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

Optimal. Leaf size=107 \[ -\frac{\cosh ^{-1}(a x)^3}{4 a^2}-\frac{3 \cosh ^{-1}(a x)}{8 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^3+\frac{3}{4} x^2 \cosh ^{-1}(a x)-\frac{3 x \sqrt{a x-1} \sqrt{a x+1}}{8 a}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.380996, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5662, 5759, 5676, 90, 52} \[ -\frac{\cosh ^{-1}(a x)^3}{4 a^2}-\frac{3 \cosh ^{-1}(a x)}{8 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^3+\frac{3}{4} x^2 \cosh ^{-1}(a x)-\frac{3 x \sqrt{a x-1} \sqrt{a x+1}}{8 a}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5676

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.0816682, size = 113, normalized size = 1.06 \[ \frac{\left (4 a^2 x^2-2\right ) \cosh ^{-1}(a x)^3+6 a^2 x^2 \cosh ^{-1}(a x)-3 \left (a x \sqrt{a x-1} \sqrt{a x+1}+\log \left (a x+\sqrt{a x-1} \sqrt{a x+1}\right )\right )-6 a x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{8 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a^2*x^2*ArcCosh[a*x] - 6*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2 + (-2 + 4*a^2*x^2)*ArcCosh[a*x]^3
- 3*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + Log[a*x + Sqrt[-1 + a*x]*Sqrt[1 + a*x]]))/(8*a^2)

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Maple [A]  time = 0.03, size = 88, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{2}{x}^{2}}{2}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax}{4}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{4}}+{\frac{3\,{a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )}{4}}-{\frac{3\,ax}{8}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\,{\rm arccosh} \left (ax\right )}{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3} - \int \frac{3 \,{\left (a^{3} x^{4} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x^{3} - a x^{2}\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2}}{2 \,{\left (a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3 - integrate(3/2*(a^3*x^4 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^
3 - a*x^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x
), x)

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Fricas [A]  time = 2.45023, size = 261, normalized size = 2.44 \begin{align*} -\frac{6 \, \sqrt{a^{2} x^{2} - 1} a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 2 \,{\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + 3 \, \sqrt{a^{2} x^{2} - 1} a x - 3 \,{\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(6*sqrt(a^2*x^2 - 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1))^2 - 2*(2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^
3 + 3*sqrt(a^2*x^2 - 1)*a*x - 3*(2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a^2

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Sympy [A]  time = 1.25823, size = 102, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acosh}^{3}{\left (a x \right )}}{2} + \frac{3 x^{2} \operatorname{acosh}{\left (a x \right )}}{4} - \frac{3 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{4 a} - \frac{3 x \sqrt{a^{2} x^{2} - 1}}{8 a} - \frac{\operatorname{acosh}^{3}{\left (a x \right )}}{4 a^{2}} - \frac{3 \operatorname{acosh}{\left (a x \right )}}{8 a^{2}} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} x^{2}}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2*acosh(a*x)**3/2 + 3*x**2*acosh(a*x)/4 - 3*x*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(4*a) - 3*x*sqrt
(a**2*x**2 - 1)/(8*a) - acosh(a*x)**3/(4*a**2) - 3*acosh(a*x)/(8*a**2), Ne(a, 0)), (-I*pi**3*x**2/16, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcosh}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*arccosh(a*x)^3, x)