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

Optimal. Leaf size=90 \[ \frac{4 x}{9 a^2}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^2-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a}+\frac{2 x^3}{27} \]

[Out]

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

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Rubi [A]  time = 0.311197, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{4 x}{9 a^2}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^2-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a}+\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(9*a^2) + (2*x^3)/27 - (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^3) - (2*x^2*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*ArcCosh[a*x])/(9*a) + (x^3*ArcCosh[a*x]^2)/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 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^2 \cosh ^{-1}(a x)^2 \, dx &=\frac{1}{3} x^3 \cosh ^{-1}(a x)^2-\frac{1}{3} (2 a) \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^2+\frac{2 \int x^2 \, dx}{9}-\frac{4 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a}\\ &=\frac{2 x^3}{27}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^2+\frac{4 \int 1 \, dx}{9 a^2}\\ &=\frac{4 x}{9 a^2}+\frac{2 x^3}{27}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.0950779, size = 64, normalized size = 0.71 \[ \frac{1}{27} \left (2 x \left (\frac{6}{a^2}+x^2\right )-\frac{6 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)}{a^3}+9 x^3 \cosh ^{-1}(a x)^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(6/a^2 + x^2) - (6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x])/a^3 + 9*x^3*ArcCosh[a*x]^2)/2
7

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Maple [A]  time = 0.031, size = 100, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{3}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax}{3}}-{\frac{2\,{\rm arccosh} \left (ax\right ){a}^{2}{x}^{2}}{9}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{4\,{\rm arccosh} \left (ax\right )}{9}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) ax}{27}}+{\frac{14\,ax}{27}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(1/3*arccosh(a*x)^2*(a*x-1)*(a*x+1)*a*x+1/3*arccosh(a*x)^2*a*x-2/9*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1
/2)*a^2*x^2-4/9*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2/27*(a*x-1)*(a*x+1)*a*x+14/27*a*x)

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Maxima [A]  time = 1.1807, size = 95, normalized size = 1.06 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcosh}\left (a x\right )^{2} - \frac{2}{9} \, 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 ) + \frac{2 \,{\left (a^{2} x^{3} + 6 \, x\right )}}{27 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.25661, size = 188, normalized size = 2.09 \begin{align*} \frac{9 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 2 \, a^{3} x^{3} - 6 \,{\left (a^{2} x^{2} + 2\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + 12 \, a x}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*a^3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^2 + 2*a^3*x^3 - 6*(a^2*x^2 + 2)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(
a^2*x^2 - 1)) + 12*a*x)/a^3

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Sympy [A]  time = 1.2952, size = 85, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acosh}^{2}{\left (a x \right )}}{3} + \frac{2 x^{3}}{27} - \frac{2 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{9 a} + \frac{4 x}{9 a^{2}} - \frac{4 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{9 a^{3}} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} x^{3}}{12} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*acosh(a*x)**2/3 + 2*x**3/27 - 2*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/(9*a) + 4*x/(9*a**2) - 4*s
qrt(a**2*x**2 - 1)*acosh(a*x)/(9*a**3), Ne(a, 0)), (-pi**2*x**3/12, True))

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Giac [A]  time = 1.32305, size = 120, normalized size = 1.33 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + \frac{2}{27} \, a{\left (\frac{a^{2} x^{3} + 6 \, x}{a^{3}} - \frac{3 \,{\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 )}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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