3.876 \(\int x^3 \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx\)

Optimal. Leaf size=341 \[ 3 i x^2 \cos (x) \text{PolyLog}\left (2,-e^{i x}\right ) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{PolyLog}\left (2,e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{PolyLog}\left (2,-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{PolyLog}\left (2,i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{PolyLog}\left (3,-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{PolyLog}\left (3,e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{PolyLog}\left (3,-i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{PolyLog}\left (3,i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i \cos (x) \text{PolyLog}\left (4,-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i \cos (x) \text{PolyLog}\left (4,e^{i x}\right ) \sqrt{a \sec ^2(x)}+x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \cos (x) \tan ^{-1}\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-2 x^3 \cos (x) \tanh ^{-1}\left (e^{i x}\right ) \sqrt{a \sec ^2(x)} \]

[Out]

x^3*Sqrt[a*Sec[x]^2] + (6*I)*x^2*ArcTan[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - 2*x^3*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[
a*Sec[x]^2] + (3*I)*x^2*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - (6*I)*x*Cos[x]*PolyLog[2, (-I)*E^(I*x)]
*Sqrt[a*Sec[x]^2] + (6*I)*x*Cos[x]*PolyLog[2, I*E^(I*x)]*Sqrt[a*Sec[x]^2] - (3*I)*x^2*Cos[x]*PolyLog[2, E^(I*x
)]*Sqrt[a*Sec[x]^2] - 6*x*Cos[x]*PolyLog[3, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*Cos[x]*PolyLog[3, (-I)*E^(I*x)]*Sqr
t[a*Sec[x]^2] - 6*Cos[x]*PolyLog[3, I*E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*x*Cos[x]*PolyLog[3, E^(I*x)]*Sqrt[a*Sec[x]
^2] - (6*I)*Cos[x]*PolyLog[4, -E^(I*x)]*Sqrt[a*Sec[x]^2] + (6*I)*Cos[x]*PolyLog[4, E^(I*x)]*Sqrt[a*Sec[x]^2]

________________________________________________________________________________________

Rubi [A]  time = 0.62853, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {6720, 2622, 321, 207, 4420, 14, 6273, 4183, 2531, 6609, 2282, 6589, 4181} \[ 3 i x^2 \cos (x) \text{PolyLog}\left (2,-e^{i x}\right ) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{PolyLog}\left (2,e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{PolyLog}\left (2,-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{PolyLog}\left (2,i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{PolyLog}\left (3,-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{PolyLog}\left (3,e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{PolyLog}\left (3,-i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{PolyLog}\left (3,i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i \cos (x) \text{PolyLog}\left (4,-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i \cos (x) \text{PolyLog}\left (4,e^{i x}\right ) \sqrt{a \sec ^2(x)}+x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \cos (x) \tan ^{-1}\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-2 x^3 \cos (x) \tanh ^{-1}\left (e^{i x}\right ) \sqrt{a \sec ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]

[Out]

x^3*Sqrt[a*Sec[x]^2] + (6*I)*x^2*ArcTan[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - 2*x^3*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[
a*Sec[x]^2] + (3*I)*x^2*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - (6*I)*x*Cos[x]*PolyLog[2, (-I)*E^(I*x)]
*Sqrt[a*Sec[x]^2] + (6*I)*x*Cos[x]*PolyLog[2, I*E^(I*x)]*Sqrt[a*Sec[x]^2] - (3*I)*x^2*Cos[x]*PolyLog[2, E^(I*x
)]*Sqrt[a*Sec[x]^2] - 6*x*Cos[x]*PolyLog[3, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*Cos[x]*PolyLog[3, (-I)*E^(I*x)]*Sqr
t[a*Sec[x]^2] - 6*Cos[x]*PolyLog[3, I*E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*x*Cos[x]*PolyLog[3, E^(I*x)]*Sqrt[a*Sec[x]
^2] - (6*I)*Cos[x]*PolyLog[4, -E^(I*x)]*Sqrt[a*Sec[x]^2] + (6*I)*Cos[x]*PolyLog[4, E^(I*x)]*Sqrt[a*Sec[x]^2]

