a)*cos(4*b*x + 4*a) + (3*I*(b*x + a)^3 - 9*I*(b*x + a)^2*a + (9*I*a^2 + 6*I)*(b*x + a) -
6*I*a)*cos(2*b*x + 2*a) + 3*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*sin(6*b*x + 6*a) - 3
*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*sin(4*b*x + 4*a) - 3*((b*x + a)^3 - 3*(b*x + a)
^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*sin(2*b*x + 2*a) + 6*I*a)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x
+ a) + 1) - (3*I*(b*x + a)^3 - 9*I*(b*x + a)^2*a + (9*I*a^2 + 6*I)*(b*x + a) + (3*I*(b*x + a)^3 - 9*I*(b*x + a
)^2*a + (9*I*a^2 + 6*I)*(b*x + a) - 6*I*a)*cos(6*b*x + 6*a) + (-3*I*(b*x + a)^3 + 9*I*(b*x + a)^2*a + (-9*I*a^
2 - 6*I)*(b*x + a) + 6*I*a)*cos(4*b*x + 4*a) + (-3*I*(b*x + a)^3 + 9*I*(b*x + a)^2*a + (-9*I*a^2 - 6*I)*(b*x +
a) + 6*I*a)*cos(2*b*x + 2*a) - 3*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*sin(6*b*x + 6*
a) + 3*((b*x + a)^3 - 3*(b*x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*sin(4*b*x + 4*a) + 3*((b*x + a)^3 - 3*(b*
x + a)^2*a + (3*a^2 + 2)*(b*x + a) - 2*a)*sin(2*b*x + 2*a) - 6*I*a)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*co
s(b*x + a) + 1) - (-6*I*(b*x + a)^2 + 12*I*(b*x + a)*a - 6*I*a^2 + (-6*I*(b*x + a)^2 + 12*I*(b*x + a)*a - 6*I*
a^2)*cos(6*b*x + 6*a) + (6*I*(b*x + a)^2 - 12*I*(b*x + a)*a + 6*I*a^2)*cos(4*b*x + 4*a) + (6*I*(b*x + a)^2 - 1
2*I*(b*x + a)*a + 6*I*a^2)*cos(2*b*x + 2*a) + 6*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*sin(6*b*x + 6*a) - 6*((b*x
+ a)^2 - 2*(b*x + a)*a + a^2)*sin(4*b*x + 4*a) - 6*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*sin(2*b*x + 2*a))*log(
cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) - (6*I*(b*x + a)^2 - 12*I*(b*x + a)*a + 6*I*a^2 + (6*I*(
b*x + a)^2 - 12*I*(b*x + a)*a + 6*I*a^2)*cos(6*b*x + 6*a) + (-6*I*(b*x + a)^2 + 12*I*(b*x + a)*a - 6*I*a^2)*co
s(4*b*x + 4*a) + (-6*I*(b*x + a)^2 + 12*I*(b*x + a)*a - 6*I*a^2)*cos(2*b*x + 2*a) - 6*((b*x + a)^2 - 2*(b*x +
a)*a + a^2)*sin(6*b*x + 6*a) + 6*((b*x + a)^2 - 2*(b*x + a)*a + a^2)*sin(4*b*x + 4*a) + 6*((b*x + a)^2 - 2*(b*
x + a)*a + a^2)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) - (36*cos(6*b*x +
6*a) - 36*cos(4*b*x + 4*a) - 36*cos(2*b*x + 2*a) + 36*I*sin(6*b*x + 6*a) - 36*I*sin(4*b*x + 4*a) - 36*I*sin(2*
b*x + 2*a) + 36)*polylog(4, -e^(I*b*x + I*a)) + (36*cos(6*b*x + 6*a) - 36*cos(4*b*x + 4*a) - 36*cos(2*b*x + 2*
a) + 36*I*sin(6*b*x + 6*a) - 36*I*sin(4*b*x + 4*a) - 36*I*sin(2*b*x + 2*a) + 36)*polylog(4, e^(I*b*x + I*a)) -
(-24*I*cos(6*b*x + 6*a) + 24*I*cos(4*b*x + 4*a) + 24*I*cos(2*b*x + 2*a) + 24*sin(6*b*x + 6*a) - 24*sin(4*b*x
