3.3.32 \(\int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [232]

Optimal. Leaf size=296 \[ \frac {a \sqrt {-1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \cosh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {a^2 \sqrt {-1+a x} \text {ArcTan}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {-1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {-1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {-1+a x} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {-1+a x} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}} \]

[Out]

a*arccosh(a*x)*(a*x-1)^(1/2)/x/(-a*x+1)^(1/2)+a^2*arccosh(a*x)^2*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-
1)^(1/2)/(-a*x+1)^(1/2)-a^2*arctan((a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-I*a^2*arccosh(a*x
)*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)+I*a^2*arccosh(a*x)*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)+I*a^2*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)-I*a^2*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)-1/2*arccosh(a*x)^2*(-a^2*x^2+1)^(1/2)/x^2

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Rubi [A]
time = 0.21, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5932, 5946, 4265, 2611, 2320, 6724, 5883, 94, 211} \begin {gather*} -\frac {a^2 \sqrt {a x-1} \text {ArcTan}\left (\sqrt {a x-1} \sqrt {a x+1}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {a x-1} \cosh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {a x-1} \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {a x-1} \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 x^2}+\frac {a \sqrt {a x-1} \cosh ^{-1}(a x)}{x \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5932

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

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]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{x^2} \, dx}{\sqrt {1-a^2 x^2}}+\frac {\left (a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {\left (a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt {1-a^2 x^2}}-\frac {\left (a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (a^3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (i a^2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a^2 \sqrt {-1+a x} \sqrt {1+a x} \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {i a^2 \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 233, normalized size = 0.79 \begin {gather*} \frac {i a^2 \sqrt {-((-1+a x) (1+a x))} \left (\frac {2 i \cosh ^{-1}(a x)}{a x}+\frac {i \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)^2}{a^2 x^2}-4 i \text {ArcTan}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )+\cosh ^{-1}(a x)^2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(a x)}\right )\right )}{2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/2)*a^2*Sqrt[-((-1 + a*x)*(1 + a*x))]*(((2*I)*ArcCosh[a*x])/(a*x) + (I*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)
*ArcCosh[a*x]^2)/(a^2*x^2) - (4*I)*ArcTan[Tanh[ArcCosh[a*x]/2]] + ArcCosh[a*x]^2*Log[1 - I/E^ArcCosh[a*x]] - A
rcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2, (-I)/E^ArcCosh[a*x]] - 2*ArcCosh[a*x]*Poly
Log[2, I/E^ArcCosh[a*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 2*PolyLog[3, I/E^ArcCosh[a*x]]))/(Sqrt[(-1 + a*
x)/(1 + a*x)]*(1 + a*x))

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Maple [F]
time = 3.33, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (a x \right )^{2}}{x^{3} \sqrt {-a^{2} x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^2/(a^2*x^5 - x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{3} \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)**2/x**3/(-a**2*x**2+1)**(1/2),x)

[Out]

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

[Out]

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

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

[Out]

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

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