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

Optimal. Leaf size=123 \[ -\frac {2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac {4}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.48, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5668, 5775, 5670, 5448, 12, 3308, 2180, 2204, 2205, 5676} \[ -\frac {2 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac {4}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

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

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Mathematica [A]  time = 0.28, size = 83, normalized size = 0.67 \[ -\frac {2 \sqrt {2 \pi } \left (\text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-\text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )\right )+\frac {4 \cosh \left (2 \cosh ^{-1}(a x)\right )}{\sqrt {\cosh ^{-1}(a x)}}+\frac {\sinh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{3/2}}}{3 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*((4*Cosh[2*ArcCosh[a*x]])/Sqrt[ArcCosh[a*x]] + 2*Sqrt[2*Pi]*(Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - Erfi[Sqrt[
2]*Sqrt[ArcCosh[a*x]]]) + Sinh[2*ArcCosh[a*x]]/ArcCosh[a*x]^(3/2))/a^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/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 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arccosh(a*x)^(5/2), x)

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maple [A]  time = 0.33, size = 122, normalized size = 0.99 \[ -\frac {\sqrt {2}\, \left (4 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x^{2} a^{2}+\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, x a +2 \mathrm {arccosh}\left (a x \right )^{2} \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \mathrm {arccosh}\left (a x \right )^{2} \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{3 \sqrt {\pi }\, a^{2} \mathrm {arccosh}\left (a x \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arccosh(a*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/acosh(a*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/acosh(a*x)**(5/2), x)

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