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

Optimal. Leaf size=182 \[ \frac{160 x}{27 a^2}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{9 a^3}+\frac{8 x \cosh ^{-1}(a x)^2}{3 a^2}-\frac{160 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4-\frac{4 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{9 a}+\frac{4}{9} x^3 \cosh ^{-1}(a x)^2-\frac{8 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a}+\frac{8 x^3}{81} \]

[Out]

(160*x)/(27*a^2) + (8*x^3)/81 - (160*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a^3) - (8*x^2*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a) + (8*x*ArcCosh[a*x]^2)/(3*a^2) + (4*x^3*ArcCosh[a*x]^2)/9 - (8*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a^3) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) + (x^3
*ArcCosh[a*x]^4)/3

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Rubi [A]  time = 0.881492, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5662, 5759, 5718, 5654, 8, 30} \[ \frac{160 x}{27 a^2}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{9 a^3}+\frac{8 x \cosh ^{-1}(a x)^2}{3 a^2}-\frac{160 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4-\frac{4 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{9 a}+\frac{4}{9} x^3 \cosh ^{-1}(a x)^2-\frac{8 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a}+\frac{8 x^3}{81} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(160*x)/(27*a^2) + (8*x^3)/81 - (160*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a^3) - (8*x^2*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a) + (8*x*ArcCosh[a*x]^2)/(3*a^2) + (4*x^3*ArcCosh[a*x]^2)/9 - (8*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a^3) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) + (x^3
*ArcCosh[a*x]^4)/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 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)^4 \, dx &=\frac{1}{3} x^3 \cosh ^{-1}(a x)^4-\frac{1}{3} (4 a) \int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4+\frac{4}{3} \int x^2 \cosh ^{-1}(a x)^2 \, dx-\frac{8 \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a}\\ &=\frac{4}{9} x^3 \cosh ^{-1}(a x)^2-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4+\frac{8 \int \cosh ^{-1}(a x)^2 \, dx}{3 a^2}-\frac{1}{9} (8 a) \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac{4}{9} x^3 \cosh ^{-1}(a x)^2-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4+\frac{8 \int x^2 \, dx}{27}-\frac{16 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}-\frac{16 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a}\\ &=\frac{8 x^3}{81}-\frac{160 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac{4}{9} x^3 \cosh ^{-1}(a x)^2-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4+\frac{16 \int 1 \, dx}{27 a^2}+\frac{16 \int 1 \, dx}{3 a^2}\\ &=\frac{160 x}{27 a^2}+\frac{8 x^3}{81}-\frac{160 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac{4}{9} x^3 \cosh ^{-1}(a x)^2-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.112116, size = 122, normalized size = 0.67 \[ \frac{8 a x \left (a^2 x^2+60\right )+27 a^3 x^3 \cosh ^{-1}(a x)^4-36 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^3+36 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)^2-24 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+20\right ) \cosh ^{-1}(a x)}{81 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*x*(60 + a^2*x^2) - 24*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(20 + a^2*x^2)*ArcCosh[a*x] + 36*a*x*(6 + a^2*x^2)*Arc
Cosh[a*x]^2 - 36*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^3 + 27*a^3*x^3*ArcCosh[a*x]^4)/(81*a^
3)

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Maple [A]  time = 0.04, size = 180, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}ax}{3}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}ax}{3}}-{\frac{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{2}{x}^{2}}{9}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{9}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{9}}+{\frac{28\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax}{9}}-{\frac{8\,{a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )}{27}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{160\,{\rm arccosh} \left (ax\right )}{27}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{ \left ( 8\,ax-8 \right ) \left ( ax+1 \right ) ax}{81}}+{\frac{488\,ax}{81}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(1/3*(a*x-1)*(a*x+1)*arccosh(a*x)^4*a*x+1/3*arccosh(a*x)^4*a*x-4/9*arccosh(a*x)^3*a^2*x^2*(a*x-1)^(1/2)*
(a*x+1)^(1/2)-8/9*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+4/9*arccosh(a*x)^2*(a*x-1)*(a*x+1)*a*x+28/9*arcco
sh(a*x)^2*a*x-8/27*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2-160/27*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^
(1/2)+8/81*(a*x-1)*(a*x+1)*a*x+488/81*a*x)

