3.879 \(\int x^3 \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx\)

Optimal. Leaf size=356 \[ \frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left (2,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left (2,e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left (3,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left (3,e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{PolyLog}\left (2,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left (4,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left (4,e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \sin ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \cos ^2(x) \tanh ^{-1}\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \sin (x) \cos (x) \sqrt{a \sec ^4(x)}-3 x \log \left (1+e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)} \]

[Out]

((3*I)/2)*x^2*Cos[x]^2*Sqrt[a*Sec[x]^4] + (x^3*Cos[x]^2*Sqrt[a*Sec[x]^4])/2 - 2*x^3*ArcTanh[E^((2*I)*x)]*Cos[x
]^2*Sqrt[a*Sec[x]^4] - 3*x*Cos[x]^2*Log[1 + E^((2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/2)*Cos[x]^2*PolyLog[2, -E^(
(2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/2)*x^2*Cos[x]^2*PolyLog[2, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - ((3*I)/2)*x^2*
Cos[x]^2*PolyLog[2, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (3*x*Cos[x]^2*PolyLog[3, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2
+ (3*x*Cos[x]^2*PolyLog[3, E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2 - ((3*I)/4)*Cos[x]^2*PolyLog[4, -E^((2*I)*x)]*Sqr
t[a*Sec[x]^4] + ((3*I)/4)*Cos[x]^2*PolyLog[4, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (3*x^2*Cos[x]*Sqrt[a*Sec[x]^4]*S
in[x])/2 + (x^3*Sqrt[a*Sec[x]^4]*Sin[x]^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.636957, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 17, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.944, Rules used = {6720, 2620, 14, 4420, 2551, 4419, 4183, 2531, 6609, 2282, 6589, 3720, 3719, 2190, 2279, 2391, 30} \[ \frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left (2,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left (2,e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left (3,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left (3,e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{PolyLog}\left (2,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left (4,-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left (4,e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \sin ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \cos ^2(x) \tanh ^{-1}\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \sin (x) \cos (x) \sqrt{a \sec ^4(x)}-3 x \log \left (1+e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]

[Out]

((3*I)/2)*x^2*Cos[x]^2*Sqrt[a*Sec[x]^4] + (x^3*Cos[x]^2*Sqrt[a*Sec[x]^4])/2 - 2*x^3*ArcTanh[E^((2*I)*x)]*Cos[x
]^2*Sqrt[a*Sec[x]^4] - 3*x*Cos[x]^2*Log[1 + E^((2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/2)*Cos[x]^2*PolyLog[2, -E^(
(2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/2)*x^2*Cos[x]^2*PolyLog[2, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - ((3*I)/2)*x^2*
Cos[x]^2*PolyLog[2, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (3*x*Cos[x]^2*PolyLog[3, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2
+ (3*x*Cos[x]^2*PolyLog[3, E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2 - ((3*I)/4)*Cos[x]^2*PolyLog[4, -E^((2*I)*x)]*Sqr
t[a*Sec[x]^4] + ((3*I)/4)*Cos[x]^2*PolyLog[4, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (3*x^2*Cos[x]*Sqrt[a*Sec[x]^4]*S
in[x])/2 + (x^3*Sqrt[a*Sec[x]^4]*Sin[x]^2)/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 2620

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

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 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 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 4419

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^3 \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx &=\left (\cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^3 \csc (x) \sec ^3(x) \, dx\\ &=x^3 \cos ^2(x) \log (\tan (x)) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^2 \left (\log (\tan (x))+\frac{\tan ^2(x)}{2}\right ) \, dx\\ &=x^3 \cos ^2(x) \log (\tan (x)) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \left (x^2 \log (\tan (x))+\frac{1}{2} x^2 \tan ^2(x)\right ) \, dx\\ &=x^3 \cos ^2(x) \log (\tan (x)) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\frac{1}{2} \left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^2 \tan ^2(x) \, dx-\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^2 \log (\tan (x)) \, dx\\ &=-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)+\left (\cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^3 \csc (x) \sec (x) \, dx+\frac{1}{2} \left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^2 \, dx+\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x \tan (x) \, dx\\ &=\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\left (6 i \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx+\left (2 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^3 \csc (2 x) \, dx\\ &=\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^2 \log \left (1-e^{2 i x}\right ) \, dx+\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \log \left (1+e^{2 i x}\right ) \, dx+\left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x^2 \log \left (1+e^{2 i x}\right ) \, dx\\ &=\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\frac{1}{2} \left (3 i \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i x}\right )-\left (3 i \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x \text{Li}_2\left (-e^{2 i x}\right ) \, dx+\left (3 i \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int x \text{Li}_2\left (e^{2 i x}\right ) \, dx\\ &=\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{Li}_3\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{Li}_3\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)+\frac{1}{2} \left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \text{Li}_3\left (-e^{2 i x}\right ) \, dx-\frac{1}{2} \left (3 \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \text{Li}_3\left (e^{2 i x}\right ) \, dx\\ &=\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{Li}_3\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{Li}_3\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)-\frac{1}{4} \left (3 i \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 i x}\right )+\frac{1}{4} \left (3 i \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt{a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{Li}_2\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{Li}_3\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{Li}_3\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{4} i \cos ^2(x) \text{Li}_4\left (-e^{2 i x}\right ) \sqrt{a \sec ^4(x)}+\frac{3}{4} i \cos ^2(x) \text{Li}_4\left (e^{2 i x}\right ) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{1}{2} x^3 \sqrt{a \sec ^4(x)} \sin ^2(x)\\ \end{align*}

