3.2.3 \(\int \frac {x^4}{\cosh ^{-1}(a x)^{5/2}} \, dx\) [103]

Optimal. Leaf size=228 \[ -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {Erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {Erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5} \]

[Out]

-1/12*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^5+1/12*erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a^5-3/8*erf(3^(1/2)*arccosh(
a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+3/8*erfi(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-5/24*erf(5^(1/2)*ar
ccosh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5+5/24*erfi(5^(1/2)*arccosh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-2/3*x^4*(a*x
-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+16/3*x^3/a^2/arccosh(a*x)^(1/2)-20/3*x^5/arccosh(a*x)^(1/2)

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Rubi [A]
time = 0.58, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5886, 5951, 5887, 5556, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {Erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {Erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^(5/2),x]

[Out]

(-2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) + (16*x^3)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (20*x^5
)/(3*Sqrt[ArcCosh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(12*a^5) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCos
h[a*x]]])/(8*a^5) - (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(24*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]
]])/(12*a^5) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(8*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcCo
sh[a*x]]])/(24*a^5)

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5886

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCosh[c
*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5951

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {x^4}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac {8 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {100}{3} \int \frac {x^4}{\sqrt {\cosh ^{-1}(a x)}} \, dx-\frac {16 \int \frac {x^2}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {x}}+\frac {\sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 \sqrt {x}}+\frac {3 \sinh (3 x)}{16 \sqrt {x}}+\frac {\sinh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {25 \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac {4 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {4 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {25 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^5}+\frac {2 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {2 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {2 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {2 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {25 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac {25 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {25 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {4 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}+\frac {4 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}-\frac {4 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}-\frac {4 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}-\frac {25 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^5}-\frac {25 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^5}+\frac {25 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}\\ \end {align*}

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Mathematica [A]
time = 1.14, size = 278, normalized size = 1.22 \begin {gather*} -\frac {2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+2 e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)+2 \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )+5 \cosh ^{-1}(a x) \left (e^{-5 \cosh ^{-1}(a x)}+e^{5 \cosh ^{-1}(a x)}-\sqrt {5} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-5 \cosh ^{-1}(a x)\right )-\sqrt {5} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},5 \cosh ^{-1}(a x)\right )\right )+3 \left (3 e^{-3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+3 e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+3 \sqrt {3} \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-3 \cosh ^{-1}(a x)\right )-3 \sqrt {3} \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \cosh ^{-1}(a x)\right )+\sinh \left (3 \cosh ^{-1}(a x)\right )\right )+\sinh \left (5 \cosh ^{-1}(a x)\right )}{24 a^5 \cosh ^{-1}(a x)^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcCosh[a*x]^(5/2),x]

[Out]

-1/24*(2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + (2*ArcCosh[a*x])/E^ArcCosh[a*x] + 2*E^ArcCosh[a*x]*ArcCosh[a*x
] + 2*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] - 2*ArcCosh[a*x]^(3/2)*Gamma[1/2, ArcCosh[a*x]] + 5*ArcC
osh[a*x]*(E^(-5*ArcCosh[a*x]) + E^(5*ArcCosh[a*x]) - Sqrt[5]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -5*ArcCosh[a*x]] -
Sqrt[5]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 5*ArcCosh[a*x]]) + 3*((3*ArcCosh[a*x])/E^(3*ArcCosh[a*x]) + 3*E^(3*ArcC
osh[a*x])*ArcCosh[a*x] + 3*Sqrt[3]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -3*ArcCosh[a*x]] - 3*Sqrt[3]*ArcCosh[a*x]^
(3/2)*Gamma[1/2, 3*ArcCosh[a*x]] + Sinh[3*ArcCosh[a*x]]) + Sinh[5*ArcCosh[a*x]])/(a^5*ArcCosh[a*x]^(3/2))

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Maple [F]
time = 6.43, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^(5/2),x)

[Out]

int(x^4/arccosh(a*x)^(5/2),x)

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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(x^4/arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^4/arccosh(a*x)^(5/2), x)

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Fricas [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(x^4/arccosh(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**(5/2),x)

[Out]

Integral(x**4/acosh(a*x)**(5/2), 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(x^4/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^4/arccosh(a*x)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/acosh(a*x)^(5/2),x)

[Out]

int(x^4/acosh(a*x)^(5/2), x)

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