3.2.42 \(\int \frac {\cosh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [142]

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

[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 = 0.12, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5932, 5946, 4265, 2317, 2438, 30} \begin {gather*} \frac {a^2 \sqrt {a x-1} \cosh ^{-1}(a x) \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )}{\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 {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 \sqrt {a x-1}}{2 x \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Rule 30

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

Rule 2317

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 2438

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 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 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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.21, size = 234, normalized size = 1.40 \begin {gather*} \frac {(1+a x) \left (a x \sqrt {\frac {-1+a x}{1+a x}}-\cosh ^{-1}(a x)+a x \cosh ^{-1}(a x)-i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right )}{2 x^2 \sqrt {1-a^2 x^2}} \end {gather*}

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 = 4.99, size = 349, normalized size = 2.09

method	result	size
default	\(-\frac {\left (\mathrm {arccosh}\left (a x \right ) a^{2} x^{2}+\sqrt {a x +1}\, \sqrt {a x -1}\, a x -\mathrm {arccosh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \dilog \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \dilog \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}\)	\(349\)

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,method=_RETURNVERBOSE)

[Out]

-1/2*(arccosh(a*x)*a^2*x^2+(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, 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)/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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}{\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)/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 = 0.00, 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)/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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {acosh}\left (a\,x\right )}{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)/(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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