Mathematica [A]  time = 1.08983, size = 191, normalized size = 0.54 \[ \frac{1}{64} \cos ^2(x) \sqrt{a \sec ^4(x)} \left (96 i x^2 \text{PolyLog}\left (2,e^{-2 i x}\right )+96 i \left (x^2+1\right ) \text{PolyLog}\left (2,-e^{2 i x}\right )+96 x \text{PolyLog}\left (3,e^{-2 i x}\right )-96 x \text{PolyLog}\left (3,-e^{2 i x}\right )-48 i \text{PolyLog}\left (4,e^{-2 i x}\right )-48 i \text{PolyLog}\left (4,-e^{2 i x}\right )+32 i x^4+96 i x^2+64 x^3 \log \left (1-e^{-2 i x}\right )-64 x^3 \log \left (1+e^{2 i x}\right )-96 x^2 \tan (x)+32 x^3 \sec ^2(x)-192 x \log \left (1+e^{2 i x}\right )-i \pi ^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]

[Out]

(Cos[x]^2*Sqrt[a*Sec[x]^4]*((-I)*Pi^4 + (96*I)*x^2 + (32*I)*x^4 + 64*x^3*Log[1 - E^((-2*I)*x)] - 192*x*Log[1 +
E^((2*I)*x)] - 64*x^3*Log[1 + E^((2*I)*x)] + (96*I)*x^2*PolyLog[2, E^((-2*I)*x)] + (96*I)*(1 + x^2)*PolyLog[2
, -E^((2*I)*x)] + 96*x*PolyLog[3, E^((-2*I)*x)] - 96*x*PolyLog[3, -E^((2*I)*x)] - (48*I)*PolyLog[4, E^((-2*I)*
x)] - (48*I)*PolyLog[4, -E^((2*I)*x)] + 32*x^3*Sec[x]^2 - 96*x^2*Tan[x]))/64

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Maple [A]  time = 0.09, size = 324, normalized size = 0.9 \begin{align*} \sqrt{{\frac{a{{\rm e}^{4\,ix}}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{4}}}}{x}^{2} \left ( 2\,x-3\,i-3\,i{{\rm e}^{-2\,ix}} \right ) -2\,i\sqrt{{\frac{a{{\rm e}^{4\,ix}}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{4}}}} \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{2} \left ( -{\frac{3\,{{\rm e}^{-2\,ix}}{x}^{2}}{2}}-{\frac{3\,i}{2}}{{\rm e}^{-2\,ix}}x\ln \left ( 1+{{\rm e}^{2\,ix}} \right ) -{\frac{3\,{{\rm e}^{-2\,ix}}{\it polylog} \left ( 2,-{{\rm e}^{2\,ix}} \right ) }{4}}-{\frac{i}{2}}{{\rm e}^{-2\,ix}}{x}^{3}\ln \left ( 1+{{\rm e}^{2\,ix}} \right ) -{\frac{3\,{{\rm e}^{-2\,ix}}{x}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,ix}} \right ) }{4}}-{\frac{3\,i}{4}}{{\rm e}^{-2\,ix}}x{\it polylog} \left ( 3,-{{\rm e}^{2\,ix}} \right ) +{\frac{3\,{{\rm e}^{-2\,ix}}{\it polylog} \left ( 4,-{{\rm e}^{2\,ix}} \right ) }{8}}+{\frac{i}{2}}{{\rm e}^{-2\,ix}}{x}^{3}\ln \left ({{\rm e}^{ix}}+1 \right ) +{\frac{3\,{{\rm e}^{-2\,ix}}{x}^{2}{\it polylog} \left ( 2,-{{\rm e}^{ix}} \right ) }{2}}+3\,i{{\rm e}^{-2\,ix}}x{\it polylog} \left ( 3,-{{\rm e}^{ix}} \right ) -3\,{{\rm e}^{-2\,ix}}{\it polylog} \left ( 4,-{{\rm e}^{ix}} \right ) +{\frac{i}{2}}{{\rm e}^{-2\,ix}}{x}^{3}\ln \left ( 1-{{\rm e}^{ix}} \right ) +{\frac{3\,{{\rm e}^{-2\,ix}}{x}^{2}{\it polylog} \left ( 2,{{\rm e}^{ix}} \right ) }{2}}+3\,i{{\rm e}^{-2\,ix}}x{\it polylog} \left ( 3,{{\rm e}^{ix}} \right ) -3\,{{\rm e}^{-2\,ix}}{\it polylog} \left ( 4,{{\rm e}^{ix}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csc(x)*sec(x)*(a*sec(x)^4)^(1/2),x)