+ 4*a) - 24*sin(2*b*x + 2*a) - 24*I)*polylog(3, I*e^(I*b*x + I*a)) - (24*I*cos(6*b*x + 6*a) - 24*I*cos(4*b*x +
4*a) - 24*I*cos(2*b*x + 2*a) - 24*sin(6*b*x + 6*a) + 24*sin(4*b*x + 4*a) + 24*sin(2*b*x + 2*a) + 24*I)*polylo
g(3, -I*e^(I*b*x + I*a)) - (-36*I*b*x*cos(6*b*x + 6*a) + 36*I*b*x*cos(4*b*x + 4*a) + 36*I*b*x*cos(2*b*x + 2*a)
+ 36*b*x*sin(6*b*x + 6*a) - 36*b*x*sin(4*b*x + 4*a) - 36*b*x*sin(2*b*x + 2*a) - 36*I*b*x)*polylog(3, -e^(I*b*
x + I*a)) - (36*I*b*x*cos(6*b*x + 6*a) - 36*I*b*x*cos(4*b*x + 4*a) - 36*I*b*x*cos(2*b*x + 2*a) - 36*b*x*sin(6*
b*x + 6*a) + 36*b*x*sin(4*b*x + 4*a) + 36*b*x*sin(2*b*x + 2*a) + 36*I*b*x)*polylog(3, e^(I*b*x + I*a)) + (12*(
b*x + a)^3 - (b*x + a)^2*(36*a + 12*I) + 12*(3*a^2 + 2*I*a)*(b*x + a) - 12*I*a^2)*sin(5*b*x + 5*a) - 8*((b*x +
a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2)*sin(3*b*x + 3*a) + (12*(b*x + a)^3 - (b*x + a)^2*(36*a - 12*I) + 12
*(3*a^2 - 2*I*a)*(b*x + a) + 12*I*a^2)*sin(b*x + a))/(-4*I*cos(6*b*x + 6*a) + 4*I*cos(4*b*x + 4*a) + 4*I*cos(2
*b*x + 2*a) + 4*sin(6*b*x + 6*a) - 4*sin(4*b*x + 4*a) - 4*sin(2*b*x + 2*a) - 4*I))/b^4
________________________________________________________________________________________
Fricas [C] time = 0.911414, size = 4625, normalized size = 11.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="fricas")
[Out]
1/4*(6*b^3*x^3*cos(b*x + a)^2 - 4*b^3*x^3 + 6*b^2*x^2*cos(b*x + a)*sin(b*x + a) + ((-9*I*b^2*x^2 - 6*I)*cos(b*
x + a)^3 + (9*I*b^2*x^2 + 6*I)*cos(b*x + a))*dilog(cos(b*x + a) + I*sin(b*x + a)) + ((9*I*b^2*x^2 + 6*I)*cos(b
*x + a)^3 + (-9*I*b^2*x^2 - 6*I)*cos(b*x + a))*dilog(cos(b*x + a) - I*sin(b*x + a)) + (12*I*b*x*cos(b*x + a)^3
- 12*I*b*x*cos(b*x + a))*dilog(I*cos(b*x + a) + sin(b*x + a)) + (12*I*b*x*cos(b*x + a)^3 - 12*I*b*x*cos(b*x +
a))*dilog(I*cos(b*x + a) - sin(b*x + a)) + (-12*I*b*x*cos(b*x + a)^3 + 12*I*b*x*cos(b*x + a))*dilog(-I*cos(b*
x + a) + sin(b*x + a)) + (-12*I*b*x*cos(b*x + a)^3 + 12*I*b*x*cos(b*x + a))*dilog(-I*cos(b*x + a) - sin(b*x +
a)) + ((-9*I*b^2*x^2 - 6*I)*cos(b*x + a)^3 + (9*I*b^2*x^2 + 6*I)*cos(b*x + a))*dilog(-cos(b*x + a) + I*sin(b*x
+ a)) + ((9*I*b^2*x^2 + 6*I)*cos(b*x + a)^3 + (-9*I*b^2*x^2 - 6*I)*cos(b*x + a))*dilog(-cos(b*x + a) - I*sin(
b*x + a)) - 3*((b^3*x^3 + 2*b*x)*cos(b*x + a)^3 - (b^3*x^3 + 2*b*x)*cos(b*x + a))*log(cos(b*x + a) + I*sin(b*x
+ a) + 1) - 6*(a^2*cos(b*x + a)^3 - a^2*cos(b*x + a))*log(cos(b*x + a) + I*sin(b*x + a) + I) - 3*((b^3*x^3 +