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

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 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 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 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]

Rubi steps

\begin{align*} \int x^3 \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx &=\left (\cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^3 \csc (x) \sec ^2(x) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}-x^3 \tanh ^{-1}(\cos (x)) \cos (x) \sqrt{a \sec ^2(x)}-\left (3 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^2 \left (-\tanh ^{-1}(\cos (x))+\sec (x)\right ) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}-x^3 \tanh ^{-1}(\cos (x)) \cos (x) \sqrt{a \sec ^2(x)}-\left (3 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int \left (-x^2 \tanh ^{-1}(\cos (x))+x^2 \sec (x)\right ) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}-x^3 \tanh ^{-1}(\cos (x)) \cos (x) \sqrt{a \sec ^2(x)}+\left (3 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^2 \tanh ^{-1}(\cos (x)) \, dx-\left (3 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^2 \sec (x) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \tan ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}+\left (\cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^3 \csc (x) \, dx+\left (6 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x \log \left (1-i e^{i x}\right ) \, dx-\left (6 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x \log \left (1+i e^{i x}\right ) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \tan ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}-2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{Li}_2\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{Li}_2\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}+\left (6 i \cos (x) \sqrt{a \sec ^2(x)}\right ) \int \text{Li}_2\left (-i e^{i x}\right ) \, dx-\left (6 i \cos (x) \sqrt{a \sec ^2(x)}\right ) \int \text{Li}_2\left (i e^{i x}\right ) \, dx-\left (3 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^2 \log \left (1-e^{i x}\right ) \, dx+\left (3 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x^2 \log \left (1+e^{i x}\right ) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \tan ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}-2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}+3 i x^2 \cos (x) \text{Li}_2\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{Li}_2\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{Li}_2\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{Li}_2\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-\left (6 i \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x \text{Li}_2\left (-e^{i x}\right ) \, dx+\left (6 i \cos (x) \sqrt{a \sec ^2(x)}\right ) \int x \text{Li}_2\left (e^{i x}\right ) \, dx+\left (6 \cos (x) \sqrt{a \sec ^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i x}\right )-\left (6 \cos (x) \sqrt{a \sec ^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i x}\right )\\ &=x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \tan ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}-2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}+3 i x^2 \cos (x) \text{Li}_2\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{Li}_2\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{Li}_2\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{Li}_2\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{Li}_3\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{Li}_3\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{Li}_3\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{Li}_3\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}+\left (6 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int \text{Li}_3\left (-e^{i x}\right ) \, dx-\left (6 \cos (x) \sqrt{a \sec ^2(x)}\right ) \int \text{Li}_3\left (e^{i x}\right ) \, dx\\ &=x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \tan ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}-2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}+3 i x^2 \cos (x) \text{Li}_2\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{Li}_2\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{Li}_2\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{Li}_2\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{Li}_3\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{Li}_3\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{Li}_3\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{Li}_3\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-\left (6 i \cos (x) \sqrt{a \sec ^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i x}\right )+\left (6 i \cos (x) \sqrt{a \sec ^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i x}\right )\\ &=x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \tan ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}-2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \cos (x) \sqrt{a \sec ^2(x)}+3 i x^2 \cos (x) \text{Li}_2\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{Li}_2\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{Li}_2\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{Li}_2\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{Li}_3\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{Li}_3\left (-i e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{Li}_3\left (i e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{Li}_3\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}-6 i \cos (x) \text{Li}_4\left (-e^{i x}\right ) \sqrt{a \sec ^2(x)}+6 i \cos (x) \text{Li}_4\left (e^{i x}\right ) \sqrt{a \sec ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.432337, size = 290, normalized size = 0.85 \[ \frac{1}{8} \sqrt{a \sec ^2(x)} \left (24 i x^2 \cos (x) \text{PolyLog}\left (2,e^{-i x}\right )+24 i x^2 \cos (x) \text{PolyLog}\left (2,-e^{i x}\right )-48 i x \cos (x) \text{PolyLog}\left (2,-i e^{i x}\right )+48 i x \cos (x) \text{PolyLog}\left (2,i e^{i x}\right )+48 x \cos (x) \text{PolyLog}\left (3,e^{-i x}\right )-48 x \cos (x) \text{PolyLog}\left (3,-e^{i x}\right )+48 \cos (x) \text{PolyLog}\left (3,-i e^{i x}\right )-48 \cos (x) \text{PolyLog}\left (3,i e^{i x}\right )-48 i \cos (x) \text{PolyLog}\left (4,e^{-i x}\right )-48 i \cos (x) \text{PolyLog}\left (4,-e^{i x}\right )+8 x^3+2 i x^4 \cos (x)+8 x^3 \log \left (1-e^{-i x}\right ) \cos (x)-8 x^3 \log \left (1+e^{i x}\right ) \cos (x)-24 x^2 \log \left (1-i e^{i x}\right ) \cos (x)+24 x^2 \log \left (1+i e^{i x}\right ) \cos (x)-i \pi ^4 \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]

