\(\int x^4 \text {arccosh}(a x) \, dx\) [1]

Optimal result

Integrand size = 8, antiderivative size = 93 \[ \int x^4 \text {arccosh}(a x) \, dx=-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x) \]

[Out]

1/5*x^5*arccosh(a*x)-8/75*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5-4/75*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/25*x^4*(a
*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 102, 12, 75} \[ \int x^4 \text {arccosh}(a x) \, dx=-\frac {8 \sqrt {a x-1} \sqrt {a x+1}}{75 a^5}-\frac {4 x^2 \sqrt {a x-1} \sqrt {a x+1}}{75 a^3}+\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{25 a} \]

[In]

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

[Out]

(-8*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^5) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^3) - (x^4*Sqrt[-1 + a*
x]*Sqrt[1 + a*x])/(25*a) + (x^5*ArcCosh[a*x])/5

Rule 12

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

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 5883

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {\int \frac {4 x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a} \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {4 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a} \\ & = -\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {4 \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{75 a^3} \\ & = -\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {8 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{75 a^3} \\ & = -\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59 \[ \int x^4 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (8+4 a^2 x^2+3 a^4 x^4\right )}{75 a^5}+\frac {1}{5} x^5 \text {arccosh}(a x) \]

[In]

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

[Out]

-1/75*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 + 4*a^2*x^2 + 3*a^4*x^4))/a^5 + (x^5*ArcCosh[a*x])/5

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52

method	result	size
parts	\(\frac {x^{5} \operatorname {arccosh}\left (a x \right )}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right )}{75 a^{5}}\)	\(48\)
derivativedivides	\(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right )}{75}}{a^{5}}\)	\(52\)
default	\(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right )}{75}}{a^{5}}\)	\(52\)

[In]

int(x^4*arccosh(a*x),x,method=_RETURNVERBOSE)

[Out]

1/5*x^5*arccosh(a*x)-1/75/a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(3*a^4*x^4+4*a^2*x^2+8)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.66 \[ \int x^4 \text {arccosh}(a x) \, dx=\frac {15 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} - 1}}{75 \, a^{5}} \]

[In]

integrate(x^4*arccosh(a*x),x, algorithm="fricas")

[Out]

1/75*(15*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1)) - (3*a^4*x^4 + 4*a^2*x^2 + 8)*sqrt(a^2*x^2 - 1))/a^5

Sympy [F]

\[ \int x^4 \text {arccosh}(a x) \, dx=\int x^{4} \operatorname {acosh}{\left (a x \right )}\, dx \]

[In]

integrate(x**4*acosh(a*x),x)

[Out]

Integral(x**4*acosh(a*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.73 \[ \int x^4 \text {arccosh}(a x) \, dx=\frac {1}{5} \, x^{5} \operatorname {arcosh}\left (a x\right ) - \frac {1}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} - 1}}{a^{6}}\right )} a \]

[In]

integrate(x^4*arccosh(a*x),x, algorithm="maxima")

[Out]

1/5*x^5*arccosh(a*x) - 1/75*(3*sqrt(a^2*x^2 - 1)*x^4/a^2 + 4*sqrt(a^2*x^2 - 1)*x^2/a^4 + 8*sqrt(a^2*x^2 - 1)/a
^6)*a

Giac [F(-2)]

Exception generated. \[ \int x^4 \text {arccosh}(a x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*arccosh(a*x),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^4 \text {arccosh}(a x) \, dx=\int x^4\,\mathrm {acosh}\left (a\,x\right ) \,d x \]

[In]

int(x^4*acosh(a*x),x)

[Out]

int(x^4*acosh(a*x), x)