3.22 \(\int x^4 \cosh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=231 \[ -\frac{272 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5625 a^3}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}-\frac{4 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{4144 \sqrt{a x-1} \sqrt{a x+1}}{5625 a^5}+\frac{16 x \cosh ^{-1}(a x)}{25 a^4}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{6 x^4 \sqrt{a x-1} \sqrt{a x+1}}{625 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{3 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{25 a} \]

[Out]

(-4144*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5625*a^5) - (272*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5625*a^3) - (6*x^4*S
qrt[-1 + a*x]*Sqrt[1 + a*x])/(625*a) + (16*x*ArcCosh[a*x])/(25*a^4) + (8*x^3*ArcCosh[a*x])/(75*a^2) + (6*x^5*A
rcCosh[a*x])/125 - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a^5) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a
*x]*ArcCosh[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a) + (x^5*ArcCosh[a*x]^
3)/5

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Rubi [A]  time = 0.771434, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{272 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5625 a^3}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}-\frac{4 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{4144 \sqrt{a x-1} \sqrt{a x+1}}{5625 a^5}+\frac{16 x \cosh ^{-1}(a x)}{25 a^4}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{6 x^4 \sqrt{a x-1} \sqrt{a x+1}}{625 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{3 x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{25 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCosh[a*x]^3,x]

[Out]

(-4144*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5625*a^5) - (272*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5625*a^3) - (6*x^4*S
qrt[-1 + a*x]*Sqrt[1 + a*x])/(625*a) + (16*x*ArcCosh[a*x])/(25*a^4) + (8*x^3*ArcCosh[a*x])/(75*a^2) + (6*x^5*A
rcCosh[a*x])/125 - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a^5) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a
*x]*ArcCosh[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a) + (x^5*ArcCosh[a*x]^
3)/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 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^4 \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{5} x^5 \cosh ^{-1}(a x)^3-\frac{1}{5} (3 a) \int \frac{x^5 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3+\frac{6}{25} \int x^4 \cosh ^{-1}(a x) \, dx-\frac{12 \int \frac{x^3 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a}\\ &=\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3-\frac{8 \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a^3}+\frac{8 \int x^2 \cosh ^{-1}(a x) \, dx}{25 a^2}-\frac{1}{125} (6 a) \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{6 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{625 a}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3+\frac{16 \int \cosh ^{-1}(a x) \, dx}{25 a^4}-\frac{6 \int \frac{4 x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{625 a}-\frac{8 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{75 a}\\ &=-\frac{8 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{225 a^3}-\frac{6 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{625 a}+\frac{16 x \cosh ^{-1}(a x)}{25 a^4}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3-\frac{8 \int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{225 a^3}-\frac{16 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a^3}-\frac{24 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{625 a}\\ &=-\frac{16 \sqrt{-1+a x} \sqrt{1+a x}}{25 a^5}-\frac{272 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5625 a^3}-\frac{6 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{625 a}+\frac{16 x \cosh ^{-1}(a x)}{25 a^4}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3-\frac{8 \int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{625 a^3}-\frac{16 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{225 a^3}\\ &=-\frac{32 \sqrt{-1+a x} \sqrt{1+a x}}{45 a^5}-\frac{272 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5625 a^3}-\frac{6 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{625 a}+\frac{16 x \cosh ^{-1}(a x)}{25 a^4}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3-\frac{16 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{625 a^3}\\ &=-\frac{4144 \sqrt{-1+a x} \sqrt{1+a x}}{5625 a^5}-\frac{272 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5625 a^3}-\frac{6 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{625 a}+\frac{16 x \cosh ^{-1}(a x)}{25 a^4}+\frac{8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \cosh ^{-1}(a x)-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.116565, size = 130, normalized size = 0.56 \[ \frac{-2 \sqrt{a x-1} \sqrt{a x+1} \left (27 a^4 x^4+136 a^2 x^2+2072\right )+1125 a^5 x^5 \cosh ^{-1}(a x)^3+30 a x \left (9 a^4 x^4+20 a^2 x^2+120\right ) \cosh ^{-1}(a x)-225 \sqrt{a x-1} \sqrt{a x+1} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cosh ^{-1}(a x)^2}{5625 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCosh[a*x]^3,x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2072 + 136*a^2*x^2 + 27*a^4*x^4) + 30*a*x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*Arc
Cosh[a*x] - 225*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^2 + 1125*a^5*x^5*ArcCosh
[a*x]^3)/(5625*a^5)

