3.24 \(\int x^2 \cosh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=155 \[ -\frac{40 \sqrt{a x-1} \sqrt{a x+1}}{27 a^3}+\frac{4 x \cosh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{3 a^3}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{27 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3+\frac{2}{9} x^3 \cosh ^{-1}(a x)-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{3 a} \]

[Out]

(-40*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (4*x*ArcCosh[a*x])
/(3*a^2) + (2*x^3*ArcCosh[a*x])/9 - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a^3) - (x^2*Sqrt[-1 + a
*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a) + (x^3*ArcCosh[a*x]^3)/3

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Rubi [A]  time = 0.4742, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5662, 5759, 5718, 5654, 74, 100, 12} \[ -\frac{40 \sqrt{a x-1} \sqrt{a x+1}}{27 a^3}+\frac{4 x \cosh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{3 a^3}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{27 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3+\frac{2}{9} x^3 \cosh ^{-1}(a x)-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCosh[a*x]^3,x]

[Out]

(-40*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (4*x*ArcCosh[a*x])
/(3*a^2) + (2*x^3*ArcCosh[a*x])/9 - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a^3) - (x^2*Sqrt[-1 + a
*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a) + (x^3*ArcCosh[a*x]^3)/3

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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

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 100

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 12

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

Rubi steps

\begin{align*} \int x^2 \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{3} x^3 \cosh ^{-1}(a x)^3-a \int \frac{x^3 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3+\frac{2}{3} \int x^2 \cosh ^{-1}(a x) \, dx-\frac{2 \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a}\\ &=\frac{2}{9} x^3 \cosh ^{-1}(a x)-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3+\frac{4 \int \cosh ^{-1}(a x) \, dx}{3 a^2}-\frac{1}{9} (2 a) \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{4 x \cosh ^{-1}(a x)}{3 a^2}+\frac{2}{9} x^3 \cosh ^{-1}(a x)-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3-\frac{2 \int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}-\frac{4 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a}\\ &=-\frac{4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a^3}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{4 x \cosh ^{-1}(a x)}{3 a^2}+\frac{2}{9} x^3 \cosh ^{-1}(a x)-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3-\frac{4 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}\\ &=-\frac{40 \sqrt{-1+a x} \sqrt{1+a x}}{27 a^3}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{4 x \cosh ^{-1}(a x)}{3 a^2}+\frac{2}{9} x^3 \cosh ^{-1}(a x)-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0925802, size = 103, normalized size = 0.66 \[ \frac{-2 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+20\right )+9 a^3 x^3 \cosh ^{-1}(a x)^3-9 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^2+6 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)}{27 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCosh[a*x]^3,x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(20 + a^2*x^2) + 6*a*x*(6 + a^2*x^2)*ArcCosh[a*x] - 9*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^2 + 9*a^3*x^3*ArcCosh[a*x]^3)/(27*a^3)

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Maple [A]  time = 0.037, size = 150, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3} \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{3}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax}{3}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{2}{x}^{2}}{3}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{3}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{2\,{\rm arccosh} \left (ax\right ) \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{9}}+{\frac{14\,ax{\rm arccosh} \left (ax\right )}{9}}-{\frac{2\,{a}^{2}{x}^{2}}{27}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{40}{27}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccosh(a*x)^3,x)

[Out]

1/a^3*(1/3*arccosh(a*x)^3*(a*x-1)*(a*x+1)*a*x+1/3*arccosh(a*x)^3*a*x-1/3*arccosh(a*x)^2*a^2*x^2*(a*x-1)^(1/2)*
(a*x+1)^(1/2)-2/3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2/9*arccosh(a*x)*(a*x-1)*(a*x+1)*a*x+14/9*a*x*arc
cosh(a*x)-2/27*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-40/27*(a*x-1)^(1/2)*(a*x+1)^(1/2))

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Maxima [A]  time = 1.252, size = 157, normalized size = 1.01 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcosh}\left (a x\right )^{3} - \frac{1}{3} \, a{\left (\frac{\sqrt{a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{a^{2} x^{2} - 1}}{a^{4}}\right )} \operatorname{arcosh}\left (a x\right )^{2} - \frac{2}{27} \, a{\left (\frac{\sqrt{a^{2} x^{2} - 1} x^{2} + \frac{20 \, \sqrt{a^{2} x^{2} - 1}}{a^{2}}}{a^{2}} - \frac{3 \,{\left (a^{2} x^{3} + 6 \, x\right )} \operatorname{arcosh}\left (a x\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccosh(a*x)^3,x, algorithm="maxima")

[Out]

1/3*x^3*arccosh(a*x)^3 - 1/3*a*(sqrt(a^2*x^2 - 1)*x^2/a^2 + 2*sqrt(a^2*x^2 - 1)/a^4)*arccosh(a*x)^2 - 2/27*a*(
(sqrt(a^2*x^2 - 1)*x^2 + 20*sqrt(a^2*x^2 - 1)/a^2)/a^2 - 3*(a^2*x^3 + 6*x)*arccosh(a*x)/a^3)

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Fricas [A]  time = 2.50902, size = 281, normalized size = 1.81 \begin{align*} \frac{9 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 9 \,{\left (a^{2} x^{2} + 2\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 6 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (a^{2} x^{2} + 20\right )} \sqrt{a^{2} x^{2} - 1}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccosh(a*x)^3,x, algorithm="fricas")

[Out]

1/27*(9*a^3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^3 - 9*(a^2*x^2 + 2)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)
)^2 + 6*(a^3*x^3 + 6*a*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a^2*x^2 + 20)*sqrt(a^2*x^2 - 1))/a^3

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Sympy [A]  time = 2.59883, size = 138, normalized size = 0.89 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acosh}^{3}{\left (a x \right )}}{3} + \frac{2 x^{3} \operatorname{acosh}{\left (a x \right )}}{9} - \frac{x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{3 a} - \frac{2 x^{2} \sqrt{a^{2} x^{2} - 1}}{27 a} + \frac{4 x \operatorname{acosh}{\left (a x \right )}}{3 a^{2}} - \frac{2 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{3 a^{3}} - \frac{40 \sqrt{a^{2} x^{2} - 1}}{27 a^{3}} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} x^{3}}{24} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acosh(a*x)**3,x)

[Out]

Piecewise((x**3*acosh(a*x)**3/3 + 2*x**3*acosh(a*x)/9 - x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(3*a) - 2*x**2*
sqrt(a**2*x**2 - 1)/(27*a) + 4*x*acosh(a*x)/(3*a**2) - 2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(3*a**3) - 40*sqrt(
a**2*x**2 - 1)/(27*a**3), Ne(a, 0)), (-I*pi**3*x**3/24, True))

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Giac [A]  time = 1.58303, size = 190, normalized size = 1.23 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - \frac{1}{27} \, a{\left (\frac{9 \,{\left ({\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{a^{4}} - \frac{2 \,{\left (3 \,{\left (a^{2} x^{3} + 6 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 21 \, \sqrt{a^{2} x^{2} - 1}}{a}\right )}}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccosh(a*x)^3,x, algorithm="giac")

[Out]

1/3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^3 - 1/27*a*(9*((a^2*x^2 - 1)^(3/2) + 3*sqrt(a^2*x^2 - 1))*log(a*x + sqrt(
a^2*x^2 - 1))^2/a^4 - 2*(3*(a^2*x^3 + 6*x)*log(a*x + sqrt(a^2*x^2 - 1)) - ((a^2*x^2 - 1)^(3/2) + 21*sqrt(a^2*x
^2 - 1))/a)/a^3)