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

Optimal. Leaf size=274 \[ \frac{1088 x^3}{16875 a^2}+\frac{16 x^3 \cosh ^{-1}(a x)^2}{75 a^2}-\frac{16 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{1088 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{5625 a^3}+\frac{16576 x}{5625 a^4}+\frac{32 x \cosh ^{-1}(a x)^2}{25 a^4}-\frac{32 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{75 a^5}-\frac{16576 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{5625 a^5}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4+\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{4 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{25 a}-\frac{24 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{625 a}+\frac{24 x^5}{3125} \]

[Out]

(16576*x)/(5625*a^4) + (1088*x^3)/(16875*a^2) + (24*x^5)/3125 - (16576*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*
x])/(5625*a^5) - (1088*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(5625*a^3) - (24*x^4*Sqrt[-1 + a*x]*Sqrt
[1 + a*x]*ArcCosh[a*x])/(625*a) + (32*x*ArcCosh[a*x]^2)/(25*a^4) + (16*x^3*ArcCosh[a*x]^2)/(75*a^2) + (12*x^5*
ArcCosh[a*x]^2)/125 - (32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(75*a^5) - (16*x^2*Sqrt[-1 + a*x]*Sqrt[
1 + a*x]*ArcCosh[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(25*a) + (x^5*ArcCosh[
a*x]^4)/5

