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

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

[Out]

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

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Rubi [A]  time = 0.471289, 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 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]

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 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 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 30

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

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*}

Mathematica [A]  time = 0.29017, size = 234, normalized size = 1.4 \[ \frac{(a x+1) \left (-i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )-i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+a x \sqrt{\frac{a x-1}{a x+1}}+a x \cosh ^{-1}(a x)-\cosh ^{-1}(a x)\right )}{2 x^2 \sqrt{1-a^2 x^2}} \]

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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Maple [A]  time = 0.206, size = 349, normalized size = 2.1 \begin{align*} -{\frac{1}{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){x}^{2}} \left ({a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )+\sqrt{ax+1}\sqrt{ax-1}ax-{\rm arccosh} \left (ax\right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{i{\rm arccosh} \left (ax\right ){a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }-{\frac{i{\rm arccosh} \left (ax\right ){a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }+{\frac{i{a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it dilog} \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }-{\frac{i{a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it dilog} \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) } \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}{a^{2} x^{5} - x^{3}}, x\right ) \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}

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)