[Out]

(a*exp(4*I*x)/(1+exp(2*I*x))^4)^(1/2)*x^2*(2*x-3*I-3*I*exp(-2*I*x))-2*I*(a*exp(4*I*x)/(1+exp(2*I*x))^4)^(1/2)*
(1+exp(2*I*x))^2*(-3/2*exp(-2*I*x)*x^2-3/2*I*exp(-2*I*x)*x*ln(1+exp(2*I*x))-3/4*exp(-2*I*x)*polylog(2,-exp(2*I
*x))-1/2*I*exp(-2*I*x)*x^3*ln(1+exp(2*I*x))-3/4*exp(-2*I*x)*x^2*polylog(2,-exp(2*I*x))-3/4*I*exp(-2*I*x)*x*pol
ylog(3,-exp(2*I*x))+3/8*exp(-2*I*x)*polylog(4,-exp(2*I*x))+1/2*I*exp(-2*I*x)*x^3*ln(exp(I*x)+1)+3/2*exp(-2*I*x
)*x^2*polylog(2,-exp(I*x))+3*I*exp(-2*I*x)*x*polylog(3,-exp(I*x))-3*exp(-2*I*x)*polylog(4,-exp(I*x))+1/2*I*exp
(-2*I*x)*x^3*ln(1-exp(I*x))+3/2*exp(-2*I*x)*x^2*polylog(2,exp(I*x))+3*I*exp(-2*I*x)*x*polylog(3,exp(I*x))-3*ex
p(-2*I*x)*polylog(4,exp(I*x)))

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Maxima [B]  time = 1.9119, size = 1175, normalized size = 3.3 \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)^4)^(1/2),x, algorithm="maxima")

[Out]

(18*x^2*cos(4*x) + 18*I*x^2*sin(4*x) - (8*x^3 + 2*(4*x^3 + 9*x)*cos(4*x) + 4*(4*x^3 + 9*x)*cos(2*x) + (8*I*x^3
+ 18*I*x)*sin(4*x) + (16*I*x^3 + 36*I*x)*sin(2*x) + 18*x)*arctan2(sin(2*x), cos(2*x) + 1) + (6*x^3*cos(4*x) +
12*x^3*cos(2*x) + 6*I*x^3*sin(4*x) + 12*I*x^3*sin(2*x) + 6*x^3)*arctan2(sin(x), cos(x) + 1) - (6*x^3*cos(4*x)
+ 12*x^3*cos(2*x) + 6*I*x^3*sin(4*x) + 12*I*x^3*sin(2*x) + 6*x^3)*arctan2(sin(x), -cos(x) + 1) - (12*I*x^3 -
18*x^2)*cos(2*x) + (12*x^2 + 3*(4*x^2 + 3)*cos(4*x) + 6*(4*x^2 + 3)*cos(2*x) - (-12*I*x^2 - 9*I)*sin(4*x) - (-
24*I*x^2 - 18*I)*sin(2*x) + 9)*dilog(-e^(2*I*x)) - (18*x^2*cos(4*x) + 36*x^2*cos(2*x) + 18*I*x^2*sin(4*x) + 36
*I*x^2*sin(2*x) + 18*x^2)*dilog(-e^(I*x)) - (18*x^2*cos(4*x) + 36*x^2*cos(2*x) + 18*I*x^2*sin(4*x) + 36*I*x^2*
sin(2*x) + 18*x^2)*dilog(e^(I*x)) - (-4*I*x^3 + (-4*I*x^3 - 9*I*x)*cos(4*x) + (-8*I*x^3 - 18*I*x)*cos(2*x) + (
4*x^3 + 9*x)*sin(4*x) + 2*(4*x^3 + 9*x)*sin(2*x) - 9*I*x)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) - (3*I
*x^3*cos(4*x) + 6*I*x^3*cos(2*x) - 3*x^3*sin(4*x) - 6*x^3*sin(2*x) + 3*I*x^3)*log(cos(x)^2 + sin(x)^2 + 2*cos(
x) + 1) - (3*I*x^3*cos(4*x) + 6*I*x^3*cos(2*x) - 3*x^3*sin(4*x) - 6*x^3*sin(2*x) + 3*I*x^3)*log(cos(x)^2 + sin
(x)^2 - 2*cos(x) + 1) - (6*cos(4*x) + 12*cos(2*x) + 6*I*sin(4*x) + 12*I*sin(2*x) + 6)*polylog(4, -e^(2*I*x)) +
(36*cos(4*x) + 72*cos(2*x) + 36*I*sin(4*x) + 72*I*sin(2*x) + 36)*polylog(4, -e^(I*x)) + (36*cos(4*x) + 72*cos
(2*x) + 36*I*sin(4*x) + 72*I*sin(2*x) + 36)*polylog(4, e^(I*x)) - (-12*I*x*cos(4*x) - 24*I*x*cos(2*x) + 12*x*s
in(4*x) + 24*x*sin(2*x) - 12*I*x)*polylog(3, -e^(2*I*x)) - (36*I*x*cos(4*x) + 72*I*x*cos(2*x) - 36*x*sin(4*x)
- 72*x*sin(2*x) + 36*I*x)*polylog(3, -e^(I*x)) - (36*I*x*cos(4*x) + 72*I*x*cos(2*x) - 36*x*sin(4*x) - 72*x*sin
(2*x) + 36*I*x)*polylog(3, e^(I*x)) + 6*(2*x^3 + 3*I*x^2)*sin(2*x))*sqrt(a)/(-6*I*cos(4*x) - 12*I*cos(2*x) + 6
*sin(4*x) + 12*sin(2*x) - 6*I)