________________________________________________________________________________________

Rubi [A]  time = 1.62758, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 19, 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} \[ \frac{1088 x^3}{16875 a^2}+\frac{16 x^3 \cosh ^{-1}(a x)^2}{75 a^2}-\frac{16 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{1088 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{5625 a^3}+\frac{16576 x}{5625 a^4}+\frac{32 x \cosh ^{-1}(a x)^2}{25 a^4}-\frac{32 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{75 a^5}-\frac{16576 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{5625 a^5}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4+\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{4 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{25 a}-\frac{24 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{625 a}+\frac{24 x^5}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16576*x)/(5625*a^4) + (1088*x^3)/(16875*a^2) + (24*x^5)/3125 - (16576*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*
x])/(5625*a^5) - (1088*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(5625*a^3) - (24*x^4*Sqrt[-1 + a*x]*Sqrt
[1 + a*x]*ArcCosh[a*x])/(625*a) + (32*x*ArcCosh[a*x]^2)/(25*a^4) + (16*x^3*ArcCosh[a*x]^2)/(75*a^2) + (12*x^5*
ArcCosh[a*x]^2)/125 - (32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(75*a^5) - (16*x^2*Sqrt[-1 + a*x]*Sqrt[
1 + a*x]*ArcCosh[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(25*a) + (x^5*ArcCosh[
a*x]^4)/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 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^4 \cosh ^{-1}(a x)^4 \, dx &=\frac{1}{5} x^5 \cosh ^{-1}(a x)^4-\frac{1}{5} (4 a) \int \frac{x^5 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{4 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4+\frac{12}{25} \int x^4 \cosh ^{-1}(a x)^2 \, dx-\frac{16 \int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a}\\ &=\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{16 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4-\frac{32 \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{75 a^3}+\frac{16 \int x^2 \cosh ^{-1}(a x)^2 \, dx}{25 a^2}-\frac{1}{125} (24 a) \int \frac{x^5 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{24 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{625 a}+\frac{16 x^3 \cosh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{32 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^5}-\frac{16 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4+\frac{24 \int x^4 \, dx}{625}+\frac{32 \int \cosh ^{-1}(a x)^2 \, dx}{25 a^4}-\frac{96 \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{625 a}-\frac{32 \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{75 a}\\ &=\frac{24 x^5}{3125}-\frac{1088 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{5625 a^3}-\frac{24 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{625 a}+\frac{32 x \cosh ^{-1}(a x)^2}{25 a^4}+\frac{16 x^3 \cosh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{32 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^5}-\frac{16 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4-\frac{64 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{625 a^3}-\frac{64 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{225 a^3}-\frac{64 \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a^3}+\frac{32 \int x^2 \, dx}{625 a^2}+\frac{32 \int x^2 \, dx}{225 a^2}\\ &=\frac{1088 x^3}{16875 a^2}+\frac{24 x^5}{3125}-\frac{16576 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{5625 a^5}-\frac{1088 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{5625 a^3}-\frac{24 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{625 a}+\frac{32 x \cosh ^{-1}(a x)^2}{25 a^4}+\frac{16 x^3 \cosh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{32 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^5}-\frac{16 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4+\frac{64 \int 1 \, dx}{625 a^4}+\frac{64 \int 1 \, dx}{225 a^4}+\frac{64 \int 1 \, dx}{25 a^4}\\ &=\frac{16576 x}{5625 a^4}+\frac{1088 x^3}{16875 a^2}+\frac{24 x^5}{3125}-\frac{16576 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{5625 a^5}-\frac{1088 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{5625 a^3}-\frac{24 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{625 a}+\frac{32 x \cosh ^{-1}(a x)^2}{25 a^4}+\frac{16 x^3 \cosh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \cosh ^{-1}(a x)^2-\frac{32 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^5}-\frac{16 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.127004, size = 158, normalized size = 0.58 \[ \frac{8 a x \left (81 a^4 x^4+680 a^2 x^2+31080\right )+16875 a^5 x^5 \cosh ^{-1}(a x)^4+900 a x \left (9 a^4 x^4+20 a^2 x^2+120\right ) \cosh ^{-1}(a x)^2-4500 \sqrt{a x-1} \sqrt{a x+1} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cosh ^{-1}(a x)^3-120 \sqrt{a x-1} \sqrt{a x+1} \left (27 a^4 x^4+136 a^2 x^2+2072\right ) \cosh ^{-1}(a x)}{84375 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*x*(31080 + 680*a^2*x^2 + 81*a^4*x^4) - 120*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2072 + 136*a^2*x^2 + 27*a^4*x^4)
*ArcCosh[a*x] + 900*a*x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x]^2 - 4500*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 +
4*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^3 + 16875*a^5*x^5*ArcCosh[a*x]^4)/(84375*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 300, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{3}{x}^{3} \left ({\rm arccosh} \left (ax\right ) \right ) ^{4} \left ( ax-1 \right ) \left ( ax+1 \right ) }{5}}+{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}ax}{5}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}ax}{5}}-{\frac{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{4}{x}^{4}}{25}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{75}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{2}{x}^{2}}{75}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{3}{x}^{3} \left ( ax-1 \right ) \left ( ax+1 \right ) }{125}}+{\frac{116\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{375}}+{\frac{596\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax}{375}}-{\frac{24\,{a}^{4}{\rm arccosh} \left (ax\right ){x}^{4}}{625}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{16576\,{\rm arccosh} \left (ax\right )}{5625}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{1088\,{a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )}{5625}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{ \left ( 24\,ax-24 \right ) \left ( ax+1 \right ){a}^{3}{x}^{3}}{3125}}+{\frac{ \left ( 6088\,ax-6088 \right ) \left ( ax+1 \right ) ax}{84375}}+{\frac{254728\,ax}{84375}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/5*a^3*x^3*arccosh(a*x)^4*(a*x-1)*(a*x+1)+1/5*(a*x-1)*(a*x+1)*arccosh(a*x)^4*a*x+1/5*arccosh(a*x)^4*a*
x-4/25*arccosh(a*x)^3*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)-32/75*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-16/
75*arccosh(a*x)^3*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+12/125*arccosh(a*x)^2*a^3*x^3*(a*x-1)*(a*x+1)+116/375*ar
ccosh(a*x)^2*(a*x-1)*(a*x+1)*a*x+596/375*arccosh(a*x)^2*a*x-24/625*arccosh(a*x)*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^
(1/2)-16576/5625*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1088/5625*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a
^2*x^2+24/3125*(a*x-1)*(a*x+1)*a^3*x^3+6088/84375*(a*x-1)*(a*x+1)*a*x+254728/84375*a*x)

