3.1.36 \(\int \frac {\sec ^{-1}(a+b x)^3}{x} \, dx\) [36]

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

[Out]

-arcsec(b*x+a)^3*ln(1+(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)+arcsec(b*x+a)^3*ln(1-a*(1/(b*x+a)+I*(1-1/(b*x+a)^
2)^(1/2))/(1-(-a^2+1)^(1/2)))+arcsec(b*x+a)^3*ln(1-a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/(1+(-a^2+1)^(1/2)))+3
/2*I*arcsec(b*x+a)^2*polylog(2,-(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)-3*I*arcsec(b*x+a)^2*polylog(2,a*(1/(b*x
+a)+I*(1-1/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))-3*I*arcsec(b*x+a)^2*polylog(2,a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^
(1/2))/(1+(-a^2+1)^(1/2)))-3/2*arcsec(b*x+a)*polylog(3,-(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)+6*arcsec(b*x+a)
*polylog(3,a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))+6*arcsec(b*x+a)*polylog(3,a*(1/(b*x+a)+I*
(1-1/(b*x+a)^2)^(1/2))/(1+(-a^2+1)^(1/2)))-3/4*I*polylog(4,-(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)+6*I*polylog
(4,a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))+6*I*polylog(4,a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2
))/(1+(-a^2+1)^(1/2)))

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Rubi [A]
time = 0.40, antiderivative size = 430, normalized size of antiderivative = 1.00, 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 = {5366, 4647, 4626, 3800, 2221, 2611, 6744, 2320, 6724, 4616} \begin {gather*} -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)}}{\sqrt {1-a^2}+1}\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)}}{\sqrt {1-a^2}+1}\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)}}{\sqrt {1-a^2}+1}\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 )+\frac {3}{2} i \sec ^{-1}(a+b x)^2 \text {Li}_2\left (-e^{2 i \sec ^{-1}(a+b x)}\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 \text {Li}_4\left (-e^{2 i \sec ^{-1}(a+b x)}\right )-\sec ^{-1}(a+b x)^3 \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right ) \end {gather*}

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3800

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 4616

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]

Rule 4626

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 4647

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 5366

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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rubi steps

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1058\) vs. \(2(430)=860\).
time = 2.66, size = 1058, normalized size = 2.46 \begin {gather*} 2 \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 {\left (-1+\sqrt {1-a^2}\right ) e^{i \sec ^{-1}(a+b x)}}{a}\right )-6 \sec ^{-1}(a+b x)^2 \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) e^{i \sec ^{-1}(a+b x)}}{a}\right )+2 \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 {\left (1+\sqrt {1-a^2}\right ) e^{i \sec ^{-1}(a+b x)}}{a}\right )+6 \sec ^{-1}(a+b x)^2 \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) e^{i \sec ^{-1}(a+b x)}}{a}\right )-3 \sec ^{-1}(a+b x)^3 \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )+2 \sec ^{-1}(a+b x)^3 \log \left (\frac {2 \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{a+b x}\right )-\sec ^{-1}(a+b x)^3 \log \left (1+\frac {a \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{-1+\sqrt {1-a^2}}\right )-\sec ^{-1}(a+b x)^3 \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{a}\right )+6 \sec ^{-1}(a+b x)^2 \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{a}\right )-\sec ^{-1}(a+b x)^3 \log \left (1-\frac {a \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{1+\sqrt {1-a^2}}\right )-\sec ^{-1}(a+b x)^3 \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{a}\right )-6 \sec ^{-1}(a+b x)^2 \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) \left (\frac {1}{a+b x}+i \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{a}\right )-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)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} i \sec ^{-1}(a+b x)^2 \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\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)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \sec ^{-1}(a+b x) \text {PolyLog}\left (3,-e^{2 i \sec ^{-1}(a+b x)}\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)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{4} i \text {PolyLog}\left (4,-e^{2 i \sec ^{-1}(a+b x)}\right ) \end {gather*}

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 = 0.90, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arcsec}\left (b x +a \right )^{3}}{x}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asec}^{3}{\left (a + b x \right )}}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(1/(a + b*x))^3/x,x)

[Out]

int(acos(1/(a + b*x))^3/x, x)

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