3.1.72 \(\int x^4 \sqrt {\cosh ^{-1}(a x)} \, dx\) [72]

Optimal. Leaf size=182 \[ \frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5} \]

[Out]

-1/1600*erf(5^(1/2)*arccosh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-1/1600*erfi(5^(1/2)*arccosh(a*x)^(1/2))*5^(1/2)*P
i^(1/2)/a^5-1/192*erf(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-1/192*erfi(3^(1/2)*arccosh(a*x)^(1/2))*
3^(1/2)*Pi^(1/2)/a^5-1/32*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^5-1/32*erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a^5+1/5*
x^5*arccosh(a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5884, 5953, 3393, 3388, 2211, 2235, 2236} \begin {gather*} -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}+\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[ArcCosh[a*x]],x]

[Out]

(x^5*Sqrt[ArcCosh[a*x]])/5 - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(32*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCos
h[a*x]]])/(64*a^5) - (Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(320*a^5) - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]
]])/(32*a^5) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(64*a^5) - (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcCosh[
a*x]]])/(320*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 3388

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 3393

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 5884

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 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int x^4 \sqrt {\cosh ^{-1}(a x)} \, dx &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx\\ &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh ^5(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 \sqrt {x}}+\frac {5 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{160 a^5}-\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{320 a^5}-\frac {\text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{320 a^5}-\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^5}-\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^5}-\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{160 a^5}-\frac {\text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{160 a^5}-\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{16 a^5}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{16 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 162, normalized size = 0.89 \begin {gather*} \frac {3 \sqrt {5} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-5 \cosh ^{-1}(a x)\right )+25 \sqrt {3} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-3 \cosh ^{-1}(a x)\right )+150 \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-\cosh ^{-1}(a x)\right )+150 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},\cosh ^{-1}(a x)\right )+25 \sqrt {3} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},3 \cosh ^{-1}(a x)\right )+3 \sqrt {5} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},5 \cosh ^{-1}(a x)\right )}{2400 a^5 \sqrt {-\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[ArcCosh[a*x]],x]

[Out]

(3*Sqrt[5]*Sqrt[ArcCosh[a*x]]*Gamma[3/2, -5*ArcCosh[a*x]] + 25*Sqrt[3]*Sqrt[ArcCosh[a*x]]*Gamma[3/2, -3*ArcCos
h[a*x]] + 150*Sqrt[ArcCosh[a*x]]*Gamma[3/2, -ArcCosh[a*x]] + 150*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, ArcCosh[a*x]]
+ 25*Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, 3*ArcCosh[a*x]] + 3*Sqrt[5]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, 5*ArcCo
sh[a*x]])/(2400*a^5*Sqrt[-ArcCosh[a*x]])

________________________________________________________________________________________

Maple [F]
time = 7.43, size = 0, normalized size = 0.00 \[\int x^{4} \sqrt {\mathrm {arccosh}\left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccosh(a*x)^(1/2),x)

[Out]

int(x^4*arccosh(a*x)^(1/2),x)

________________________________________________________________________________________

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)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4*sqrt(arccosh(a*x)), x)

________________________________________________________________________________________

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)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acosh(a*x)**(1/2),x)

[Out]

Integral(x**4*sqrt(acosh(a*x)), x)

________________________________________________________________________________________

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)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4*sqrt(arccosh(a*x)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acosh(a*x)^(1/2),x)

[Out]

int(x^4*acosh(a*x)^(1/2), x)

________________________________________________________________________________________