3.1.20 \(\int \frac {\cosh ^{-1}(a x)^2}{x^4} \, dx\) [20]

Optimal. Leaf size=114 \[ \frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cosh ^{-1}(a x) \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+\frac {1}{3} i a^3 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right ) \]

[Out]

1/3*a^2/x-1/3*arccosh(a*x)^2/x^3+2/3*a^3*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-1/3*I*a^3*polylo
g(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*I*a^3*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*a*arcco
sh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^2

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Rubi [A]
time = 0.26, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5933, 5947, 4265, 2317, 2438, 30} \begin {gather*} \frac {2}{3} a^3 \cosh ^{-1}(a x) \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\frac {a^2}{3 x}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^2/x^4,x]

[Out]

a^2/(3*x) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*x^2) - ArcCosh[a*x]^2/(3*x^3) + (2*a^3*ArcCosh[a*
x]*ArcTan[E^ArcCosh[a*x]])/3 - (I/3)*a^3*PolyLog[2, (-I)*E^ArcCosh[a*x]] + (I/3)*a^3*PolyLog[2, I*E^ArcCosh[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 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 5933

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*d
2*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/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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*ArcCosh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\cosh ^{-1}(a x)}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx+\frac {1}{3} a^3 \int \frac {\cosh ^{-1}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \text {Subst}\left (\int x \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=\frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac {\cosh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 144, normalized size = 1.26 \begin {gather*} \frac {1}{3} a^3 \left (\frac {1}{a x}+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)}{a^2 x^2}-\frac {\cosh ^{-1}(a x)^2}{a^3 x^3}-i \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+i \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-i \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^2/x^4,x]

[Out]

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

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Maple [A]
time = 4.67, size = 171, normalized size = 1.50

method	result	size
derivativedivides	\(a^{3} \left (-\frac {-a x \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+\mathrm {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {i \mathrm {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \mathrm {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}-\frac {i \dilog \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \dilog \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}\right )\)	\(171\)
default	\(a^{3} \left (-\frac {-a x \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+\mathrm {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {i \mathrm {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \mathrm {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}-\frac {i \dilog \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \dilog \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}\right )\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

a^3*(-1/3*(-a*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+arccosh(a*x)^2-a^2*x^2)/a^3/x^3-1/3*I*arccosh(a*x)*ln
(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*I*arccosh(a*x)*ln(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-1/3*I*dil
og(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*I*dilog(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))))

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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)^2/x^4,x, algorithm="maxima")

[Out]

-1/3*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/x^3 + integrate(2/3*(a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x
- a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/(a^3*x^6 - a*x^4 + (a^2*x^5 - x^3)*sqrt(a*x + 1)*sqrt(a*x - 1)),
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^4,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^2/x^4, 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^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**2/x**4,x)

[Out]

Integral(acosh(a*x)**2/x**4, 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^4,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^2/x^4, 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 )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^2/x^4,x)

[Out]

int(acosh(a*x)^2/x^4, x)

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