3.36 \(\int \frac{\sec ^{-1}(a+b x)^3}{x} \, dx\)

Optimal. Leaf size=430 \[ -3 i \sec ^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-3 i \sec ^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 \sec ^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 \sec ^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 i \text{PolyLog}\left (4,\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 i \text{PolyLog}\left (4,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+\frac{3}{2} i \sec ^{-1}(a+b x)^2 \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )-\frac{3}{2} \sec ^{-1}(a+b x) \text{PolyLog}\left (3,-e^{2 i \sec ^{-1}(a+b x)}\right )-\frac{3}{4} i \text{PolyLog}\left (4,-e^{2 i \sec ^{-1}(a+b x)}\right )+\sec ^{-1}(a+b x)^3 \log \left (1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\sec ^{-1}(a+b x)^3 \log \left (1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\sec ^{-1}(a+b x)^3 \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right ) \]

[Out]

ArcSec[a + b*x]^3*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcSec[a + b*x]^3*Log[1 - (a*E^(I*A
rcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcSec[a + b*x]^3*Log[1 + E^((2*I)*ArcSec[a + b*x])] - (3*I)*ArcSec[a
+ b*x]^2*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] - (3*I)*ArcSec[a + b*x]^2*PolyLog[2, (a*E^(
I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + ((3*I)/2)*ArcSec[a + b*x]^2*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])]
+ 6*ArcSec[a + b*x]*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + 6*ArcSec[a + b*x]*PolyLog[3, (
a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3*ArcSec[a + b*x]*PolyLog[3, -E^((2*I)*ArcSec[a + b*x])])/2 +
(6*I)*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + (6*I)*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/
(1 + Sqrt[1 - a^2])] - ((3*I)/4)*PolyLog[4, -E^((2*I)*ArcSec[a + b*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.53309, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5258, 4551, 4530, 3719, 2190, 2531, 6609, 2282, 6589, 4520} \[ -3 i \sec ^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-3 i \sec ^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 \sec ^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 \sec ^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 i \text{PolyLog}\left (4,\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 i \text{PolyLog}\left (4,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+\frac{3}{2} i \sec ^{-1}(a+b x)^2 \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )-\frac{3}{2} \sec ^{-1}(a+b x) \text{PolyLog}\left (3,-e^{2 i \sec ^{-1}(a+b x)}\right )-\frac{3}{4} i \text{PolyLog}\left (4,-e^{2 i \sec ^{-1}(a+b x)}\right )+\sec ^{-1}(a+b x)^3 \log \left (1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\sec ^{-1}(a+b x)^3 \log \left (1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\sec ^{-1}(a+b x)^3 \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSec[a + b*x]^3/x,x]

[Out]

ArcSec[a + b*x]^3*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcSec[a + b*x]^3*Log[1 - (a*E^(I*A
rcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcSec[a + b*x]^3*Log[1 + E^((2*I)*ArcSec[a + b*x])] - (3*I)*ArcSec[a
+ b*x]^2*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] - (3*I)*ArcSec[a + b*x]^2*PolyLog[2, (a*E^(
I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + ((3*I)/2)*ArcSec[a + b*x]^2*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])]
+ 6*ArcSec[a + b*x]*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + 6*ArcSec[a + b*x]*PolyLog[3, (
a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3*ArcSec[a + b*x]*PolyLog[3, -E^((2*I)*ArcSec[a + b*x])])/2 +
(6*I)*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + (6*I)*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/
(1 + Sqrt[1 - a^2])] - ((3*I)/4)*PolyLog[4, -E^((2*I)*ArcSec[a + b*x])]

Rule 5258

Int[((a_.) + ArcSec[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d^(m + 1),
Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d*e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]], x] /; FreeQ[{a, b, c,
d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rule 4551

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
ec[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[((e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Cos[c +
d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]

Rule 4530

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tan[(c_.) + (d_.)*(x_)]^(n_.))/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Tan[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sin[c + d*x]*Tan[c + d*x]^
(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 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 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 4520

