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

Optimal. Leaf size=157 \[ -\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}-\frac {15 \sqrt {\cosh ^{-1}(a x)}}{64 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}+\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{8 a} \]

[Out]

-1/4*arccosh(a*x)^(5/2)/a^2+1/2*x^2*arccosh(a*x)^(5/2)-15/512*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)
/a^2-15/512*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2-5/8*x*arccosh(a*x)^(3/2)*(a*x-1)^(1/2)*(a*x+
1)^(1/2)/a-15/64*arccosh(a*x)^(1/2)/a^2+15/32*x^2*arccosh(a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.71, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5664, 5759, 5676, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac {15 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}-\frac {15 \sqrt {\cosh ^{-1}(a x)}}{64 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}+\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[ArcCosh[a*x]])/(64*a^2) + (15*x^2*Sqrt[ArcCosh[a*x]])/32 - (5*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh
[a*x]^(3/2))/(8*a) - ArcCosh[a*x]^(5/2)/(4*a^2) + (x^2*ArcCosh[a*x]^(5/2))/2 - (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt
[ArcCosh[a*x]]])/(256*a^2) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a^2)

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 3307

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

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5664

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCosh[c*x])^n)/
(m + 1), x] - Dist[(b*c*n)/(m + 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] && GtQ[n, 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 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rubi steps

\begin {align*} \int x \cosh ^{-1}(a x)^{5/2} \, dx &=\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {1}{4} (5 a) \int \frac {x^2 \cosh ^{-1}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}+\frac {15}{16} \int x \sqrt {\cosh ^{-1}(a x)} \, dx-\frac {5 \int \frac {\cosh ^{-1}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a}\\ &=\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {1}{64} (15 a) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx\\ &=\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}\\ &=\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}\\ &=-\frac {15 \sqrt {\cosh ^{-1}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a^2}\\ &=-\frac {15 \sqrt {\cosh ^{-1}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a^2}-\frac {15 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a^2}\\ &=-\frac {15 \sqrt {\cosh ^{-1}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^2}-\frac {15 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^2}\\ &=-\frac {15 \sqrt {\cosh ^{-1}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\cosh ^{-1}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac {\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 92, normalized size = 0.59 \[ \frac {-15 \sqrt {2 \pi } \left (\text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )\right )+8 \left (16 \cosh ^{-1}(a x)^2+15\right ) \cosh \left (2 \cosh ^{-1}(a x)\right ) \sqrt {\cosh ^{-1}(a x)}-160 \cosh ^{-1}(a x)^{3/2} \sinh \left (2 \cosh ^{-1}(a x)\right )}{512 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*Sqrt[ArcCosh[a*x]]*(15 + 16*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] - 15*Sqrt[2*Pi]*(Erf[Sqrt[2]*Sqrt[ArcCosh[
a*x]]] + Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]) - 160*ArcCosh[a*x]^(3/2)*Sinh[2*ArcCosh[a*x]])/(512*a^2)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.27, size = 139, normalized size = 0.89 \[ \frac {\sqrt {2}\, \left (128 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, x^{2} a^{2}-160 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, x a -64 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }+120 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, x^{2} a^{2}-60 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }-15 \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-15 \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{512 \sqrt {\pi }\, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*2^(1/2)*(128*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*x^2*a^2-160*arccosh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*(a*x+1)
^(1/2)*(a*x-1)^(1/2)*x*a-64*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)+120*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*x^2*a^
2-60*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)-15*Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-15*Pi*erfi(2^(1/2)*arccosh(a*x)
^(1/2)))/Pi^(1/2)/a^2

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________