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Fricas [C]  time = 4.53984, size = 2446, normalized size = 6.87 \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)^4)^(1/2),x, algorithm="fricas")

[Out]

3*x*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, cos(x) + I*sin(x)) + 3*x*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, cos(x)
- I*sin(x)) - 3*x*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, I*cos(x) + sin(x)) - 3*x*sqrt(a/cos(x)^4)*cos(x)^2*poly
log(3, I*cos(x) - sin(x)) - 3*x*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, -I*cos(x) + sin(x)) - 3*x*sqrt(a/cos(x)^4
)*cos(x)^2*polylog(3, -I*cos(x) - sin(x)) + 3*x*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, -cos(x) + I*sin(x)) + 3*x
*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, -cos(x) - I*sin(x)) + 3*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, cos(x) +
I*sin(x)) - 3*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, cos(x) - I*sin(x)) + 3*I*sqrt(a/cos(x)^4)*cos(x)^2*polylo
g(4, I*cos(x) + sin(x)) - 3*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, I*cos(x) - sin(x)) - 3*I*sqrt(a/cos(x)^4)*c
os(x)^2*polylog(4, -I*cos(x) + sin(x)) + 3*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, -I*cos(x) - sin(x)) - 3*I*sq
rt(a/cos(x)^4)*cos(x)^2*polylog(4, -cos(x) + I*sin(x)) + 3*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, -cos(x) - I*
sin(x)) + 1/2*(x^3*cos(x)^2*log(cos(x) + I*sin(x) + 1) + x^3*cos(x)^2*log(cos(x) - I*sin(x) + 1) + x^3*cos(x)^
2*log(-cos(x) + I*sin(x) + 1) + x^3*cos(x)^2*log(-cos(x) - I*sin(x) + 1) - 3*I*x^2*cos(x)^2*dilog(cos(x) + I*s
in(x)) + 3*I*x^2*cos(x)^2*dilog(cos(x) - I*sin(x)) + 3*I*x^2*cos(x)^2*dilog(-cos(x) + I*sin(x)) - 3*I*x^2*cos(
x)^2*dilog(-cos(x) - I*sin(x)) + (-3*I*x^2 - 3*I)*cos(x)^2*dilog(I*cos(x) + sin(x)) + (3*I*x^2 + 3*I)*cos(x)^2
*dilog(I*cos(x) - sin(x)) + (3*I*x^2 + 3*I)*cos(x)^2*dilog(-I*cos(x) + sin(x)) + (-3*I*x^2 - 3*I)*cos(x)^2*dil
og(-I*cos(x) - sin(x)) - (x^3 + 3*x)*cos(x)^2*log(I*cos(x) + sin(x) + 1) - (x^3 + 3*x)*cos(x)^2*log(I*cos(x) -
sin(x) + 1) - (x^3 + 3*x)*cos(x)^2*log(-I*cos(x) + sin(x) + 1) - (x^3 + 3*x)*cos(x)^2*log(-I*cos(x) - sin(x)
+ 1) - 3*x^2*cos(x)*sin(x) + x^3)*sqrt(a/cos(x)^4)

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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)**4)**(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 )^{4}} 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)^4)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sec(x)^4)*x^3*csc(x)*sec(x), x)