3.105 \(\int \frac {x^2}{\cosh ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {8 x}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

[Out]

-1/6*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3+1/6*erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3-1/2*erf(3^(1/2)*arccosh(a*
x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+1/2*erfi(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-2/3*x^2*(a*x-1)^(1/2)
*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+8/3*x/a^2/arccosh(a*x)^(1/2)-4*x^3/arccosh(a*x)^(1/2)

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Rubi [A]  time = 0.63, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205, 5658} \[ -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {8 x}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) + (8*x)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (4*x^3)/Sq
rt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(6*a^3) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(
2*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(2*a^3)

Rule 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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 3308

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 5448

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 5658

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

Rule 5668

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 5670

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

Rule 5775

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*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), 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, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.74, size = 194, normalized size = 1.17 \[ \frac {-\sqrt {\frac {a x-1}{a x+1}} (a x+1)-3 e^{-3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)-e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)-3 e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-\sinh \left (3 \cosh ^{-1}(a x)\right )-3 \sqrt {3} \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-3 \cosh ^{-1}(a x)\right )-\left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )+3 \sqrt {3} \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \cosh ^{-1}(a x)\right )}{6 a^3 \cosh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^2/arccosh(a*x)^(5/2), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/acosh(a*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/acosh(a*x)**(5/2), x)

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