\(\int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [259]

Optimal result

Integrand size = 24, antiderivative size = 166 \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 a \sqrt {-1+a x} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5917, 5882, 3799, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {a x-1} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 a \sqrt {a x-1} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {a \sqrt {a x-1} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )}{\sqrt {1-a x}} \]

[In]

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

[Out]

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

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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 5917

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[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*A
rcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m
+ 2*p + 3, 0] && NeQ[m, -1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {\left (3 a \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)^2}{x} \, dx}{\sqrt {1-a x}} \\ & = -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {\left (3 a \sqrt {-1+a x}\right ) \text {Subst}\left (\int x^2 \tanh (x) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {\left (6 a \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (6 a \sqrt {-1+a x}\right ) \text {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (3 a \sqrt {-1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (3 a \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 \text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 a \sqrt {-1+a x} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83 \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (2 \text {arccosh}(a x)^2 \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )\right )}{2 \sqrt {-((-1+a x) (1+a x))}} \]

[In]

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

[Out]

(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(2*ArcCosh[a*x]^2*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*
x)*ArcCosh[a*x])/(a*x) - 3*Log[1 + E^(-2*ArcCosh[a*x])]) + 6*ArcCosh[a*x]*PolyLog[2, -E^(-2*ArcCosh[a*x])] + 3
*PolyLog[3, -E^(-2*ArcCosh[a*x])]))/(2*Sqrt[-((-1 + a*x)*(1 + a*x))])

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.89

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \operatorname {arccosh}\left (a x \right )^{3}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{3} a}{a^{2} x^{2}-1}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{2 \left (a^{2} x^{2}-1\right )}\)	\(313\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3/(a^2*x^4 - x^2), x)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]

[In]

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

[Out]

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