3.1.31 \(\int \frac {\cosh ^{-1}(a x)^3}{x^5} \, dx\) [31]

Optimal. Leaf size=174 \[ -\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{2} a^4 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right ) \]

[Out]

1/4*a^2*arccosh(a*x)/x^2+1/2*a^4*arccosh(a*x)^2-1/4*arccosh(a*x)^3/x^4-a^4*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2
)*(a*x+1)^(1/2))^2)-1/2*a^4*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-1/4*a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2
)/x+1/4*a*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^3+1/2*a^3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5883, 5933, 5918, 5882, 3799, 2221, 2317, 2438, 97} \begin {gather*} -\frac {1}{2} a^4 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2-a^4 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {a^3 \sqrt {a x-1} \sqrt {a x+1}}{4 x}+\frac {a^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/x^5,x]

[Out]

-1/4*(a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/x + (a^2*ArcCosh[a*x])/(4*x^2) + (a^4*ArcCosh[a*x]^2)/2 + (a*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(4*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x) - ArcCos
h[a*x]^3/(4*x^4) - a^4*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] - (a^4*PolyLog[2, -E^(2*ArcCosh[a*x])])/2

Rule 97

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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 5918

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[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, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] &&
NeQ[p, -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]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\cosh ^{-1}(a x)^2}{x^4 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-\frac {1}{2} a^2 \int \frac {\cosh ^{-1}(a x)}{x^3} \, dx+\frac {1}{2} a^3 \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-\frac {1}{4} a^3 \int \frac {1}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx-a^4 \int \frac {\cosh ^{-1}(a x)}{x} \, dx\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \text {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-\left (2 a^4\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+a^4 \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {1}{2} a^4 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{2} a^4 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.45, size = 220, normalized size = 1.26 \begin {gather*} \frac {a^3 x^3-a^5 x^5-a x (1+a x) \left (1-a x+2 a^2 x^2+2 a^3 x^3 \left (-1+\sqrt {\frac {-1+a x}{1+a x}}\right )\right ) \cosh ^{-1}(a x)^2-\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3-a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x) \left (-1+4 a^2 x^2 \log \left (1+e^{-2 \cosh ^{-1}(a x)}\right )\right )+2 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )}{4 x^4 \sqrt {-1+a x} \sqrt {1+a x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/x^5,x]

[Out]

(a^3*x^3 - a^5*x^5 - a*x*(1 + a*x)*(1 - a*x + 2*a^2*x^2 + 2*a^3*x^3*(-1 + Sqrt[(-1 + a*x)/(1 + a*x)]))*ArcCosh
[a*x]^2 - Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3 - a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a
*x]*(-1 + 4*a^2*x^2*Log[1 + E^(-2*ArcCosh[a*x])]) + 2*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*PolyLog[2,
-E^(-2*ArcCosh[a*x])])/(4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

________________________________________________________________________________________

Maple [A]
time = 4.01, size = 191, normalized size = 1.10

method	result	size
derivativedivides	\(a^{4} \left (-\frac {-2 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}+2 \mathrm {arccosh}\left (a x \right )^{2} a^{4} x^{4}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x \mathrm {arccosh}\left (a x \right )^{2}+\sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-a^{4} x^{4}+\mathrm {arccosh}\left (a x \right )^{3}-a^{2} x^{2} \mathrm {arccosh}\left (a x \right )}{4 a^{4} x^{4}}+\mathrm {arccosh}\left (a x \right )^{2}-\mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {\polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\)	\(191\)
default	\(a^{4} \left (-\frac {-2 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}+2 \mathrm {arccosh}\left (a x \right )^{2} a^{4} x^{4}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x \mathrm {arccosh}\left (a x \right )^{2}+\sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-a^{4} x^{4}+\mathrm {arccosh}\left (a x \right )^{3}-a^{2} x^{2} \mathrm {arccosh}\left (a x \right )}{4 a^{4} x^{4}}+\mathrm {arccosh}\left (a x \right )^{2}-\mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {\polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

a^4*(-1/4*(-2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^3*x^3+2*arccosh(a*x)^2*a^4*x^4-(a*x-1)^(1/2)*(a*x+1
)^(1/2)*a*x*arccosh(a*x)^2+(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^3*x^3-a^4*x^4+arccosh(a*x)^3-a^2*x^2*arccosh(a*x))/a^
4/x^4+arccosh(a*x)^2-arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-1/2*polylog(2,-(a*x+(a*x-1)^(1/2)*
(a*x+1)^(1/2))^2))

________________________________________________________________________________________

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)^3/x^5,x, algorithm="maxima")

[Out]

-1/4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^4 + integrate(3/4*(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))^2/(a^3*x^7 - a*x^5 + (a^2*x^6 - x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))
, x)

________________________________________________________________________________________

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)^3/x^5,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^3/x^5, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/x**5,x)

[Out]

Integral(acosh(a*x)**3/x**5, x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3/x^5,x)

[Out]

int(acosh(a*x)^3/x^5, x)

________________________________________________________________________________________