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

Rubi steps

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

Mathematica [B]  time = 3.4814, size = 1058, normalized size = 2.46 \[ 2 \log \left (\frac{e^{i \sec ^{-1}(a+b x)} a}{\sqrt{1-a^2}-1}+1\right ) \sec ^{-1}(a+b x)^3+\log \left (\frac{e^{i \sec ^{-1}(a+b x)} \left (\sqrt{1-a^2}-1\right )}{a}+1\right ) \sec ^{-1}(a+b x)^3+2 \log \left (1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \sec ^{-1}(a+b x)^3+\log \left (1-\frac{\left (\sqrt{1-a^2}+1\right ) e^{i \sec ^{-1}(a+b x)}}{a}\right ) \sec ^{-1}(a+b x)^3-3 \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right ) \sec ^{-1}(a+b x)^3+2 \log \left (\frac{2 \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{a+b x}\right ) \sec ^{-1}(a+b x)^3-\log \left (\frac{a \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{\sqrt{1-a^2}-1}+1\right ) \sec ^{-1}(a+b x)^3-\log \left (\frac{\left (\sqrt{1-a^2}-1\right ) \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{a}+1\right ) \sec ^{-1}(a+b x)^3-\log \left (1-\frac{a \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{\sqrt{1-a^2}+1}\right ) \sec ^{-1}(a+b x)^3-\log \left (1-\frac{\left (\sqrt{1-a^2}+1\right ) \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{a}\right ) \sec ^{-1}(a+b x)^3-6 \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right ) \log \left (\frac{e^{i \sec ^{-1}(a+b x)} \left (\sqrt{1-a^2}-1\right )}{a}+1\right ) \sec ^{-1}(a+b x)^2+6 \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right ) \log \left (1-\frac{\left (\sqrt{1-a^2}+1\right ) e^{i \sec ^{-1}(a+b x)}}{a}\right ) \sec ^{-1}(a+b x)^2+6 \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right ) \log \left (\frac{\left (\sqrt{1-a^2}-1\right ) \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{a}+1\right ) \sec ^{-1}(a+b x)^2-6 \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right ) \log \left (1-\frac{\left (\sqrt{1-a^2}+1\right ) \left (i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right )}{a}\right ) \sec ^{-1}(a+b x)^2-3 i \text{PolyLog}\left (2,-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right ) \sec ^{-1}(a+b x)^2-3 i \text{PolyLog}\left (2,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \sec ^{-1}(a+b x)^2+\frac{3}{2} i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right ) \sec ^{-1}(a+b x)^2+6 \text{PolyLog}\left (3,-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right ) \sec ^{-1}(a+b x)+6 \text{PolyLog}\left (3,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \sec ^{-1}(a+b x)-\frac{3}{2} \text{PolyLog}\left (3,-e^{2 i \sec ^{-1}(a+b x)}\right ) \sec ^{-1}(a+b x)+6 i \text{PolyLog}\left (4,-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )+6 i \text{PolyLog}\left (4,\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\frac{3}{4} i \text{PolyLog}\left (4,-e^{2 i \sec ^{-1}(a+b x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSec[a + b*x]^3/x,x]

[Out]

2*ArcSec[a + b*x]^3*Log[1 + (a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2])] + ArcSec[a + b*x]^3*Log[1 + ((-1 +
Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - 6*ArcSec[a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + ((-1
+ Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + 2*ArcSec[a + b*x]^3*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1
- a^2])] + ArcSec[a + b*x]^3*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + 6*ArcSec[a + b*x]^2*Arc
Sin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - 3*ArcSec[a + b*x]^3*Log
[1 + E^((2*I)*ArcSec[a + b*x])] + 2*ArcSec[a + b*x]^3*Log[(2*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(a
+ b*x)] - ArcSec[a + b*x]^3*Log[1 + (a*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(-1 + Sqrt[1 - a^2])] -
ArcSec[a + b*x]^3*Log[1 + ((-1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] + 6*ArcSec[
a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + ((-1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*
x)^(-2)]))/a] - ArcSec[a + b*x]^3*Log[1 - (a*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(1 + Sqrt[1 - a^2]
)] - ArcSec[a + b*x]^3*Log[1 - ((1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - 6*ArcS
ec[a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - ((1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a +
b*x)^(-2)]))/a] - (3*I)*ArcSec[a + b*x]^2*PolyLog[2, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))] - (3*I
)*ArcSec[a + b*x]^2*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + ((3*I)/2)*ArcSec[a + b*x]^2*Po
lyLog[2, -E^((2*I)*ArcSec[a + b*x])] + 6*ArcSec[a + b*x]*PolyLog[3, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 -
a^2]))] + 6*ArcSec[a + b*x]*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3*ArcSec[a + b*x]*Po
lyLog[3, -E^((2*I)*ArcSec[a + b*x])])/2 + (6*I)*PolyLog[4, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))]
+ (6*I)*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - ((3*I)/4)*PolyLog[4, -E^((2*I)*ArcSec[a +
b*x])]

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Maple [F]  time = 0.697, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arcsec} \left (bx+a\right ) \right ) ^{3}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsec(b*x+a)^3/x,x)

[Out]

int(arcsec(b*x+a)^3/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcsec}\left (b x + a\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(b*x+a)^3/x,x, algorithm="maxima")

[Out]

integrate(arcsec(b*x + a)^3/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcsec}\left (b x + a\right )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(b*x+a)^3/x,x, algorithm="fricas")

[Out]

integral(arcsec(b*x + a)^3/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asec}^{3}{\left (a + b x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asec(b*x+a)**3/x,x)

[Out]

Integral(asec(a + b*x)**3/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcsec}\left (b x + a\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(b*x+a)^3/x,x, algorithm="giac")

[Out]

integrate(arcsec(b*x + a)^3/x, x)