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

Optimal. Leaf size=93 \[ -\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{75 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x) \]

[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

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Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 102, 12, 75} \begin {gather*} -\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 \cosh ^{-1}(a x)-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{25 a} \end {gather*}

Antiderivative was successfully verified.

[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*} \int x^4 \cosh ^{-1}(a x) \, dx &=\frac {1}{5} x^5 \cosh ^{-1}(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 \cosh ^{-1}(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 \cosh ^{-1}(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 \cosh ^{-1}(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 \cosh ^{-1}(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 \cosh ^{-1}(a x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.59 \begin {gather*} -\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 \cosh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [A]
time = 1.46, size = 52, normalized size = 0.56

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 68, normalized size = 0.73 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.34, size = 61, normalized size = 0.66 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [C] Result contains complex when optimal does not.
time = 0.32, size = 76, normalized size = 0.82 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acosh}{\left (a x \right )}}{5} - \frac {x^{4} \sqrt {a^{2} x^{2} - 1}}{25 a} - \frac {4 x^{2} \sqrt {a^{2} x^{2} - 1}}{75 a^{3}} - \frac {8 \sqrt {a^{2} x^{2} - 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{5}}{10} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*acosh(a*x)/5 - x**4*sqrt(a**2*x**2 - 1)/(25*a) - 4*x**2*sqrt(a**2*x**2 - 1)/(75*a**3) - 8*sqrt
(a**2*x**2 - 1)/(75*a**5), Ne(a, 0)), (I*pi*x**5/10, True))

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Giac [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),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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\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),x)

[Out]

int(x^4*acosh(a*x), x)

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