[Out]

((8*x^3 - I*Pi^4*Cos[x] + (2*I)*x^4*Cos[x] + 8*x^3*Cos[x]*Log[1 - E^((-I)*x)] - 24*x^2*Cos[x]*Log[1 - I*E^(I*x
)] + 24*x^2*Cos[x]*Log[1 + I*E^(I*x)] - 8*x^3*Cos[x]*Log[1 + E^(I*x)] + (24*I)*x^2*Cos[x]*PolyLog[2, E^((-I)*x
)] + (24*I)*x^2*Cos[x]*PolyLog[2, -E^(I*x)] - (48*I)*x*Cos[x]*PolyLog[2, (-I)*E^(I*x)] + (48*I)*x*Cos[x]*PolyL
og[2, I*E^(I*x)] + 48*x*Cos[x]*PolyLog[3, E^((-I)*x)] - 48*x*Cos[x]*PolyLog[3, -E^(I*x)] + 48*Cos[x]*PolyLog[3
, (-I)*E^(I*x)] - 48*Cos[x]*PolyLog[3, I*E^(I*x)] - (48*I)*Cos[x]*PolyLog[4, E^((-I)*x)] - (48*I)*Cos[x]*PolyL
og[4, -E^(I*x)])*Sqrt[a*Sec[x]^2])/8

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Maple [A]  time = 0.226, size = 250, normalized size = 0.7 \begin{align*} 2\,\sqrt{{\frac{a{{\rm e}^{2\,ix}}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{2}}}}{x}^{3}+4\,\sqrt{{\frac{a{{\rm e}^{2\,ix}}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{2}}}} \left ( -3/2\,{x}^{2}\ln \left ( 1-i{{\rm e}^{ix}} \right ) +3\,ix{\it polylog} \left ( 2,i{{\rm e}^{ix}} \right ) -3\,{\it polylog} \left ( 3,i{{\rm e}^{ix}} \right ) +3/2\,{x}^{2}\ln \left ( 1+i{{\rm e}^{ix}} \right ) -3\,ix{\it polylog} \left ( 2,-i{{\rm e}^{ix}} \right ) +3\,{\it polylog} \left ( 3,-i{{\rm e}^{ix}} \right ) +i/2 \left ( 1/4\,{x}^{4}+i{x}^{3}\ln \left ({{\rm e}^{ix}}+1 \right ) +3\,{x}^{2}{\it polylog} \left ( 2,-{{\rm e}^{ix}} \right ) +6\,ix{\it polylog} \left ( 3,-{{\rm e}^{ix}} \right ) -6\,{\it polylog} \left ( 4,-{{\rm e}^{ix}} \right ) \right ) +i/2 \left ( -1/4\,{x}^{4}-i{x}^{3}\ln \left ( 1-{{\rm e}^{ix}} \right ) -3\,{x}^{2}{\it polylog} \left ( 2,{{\rm e}^{ix}} \right ) -6\,ix{\it polylog} \left ( 3,{{\rm e}^{ix}} \right ) +6\,{\it polylog} \left ( 4,{{\rm e}^{ix}} \right ) \right ) \right ) \cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csc(x)*sec(x)*(a*sec(x)^2)^(1/2),x)

