3.3.58 \(\int \frac {\cosh ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\) [258]

Optimal. Leaf size=265 \[ \frac {2 \sqrt {-1+a x} \cosh ^{-1}(a x)^3 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i \sqrt {-1+a x} \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i \sqrt {-1+a x} \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {6 i \sqrt {-1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 i \sqrt {-1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 i \sqrt {-1+a x} \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {6 i \sqrt {-1+a x} \text {PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}} \]

[Out]

2*arccosh(a*x)^3*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*I*arccosh(a*x)^2*polyl
og(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+3*I*arccosh(a*x)^2*polylog(2,I*(a*x+(a
*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+6*I*arccosh(a*x)*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x
+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-6*I*arccosh(a*x)*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-
1)^(1/2)/(-a*x+1)^(1/2)-6*I*polylog(4,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+6*I*p
olylog(4,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5946, 4265, 2611, 6744, 2320, 6724} \begin {gather*} \frac {2 \sqrt {a x-1} \cosh ^{-1}(a x)^3 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i \sqrt {a x-1} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i \sqrt {a x-1} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {6 i \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 i \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 i \sqrt {a x-1} \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {6 i \sqrt {a x-1} \text {Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

(2*Sqrt[-1 + a*x]*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((3*I)*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*
PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + ((3*I)*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a
*x]])/Sqrt[1 - a*x] + ((6*I)*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((6*
I)*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((6*I)*Sqrt[-1 + a*x]*PolyLog[4,
(-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + ((6*I)*Sqrt[-1 + a*x]*PolyLog[4, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]

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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5946

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c
*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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 {\cosh ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^3}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (3 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (3 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (6 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {3 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {6 i \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.46, size = 488, normalized size = 1.84 \begin {gather*} \frac {i \sqrt {-((-1+a x) (1+a x))} \left (7 \pi ^4+8 i \pi ^3 \cosh ^{-1}(a x)+24 \pi ^2 \cosh ^{-1}(a x)^2-32 i \pi \cosh ^{-1}(a x)^3-16 \cosh ^{-1}(a x)^4+8 i \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+48 \pi ^2 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-96 i \pi \cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \cosh ^{-1}(a x) \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+96 i \pi \cosh ^{-1}(a x)^2 \log \left (1-i e^{\cosh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )-48 \left (\pi -2 i \cosh ^{-1}(a x)\right )^2 \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+192 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-48 \pi ^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-192 i \pi \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (4,-i e^{-\cosh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )\right )}{64 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

((I/64)*Sqrt[-((-1 + a*x)*(1 + a*x))]*(7*Pi^4 + (8*I)*Pi^3*ArcCosh[a*x] + 24*Pi^2*ArcCosh[a*x]^2 - (32*I)*Pi*A
rcCosh[a*x]^3 - 16*ArcCosh[a*x]^4 + (8*I)*Pi^3*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*ArcCosh[a*x]*Log[1 + I/E^Ar
cCosh[a*x]] - (96*I)*Pi*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]]
- 48*Pi^2*ArcCosh[a*x]*Log[1 - I*E^ArcCosh[a*x]] + (96*I)*Pi*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I)
*Pi^3*Log[1 + I*E^ArcCosh[a*x]] + 64*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)
*ArcCosh[a*x])/4]] - 48*(Pi - (2*I)*ArcCosh[a*x])^2*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 192*ArcCosh[a*x]^2*PolyL
og[2, (-I)*E^ArcCosh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*ArcCosh[a*x]*PolyLog[2, I*E^Arc
Cosh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcCosh[a*x]] + 384*ArcCosh[a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 3
84*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcCosh[a*x]] + 384*PolyLog[4, (-I
)/E^ArcCosh[a*x]] + 384*PolyLog[4, (-I)*E^ArcCosh[a*x]]))/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (a x \right )^{3}}{x \sqrt {-a^{2} x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x)

[Out]

int(arccosh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x)

________________________________________________________________________________________

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(arccosh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x), x)

________________________________________________________________________________________

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(arccosh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3/(a^2*x^3 - x), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/x/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(acosh(a*x)**3/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)

________________________________________________________________________________________

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(arccosh(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3/(x*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(acosh(a*x)^3/(x*(1 - a^2*x^2)^(1/2)), x)

________________________________________________________________________________________