3.142 \(\int \frac {\cosh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=167 \[ -\frac {i a^2 \sqrt {a x-1} \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {i a^2 \sqrt {a x-1} \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{2 x^2}+\frac {a^2 \sqrt {a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {a \sqrt {a x-1}}{2 x \sqrt {1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.47, antiderivative size = 230, normalized size of antiderivative = 1.38, number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5798, 5748, 5761, 4180, 2279, 2391, 30} \[ -\frac {i a^2 \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {i a^2 \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x \sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {a^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4180

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 5748

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b
*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 5798

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

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 234, normalized size = 1.40 \[ \frac {(a x+1) \left (-i a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )-i a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+a x \sqrt {\frac {a x-1}{a x+1}}+a x \cosh ^{-1}(a x)-\cosh ^{-1}(a x)\right )}{2 x^2 \sqrt {1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 2.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}{a^{2} x^{5} - x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.61, size = 349, normalized size = 2.09 \[ -\frac {\left (a^{2} x^{2} \mathrm {arccosh}\left (a x \right )+\sqrt {a x +1}\, \sqrt {a x -1}\, a x -\mathrm {arccosh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \dilog \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \dilog \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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