[Out]

2*(a*exp(2*I*x)/(1+exp(2*I*x))^2)^(1/2)*x^3+4*(a*exp(2*I*x)/(1+exp(2*I*x))^2)^(1/2)*(-3/2*x^2*ln(1-I*exp(I*x))
+3*I*x*polylog(2,I*exp(I*x))-3*polylog(3,I*exp(I*x))+3/2*x^2*ln(1+I*exp(I*x))-3*I*x*polylog(2,-I*exp(I*x))+3*p
olylog(3,-I*exp(I*x))+1/2*I*(1/4*x^4+I*x^3*ln(exp(I*x)+1)+3*x^2*polylog(2,-exp(I*x))+6*I*x*polylog(3,-exp(I*x)
)-6*polylog(4,-exp(I*x)))+1/2*I*(-1/4*x^4-I*x^3*ln(1-exp(I*x))-3*x^2*polylog(2,exp(I*x))-6*I*x*polylog(3,exp(I
*x))+6*polylog(4,exp(I*x))))*cos(x)

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Maxima [B]  time = 1.65314, size = 790, normalized size = 2.32 \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(x)*sec(x)*(a*sec(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-(4*I*x^3*cos(x) - 4*x^3*sin(x) - (6*x^2*cos(2*x) + 6*I*x^2*sin(2*x) + 6*x^2)*arctan2(cos(x), sin(x) + 1) - (6
*x^2*cos(2*x) + 6*I*x^2*sin(2*x) + 6*x^2)*arctan2(cos(x), -sin(x) + 1) + (2*x^3*cos(2*x) + 2*I*x^3*sin(2*x) +
2*x^3)*arctan2(sin(x), cos(x) + 1) + (2*x^3*cos(2*x) + 2*I*x^3*sin(2*x) + 2*x^3)*arctan2(sin(x), -cos(x) + 1)
- (12*x*cos(2*x) + 12*I*x*sin(2*x) + 12*x)*dilog(I*e^(I*x)) + (12*x*cos(2*x) + 12*I*x*sin(2*x) + 12*x)*dilog(-
I*e^(I*x)) - (6*x^2*cos(2*x) + 6*I*x^2*sin(2*x) + 6*x^2)*dilog(-e^(I*x)) + (6*x^2*cos(2*x) + 6*I*x^2*sin(2*x)
+ 6*x^2)*dilog(e^(I*x)) + (-I*x^3*cos(2*x) + x^3*sin(2*x) - I*x^3)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + (
I*x^3*cos(2*x) - x^3*sin(2*x) + I*x^3)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + (-3*I*x^2*cos(2*x) + 3*x^2*si
n(2*x) - 3*I*x^2)*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + (3*I*x^2*cos(2*x) - 3*x^2*sin(2*x) + 3*I*x^2)*log(
cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) + 12*(cos(2*x) + I*sin(2*x) + 1)*polylog(4, -e^(I*x)) - 12*(cos(2*x) + I*s
in(2*x) + 1)*polylog(4, e^(I*x)) + (-12*I*cos(2*x) + 12*sin(2*x) - 12*I)*polylog(3, I*e^(I*x)) + (12*I*cos(2*x
) - 12*sin(2*x) + 12*I)*polylog(3, -I*e^(I*x)) + (-12*I*x*cos(2*x) + 12*x*sin(2*x) - 12*I*x)*polylog(3, -e^(I*
x)) + (12*I*x*cos(2*x) - 12*x*sin(2*x) + 12*I*x)*polylog(3, e^(I*x)))*sqrt(a)/(-2*I*cos(2*x) + 2*sin(2*x) - 2*
I)

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Fricas [C]  time = 3.03825, size = 1906, normalized size = 5.59 \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(x)*sec(x)*(a*sec(x)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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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(x)*sec(x)*(a*sec(x)**2)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sec \left (x\right )^{2}} x^{3} \csc \left (x\right ) \sec \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csc(x)*sec(x)*(a*sec(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sec(x)^2)*x^3*csc(x)*sec(x), x)