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Maxima [A]  time = 1.2046, size = 193, normalized size = 1.06 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcosh}\left (a x\right )^{4} - \frac{4}{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 )^{3} - \frac{4}{81} \,{\left (2 \, a{\left (\frac{3 \,{\left (\sqrt{a^{2} x^{2} - 1} x^{2} + \frac{20 \, \sqrt{a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname{arcosh}\left (a x\right )}{a^{3}} - \frac{a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac{9 \,{\left (a^{2} x^{3} + 6 \, x\right )} \operatorname{arcosh}\left (a x\right )^{2}}{a^{3}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccosh(a*x)^4 - 4/9*a*(sqrt(a^2*x^2 - 1)*x^2/a^2 + 2*sqrt(a^2*x^2 - 1)/a^4)*arccosh(a*x)^3 - 4/81*(2*
a*(3*(sqrt(a^2*x^2 - 1)*x^2 + 20*sqrt(a^2*x^2 - 1)/a^2)*arccosh(a*x)/a^3 - (a^2*x^3 + 60*x)/a^4) - 9*(a^2*x^3
+ 6*x)*arccosh(a*x)^2/a^3)*a

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Fricas [A]  time = 2.35788, size = 358, normalized size = 1.97 \begin{align*} \frac{27 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{4} + 8 \, a^{3} x^{3} - 36 \,{\left (a^{2} x^{2} + 2\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + 36 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 24 \,{\left (a^{2} x^{2} + 20\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + 480 \, a x}{81 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/81*(27*a^3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^4 + 8*a^3*x^3 - 36*(a^2*x^2 + 2)*sqrt(a^2*x^2 - 1)*log(a*x + sqr
t(a^2*x^2 - 1))^3 + 36*(a^3*x^3 + 6*a*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 24*(a^2*x^2 + 20)*sqrt(a^2*x^2 - 1)*
log(a*x + sqrt(a^2*x^2 - 1)) + 480*a*x)/a^3

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Sympy [A]  time = 5.03241, size = 165, normalized size = 0.91 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acosh}^{4}{\left (a x \right )}}{3} + \frac{4 x^{3} \operatorname{acosh}^{2}{\left (a x \right )}}{9} + \frac{8 x^{3}}{81} - \frac{4 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{9 a} - \frac{8 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{27 a} + \frac{8 x \operatorname{acosh}^{2}{\left (a x \right )}}{3 a^{2}} + \frac{160 x}{27 a^{2}} - \frac{8 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{9 a^{3}} - \frac{160 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{27 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{3}}{48} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*acosh(a*x)**4/3 + 4*x**3*acosh(a*x)**2/9 + 8*x**3/81 - 4*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**
3/(9*a) - 8*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/(27*a) + 8*x*acosh(a*x)**2/(3*a**2) + 160*x/(27*a**2) - 8*sqrt
(a**2*x**2 - 1)*acosh(a*x)**3/(9*a**3) - 160*sqrt(a**2*x**2 - 1)*acosh(a*x)/(27*a**3), Ne(a, 0)), (pi**4*x**3/
48, True))

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Giac [A]  time = 1.77945, size = 230, normalized size = 1.26 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{4} - \frac{4}{81} \, 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 )^{3}}{a^{4}} - \frac{2 \, a^{2} x^{3} + 9 \,{\left (a^{2} x^{3} + 6 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 120 \, x - \frac{6 \,{\left ({\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 21 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^4 - 4/81*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))^3/a^4 - (2*a^2*x^3 + 9*(a^2*x^3 + 6*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 + 120*x - 6*((a^2*x^2 - 1)
^(3/2) + 21*sqrt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a)/a^3)