2*b*x)*cos(b*x + a)^3 - (b^3*x^3 + 2*b*x)*cos(b*x + a))*log(cos(b*x + a) - I*sin(b*x + a) + 1) + 6*(a^2*cos(b*
x + a)^3 - a^2*cos(b*x + a))*log(cos(b*x + a) - I*sin(b*x + a) + I) - 6*((b^2*x^2 - a^2)*cos(b*x + a)^3 - (b^2
*x^2 - a^2)*cos(b*x + a))*log(I*cos(b*x + a) + sin(b*x + a) + 1) + 6*((b^2*x^2 - a^2)*cos(b*x + a)^3 - (b^2*x^
2 - a^2)*cos(b*x + a))*log(I*cos(b*x + a) - sin(b*x + a) + 1) - 6*((b^2*x^2 - a^2)*cos(b*x + a)^3 - (b^2*x^2 -
a^2)*cos(b*x + a))*log(-I*cos(b*x + a) + sin(b*x + a) + 1) + 6*((b^2*x^2 - a^2)*cos(b*x + a)^3 - (b^2*x^2 - a
^2)*cos(b*x + a))*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - 3*((a^3 + 2*a)*cos(b*x + a)^3 - (a^3 + 2*a)*cos(b*
x + a))*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - 3*((a^3 + 2*a)*cos(b*x + a)^3 - (a^3 + 2*a)*cos(b*
x + a))*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) + 3*((b^3*x^3 + a^3 + 2*b*x + 2*a)*cos(b*x + a)^3 -
(b^3*x^3 + a^3 + 2*b*x + 2*a)*cos(b*x + a))*log(-cos(b*x + a) + I*sin(b*x + a) + 1) - 6*(a^2*cos(b*x + a)^3 -
a^2*cos(b*x + a))*log(-cos(b*x + a) + I*sin(b*x + a) + I) + 3*((b^3*x^3 + a^3 + 2*b*x + 2*a)*cos(b*x + a)^3 -
(b^3*x^3 + a^3 + 2*b*x + 2*a)*cos(b*x + a))*log(-cos(b*x + a) - I*sin(b*x + a) + 1) + 6*(a^2*cos(b*x + a)^3 -
a^2*cos(b*x + a))*log(-cos(b*x + a) - I*sin(b*x + a) + I) + (18*I*cos(b*x + a)^3 - 18*I*cos(b*x + a))*polylog(
4, cos(b*x + a) + I*sin(b*x + a)) + (-18*I*cos(b*x + a)^3 + 18*I*cos(b*x + a))*polylog(4, cos(b*x + a) - I*sin
(b*x + a)) + (18*I*cos(b*x + a)^3 - 18*I*cos(b*x + a))*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) + (-18*I*cos
(b*x + a)^3 + 18*I*cos(b*x + a))*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) + 18*(b*x*cos(b*x + a)^3 - b*x*cos
(b*x + a))*polylog(3, cos(b*x + a) + I*sin(b*x + a)) + 18*(b*x*cos(b*x + a)^3 - b*x*cos(b*x + a))*polylog(3, c
os(b*x + a) - I*sin(b*x + a)) + 12*(cos(b*x + a)^3 - cos(b*x + a))*polylog(3, I*cos(b*x + a) + sin(b*x + a)) -
12*(cos(b*x + a)^3 - cos(b*x + a))*polylog(3, I*cos(b*x + a) - sin(b*x + a)) + 12*(cos(b*x + a)^3 - cos(b*x +
a))*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 12*(cos(b*x + a)^3 - cos(b*x + a))*polylog(3, -I*cos(b*x + a
) - sin(b*x + a)) - 18*(b*x*cos(b*x + a)^3 - b*x*cos(b*x + a))*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) - 18
*(b*x*cos(b*x + a)^3 - b*x*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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*csc(b*x+a)**3*sec(b*x+a)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="giac")
[Out]
integrate(x^3*csc(b*x + a)^3*sec(b*x + a)^2, x)