Integrand size = 14, antiderivative size = 128 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right ) \]
1/4*I*(a+b*arcsec(c*x))^4/b-(a+b*arcsec(c*x))^3*ln(1+(1/c/x+I*(1-1/c^2/x^2 )^(1/2))^2)+3/2*I*b*(a+b*arcsec(c*x))^2*polylog(2,-(1/c/x+I*(1-1/c^2/x^2)^ (1/2))^2)-3/2*b^2*(a+b*arcsec(c*x))*polylog(3,-(1/c/x+I*(1-1/c^2/x^2)^(1/2 ))^2)-3/4*I*b^3*polylog(4,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)
Time = 0.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\frac {1}{4} \left (6 i a^2 b \sec ^{-1}(c x)^2+4 i a b^2 \sec ^{-1}(c x)^3+i b^3 \sec ^{-1}(c x)^4-12 a^2 b \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-12 a b^2 \sec ^{-1}(c x)^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-4 b^3 \sec ^{-1}(c x)^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+4 a^3 \log (c x)+6 i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-6 b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right )\right ) \]
((6*I)*a^2*b*ArcSec[c*x]^2 + (4*I)*a*b^2*ArcSec[c*x]^3 + I*b^3*ArcSec[c*x] ^4 - 12*a^2*b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 12*a*b^2*ArcSec [c*x]^2*Log[1 + E^((2*I)*ArcSec[c*x])] - 4*b^3*ArcSec[c*x]^3*Log[1 + E^((2 *I)*ArcSec[c*x])] + 4*a^3*Log[c*x] + (6*I)*b*(a + b*ArcSec[c*x])^2*PolyLog [2, -E^((2*I)*ArcSec[c*x])] - 6*b^2*(a + b*ArcSec[c*x])*PolyLog[3, -E^((2* I)*ArcSec[c*x])] - (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcSec[c*x])])/4
Time = 0.64 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5745, 3042, 4202, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 5745 |
| \(\displaystyle \int c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^3d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan \left (\sec ^{-1}(c x)\right ) \left (a+b \sec ^{-1}(c x)\right )^3d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \int \frac {e^{2 i \sec ^{-1}(c x)} \left (a+b \sec ^{-1}(c x)\right )^3}{1+e^{2 i \sec ^{-1}(c x)}}d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \left (\frac {3}{2} i b \int \left (a+b \sec ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )d\sec ^{-1}(c x)-\frac {1}{2} i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^3\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-i b \int \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )d\sec ^{-1}(c x)\right )-\frac {1}{2} i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^3\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-i b \left (\frac {1}{2} i b \int \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )d\sec ^{-1}(c x)-\frac {1}{2} i \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^3\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-i b \left (\frac {1}{4} b \int e^{-2 i \sec ^{-1}(c x)} \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )de^{2 i \sec ^{-1}(c x)}-\frac {1}{2} i \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^3\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^3\right )\) |
((I/4)*(a + b*ArcSec[c*x])^4)/b - (2*I)*((-1/2*I)*(a + b*ArcSec[c*x])^3*Lo g[1 + E^((2*I)*ArcSec[c*x])] + ((3*I)/2)*b*((I/2)*(a + b*ArcSec[c*x])^2*Po lyLog[2, -E^((2*I)*ArcSec[c*x])] - I*b*((-1/2*I)*(a + b*ArcSec[c*x])*PolyL og[3, -E^((2*I)*ArcSec[c*x])] + (b*PolyLog[4, -E^((2*I)*ArcSec[c*x])])/4)) )
3.1.28.3.1 Defintions of rubi rules used
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] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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] && IGt Q[m, 0]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /c^(m + 1) Subst[Int[(a + b*x)^n*Sec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x ]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] | | LtQ[m, -1])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (177 ) = 354\).
Time = 0.86 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.84
| method | result | size |
| parts | \(a^{3} \ln \left (x \right )+b^{3} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{4}}{4}-\operatorname {arcsec}\left (c x \right )^{3} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {3 i \operatorname {arcsec}\left (c x \right )^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 \,\operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 i \operatorname {polylog}\left (4, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{3}}{3}-\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )-\frac {\operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {i \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(364\) |
| derivativedivides | \(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{4}}{4}-\operatorname {arcsec}\left (c x \right )^{3} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {3 i \operatorname {arcsec}\left (c x \right )^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 \,\operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 i \operatorname {polylog}\left (4, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{3}}{3}-\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )-\frac {\operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {i \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(366\) |
| default | \(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{4}}{4}-\operatorname {arcsec}\left (c x \right )^{3} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {3 i \operatorname {arcsec}\left (c x \right )^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 \,\operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 i \operatorname {polylog}\left (4, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{3}}{3}-\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )-\frac {\operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {i \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(366\) |
a^3*ln(x)+b^3*(1/4*I*arcsec(c*x)^4-arcsec(c*x)^3*ln(1+(1/c/x+I*(1-1/c^2/x^ 2)^(1/2))^2)+3/2*I*arcsec(c*x)^2*polylog(2,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^ 2)-3/2*arcsec(c*x)*polylog(3,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)-3/4*I*polyl og(4,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2))+3*a*b^2*(1/3*I*arcsec(c*x)^3-arcse c(c*x)^2*ln(1+(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)+I*arcsec(c*x)*polylog(2,-(1 /c/x+I*(1-1/c^2/x^2)^(1/2))^2)-1/2*polylog(3,-(1/c/x+I*(1-1/c^2/x^2)^(1/2) )^2))+3*a^2*b*(1/2*I*arcsec(c*x)^2-arcsec(c*x)*ln(1+(1/c/x+I*(1-1/c^2/x^2) ^(1/2))^2)+1/2*I*polylog(2,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2))
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x}\, dx \]
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
-3/2*a*b^2*c^2*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*log(c)^2 - 12*b^3*c^2 *integrate(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*x^3 - x), x)*l og(c)^2 + 12*b^3*c^2*integrate(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) *log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) - 24*b^3*c^2*integrate(1/4*x^2*arct an(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)/(c^2*x^3 - x), x)*log(c) + 12*a*b^2 *c^2*integrate(1/4*x^2*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) - 24*a*b^2*c^ 2*integrate(1/4*x^2*log(x)/(c^2*x^3 - x), x)*log(c) + b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3*log(x) - 3/4*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) *log(c^2*x^2)^2*log(x) + 24*b^3*c^2*integrate(1/4*x^2*arctan(sqrt(c*x + 1) *sqrt(c*x - 1))*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) - 12*b^3*c^2*integra te(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)^2/(c^2*x^3 - x), x) + 12*a*b^2*c^2*integrate(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^ 2*x^3 - x), x) - 3*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^3 - x ), x) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) - 12*a*b^2*c^2*integrate(1/4*x^2*log(x)^2/(c^2*x^3 - x), x) + 12*a^2*b* c^2*integrate(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*x^3 - x), x ) + 3/2*a*b^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log(c)^2 + 12*b^3*i ntegrate(1/4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*x^3 - x), x)*log(c)^ 2 - 12*b^3*integrate(1/4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)/ (c^2*x^3 - x), x)*log(c) + 24*b^3*integrate(1/4*arctan(sqrt(c*x + 1)*sq...
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x} \,d x \]