3.12 \(\int x^4 \cosh ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=132 \[ \frac{8 x^3}{225 a^2}-\frac{8 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{75 a^3}+\frac{16 x}{75 a^4}-\frac{16 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{75 a^5}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^2-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{25 a}+\frac{2 x^5}{125} \]

[Out]

(16*x)/(75*a^4) + (8*x^3)/(225*a^2) + (2*x^5)/125 - (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^5) -
(8*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^3) - (2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])
/(25*a) + (x^5*ArcCosh[a*x]^2)/5

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Rubi [A]  time = 0.489759, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5662, 5759, 5718, 8, 30} \[ \frac{8 x^3}{225 a^2}-\frac{8 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{75 a^3}+\frac{16 x}{75 a^4}-\frac{16 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{75 a^5}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^2-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{25 a}+\frac{2 x^5}{125} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCosh[a*x]^2,x]

[Out]

(16*x)/(75*a^4) + (8*x^3)/(225*a^2) + (2*x^5)/125 - (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^5) -
(8*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^3) - (2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])
/(25*a) + (x^5*ArcCosh[a*x]^2)/5

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^4 \cosh ^{-1}(a x)^2 \, dx &=\frac{1}{5} x^5 \cosh ^{-1}(a x)^2-\frac{1}{5} (2 a) \int \frac{x^5 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^2+\frac{2 \int x^4 \, dx}{25}-\frac{8 \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a}\\ &=\frac{2 x^5}{125}-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^2-\frac{16 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{75 a^3}+\frac{8 \int x^2 \, dx}{75 a^2}\\ &=\frac{8 x^3}{225 a^2}+\frac{2 x^5}{125}-\frac{16 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^2+\frac{16 \int 1 \, dx}{75 a^4}\\ &=\frac{16 x}{75 a^4}+\frac{8 x^3}{225 a^2}+\frac{2 x^5}{125}-\frac{16 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.106839, size = 80, normalized size = 0.61 \[ \frac{\frac{40 x^3}{a^2}-\frac{30 \sqrt{a x-1} \sqrt{a x+1} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cosh ^{-1}(a x)}{a^5}+\frac{240 x}{a^4}+225 x^5 \cosh ^{-1}(a x)^2+18 x^5}{1125} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCosh[a*x]^2,x]

[Out]

((240*x)/a^4 + (40*x^3)/a^2 + 18*x^5 - (30*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*
x])/a^5 + 225*x^5*ArcCosh[a*x]^2)/1125

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Maple [A]  time = 0.04, size = 168, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{3}{x}^{3} \left ( ax-1 \right ) \left ( ax+1 \right ) }{5}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{5}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax}{5}}-{\frac{2\,{\rm arccosh} \left (ax\right ){a}^{4}{x}^{4}}{25}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{16\,{\rm arccosh} \left (ax\right )}{75}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{8\,{\rm arccosh} \left (ax\right ){a}^{2}{x}^{2}}{75}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ){a}^{3}{x}^{3}}{125}}+{\frac{ \left ( 58\,ax-58 \right ) \left ( ax+1 \right ) ax}{1125}}+{\frac{298\,ax}{1125}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/5*arccosh(a*x)^2*a^3*x^3*(a*x-1)*(a*x+1)+1/5*arccosh(a*x)^2*(a*x-1)*(a*x+1)*a*x+1/5*arccosh(a*x)^2*a*
x-2/25*arccosh(a*x)*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)-16/75*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-8/75*ar
ccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2+2/125*(a*x-1)*(a*x+1)*a^3*x^3+58/1125*(a*x-1)*(a*x+1)*a*x+298/1
125*a*x)

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Maxima [A]  time = 1.28225, size = 134, normalized size = 1.02 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arcosh}\left (a x\right )^{2} - \frac{2}{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 \operatorname{arcosh}\left (a x\right ) + \frac{2 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^2,x, algorithm="maxima")

[Out]

1/5*x^5*arccosh(a*x)^2 - 2/75*(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*arccosh(a*x) + 2/1125*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)/a^4

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Fricas [A]  time = 2.53045, size = 234, normalized size = 1.77 \begin{align*} \frac{225 \, a^{5} x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 18 \, a^{5} x^{5} + 40 \, a^{3} x^{3} - 30 \,{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + 240 \, a x}{1125 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^2,x, algorithm="fricas")

[Out]

1/1125*(225*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^2 + 18*a^5*x^5 + 40*a^3*x^3 - 30*(3*a^4*x^4 + 4*a^2*x^2 + 8)*
sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)) + 240*a*x)/a^5

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Sympy [A]  time = 4.91408, size = 122, normalized size = 0.92 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acosh}^{2}{\left (a x \right )}}{5} + \frac{2 x^{5}}{125} - \frac{2 x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{25 a} + \frac{8 x^{3}}{225 a^{2}} - \frac{8 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{75 a^{3}} + \frac{16 x}{75 a^{4}} - \frac{16 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{75 a^{5}} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} x^{5}}{20} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*acosh(a*x)**2/5 + 2*x**5/125 - 2*x**4*sqrt(a**2*x**2 - 1)*acosh(a*x)/(25*a) + 8*x**3/(225*a**2
) - 8*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/(75*a**3) + 16*x/(75*a**4) - 16*sqrt(a**2*x**2 - 1)*acosh(a*x)/(75*a
**5), Ne(a, 0)), (-pi**2*x**5/20, True))

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Giac [A]  time = 1.33187, size = 153, normalized size = 1.16 \begin{align*} \frac{1}{5} \, x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + \frac{2}{1125} \, a{\left (\frac{9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x}{a^{5}} - \frac{15 \,{\left (3 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^2,x, algorithm="giac")

[Out]

1/5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^2 + 2/1125*a*((9*a^4*x^5 + 20*a^2*x^3 + 120*x)/a^5 - 15*(3*(a^2*x^2 - 1)^
(5/2) + 10*(a^2*x^2 - 1)^(3/2) + 15*sqrt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a^6)