________________________________________________________________________________________

Maxima [A]  time = 1.23144, size = 271, normalized size = 0.99 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arcosh}\left (a x\right )^{4} - \frac{4}{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 )^{3} - \frac{4}{84375} \,{\left (2 \, a{\left (\frac{15 \,{\left (27 \, \sqrt{a^{2} x^{2} - 1} a^{2} x^{4} + 136 \, \sqrt{a^{2} x^{2} - 1} x^{2} + \frac{2072 \, \sqrt{a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname{arcosh}\left (a x\right )}{a^{5}} - \frac{81 \, a^{4} x^{5} + 680 \, a^{2} x^{3} + 31080 \, x}{a^{6}}\right )} - \frac{225 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \operatorname{arcosh}\left (a x\right )^{2}}{a^{5}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccosh(a*x)^4 - 4/75*(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)^3 - 4/84375*(2*a*(15*(27*sqrt(a^2*x^2 - 1)*a^2*x^4 + 136*sqrt(a^2*x^2 - 1)*x^2 + 2072*sqr
t(a^2*x^2 - 1)/a^2)*arccosh(a*x)/a^5 - (81*a^4*x^5 + 680*a^2*x^3 + 31080*x)/a^6) - 225*(9*a^4*x^5 + 20*a^2*x^3
+ 120*x)*arccosh(a*x)^2/a^5)*a

________________________________________________________________________________________

Fricas [A]  time = 2.61499, size = 466, normalized size = 1.7 \begin{align*} \frac{16875 \, a^{5} x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{4} + 648 \, a^{5} x^{5} + 5440 \, a^{3} x^{3} - 4500 \,{\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 )^{3} + 900 \,{\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 120 \,{\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + 248640 \, a x}{84375 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/84375*(16875*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^4 + 648*a^5*x^5 + 5440*a^3*x^3 - 4500*(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))^3 + 900*(9*a^5*x^5 + 20*a^3*x^3 + 120*a*x)*log(a*x + sq
rt(a^2*x^2 - 1))^2 - 120*(27*a^4*x^4 + 136*a^2*x^2 + 2072)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)) + 24
8640*a*x)/a^5

________________________________________________________________________________________

Sympy [A]  time = 14.6522, size = 248, normalized size = 0.91 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acosh}^{4}{\left (a x \right )}}{5} + \frac{12 x^{5} \operatorname{acosh}^{2}{\left (a x \right )}}{125} + \frac{24 x^{5}}{3125} - \frac{4 x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{25 a} - \frac{24 x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{625 a} + \frac{16 x^{3} \operatorname{acosh}^{2}{\left (a x \right )}}{75 a^{2}} + \frac{1088 x^{3}}{16875 a^{2}} - \frac{16 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{75 a^{3}} - \frac{1088 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{5625 a^{3}} + \frac{32 x \operatorname{acosh}^{2}{\left (a x \right )}}{25 a^{4}} + \frac{16576 x}{5625 a^{4}} - \frac{32 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{75 a^{5}} - \frac{16576 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{5625 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{5}}{80} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*acosh(a*x)**4/5 + 12*x**5*acosh(a*x)**2/125 + 24*x**5/3125 - 4*x**4*sqrt(a**2*x**2 - 1)*acosh(
a*x)**3/(25*a) - 24*x**4*sqrt(a**2*x**2 - 1)*acosh(a*x)/(625*a) + 16*x**3*acosh(a*x)**2/(75*a**2) + 1088*x**3/
(16875*a**2) - 16*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**3/(75*a**3) - 1088*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/
(5625*a**3) + 32*x*acosh(a*x)**2/(25*a**4) + 16576*x/(5625*a**4) - 32*sqrt(a**2*x**2 - 1)*acosh(a*x)**3/(75*a*
*5) - 16576*sqrt(a**2*x**2 - 1)*acosh(a*x)/(5625*a**5), Ne(a, 0)), (pi**4*x**5/80, True))

________________________________________________________________________________________

Giac [A]  time = 1.84988, size = 293, normalized size = 1.07 \begin{align*} \frac{1}{5} \, x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{4} - \frac{4}{84375} \, a{\left (\frac{1125 \,{\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 )^{3}}{a^{6}} - \frac{162 \, a^{4} x^{5} + 1360 \, a^{2} x^{3} + 225 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 62160 \, x - \frac{30 \,{\left (27 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 190 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 2235 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^4 - 4/84375*a*(1125*(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))^3/a^6 - (162*a^4*x^5 + 1360*a^2*x^3 + 225*(9*a^4*x^5 + 20*a^2*
x^3 + 120*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 + 62160*x - 30*(27*(a^2*x^2 - 1)^(5/2) + 190*(a^2*x^2 - 1)^(3/2) +
2235*sqrt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a)/a^5)