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Maple [A]  time = 0.048, size = 246, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3} \left ( ax-1 \right ) \left ( ax+1 \right ) }{5}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3} \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{5}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax}{5}}-{\frac{3\,{a}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{x}^{4}}{25}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{25}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{2}{x}^{2}}{25}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{6\,{\rm arccosh} \left (ax\right ) \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{3}{x}^{3}}{125}}+{\frac{58\,{\rm arccosh} \left (ax\right ) \left ( ax-1 \right ) \left ( ax+1 \right ) ax}{375}}+{\frac{298\,ax{\rm arccosh} \left (ax\right )}{375}}-{\frac{6\,{x}^{4}{a}^{4}}{625}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{4144}{5625}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{272\,{a}^{2}{x}^{2}}{5625}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccosh(a*x)^3,x)

[Out]

1/a^5*(1/5*arccosh(a*x)^3*a^3*x^3*(a*x-1)*(a*x+1)+1/5*arccosh(a*x)^3*(a*x-1)*(a*x+1)*a*x+1/5*arccosh(a*x)^3*a*
x-3/25*arccosh(a*x)^2*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)-8/25*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-4/25
*arccosh(a*x)^2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6/125*arccosh(a*x)*(a*x-1)*(a*x+1)*a^3*x^3+58/375*arccosh(
a*x)*(a*x-1)*(a*x+1)*a*x+298/375*a*x*arccosh(a*x)-6/625*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)-4144/5625*(a*x-1)^
(1/2)*(a*x+1)^(1/2)-272/5625*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2))

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Maxima [A]  time = 1.19804, size = 223, normalized size = 0.97 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arcosh}\left (a x\right )^{3} - \frac{1}{25} \,{\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 )^{2} - \frac{2}{5625} \, a{\left (\frac{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}}}{a^{4}} - \frac{15 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \operatorname{arcosh}\left (a x\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^3,x, algorithm="maxima")

[Out]

1/5*x^5*arccosh(a*x)^3 - 1/25*(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)^2 - 2/5625*a*((27*sqrt(a^2*x^2 - 1)*a^2*x^4 + 136*sqrt(a^2*x^2 - 1)*x^2 + 2072*sqrt(a^2*x
^2 - 1)/a^2)/a^4 - 15*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*arccosh(a*x)/a^5)

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Fricas [A]  time = 2.41865, size = 359, normalized size = 1.55 \begin{align*} \frac{1125 \, a^{5} x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 225 \,{\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 )^{2} + 30 \,{\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 \,{\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \sqrt{a^{2} x^{2} - 1}}{5625 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^3,x, algorithm="fricas")

[Out]

1/5625*(1125*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^3 - 225*(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))^2 + 30*(9*a^5*x^5 + 20*a^3*x^3 + 120*a*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(27*a^4*x^4
+ 136*a^2*x^2 + 2072)*sqrt(a^2*x^2 - 1))/a^5

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Sympy [A]  time = 8.78706, size = 206, normalized size = 0.89 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acosh}^{3}{\left (a x \right )}}{5} + \frac{6 x^{5} \operatorname{acosh}{\left (a x \right )}}{125} - \frac{3 x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{25 a} - \frac{6 x^{4} \sqrt{a^{2} x^{2} - 1}}{625 a} + \frac{8 x^{3} \operatorname{acosh}{\left (a x \right )}}{75 a^{2}} - \frac{4 x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{25 a^{3}} - \frac{272 x^{2} \sqrt{a^{2} x^{2} - 1}}{5625 a^{3}} + \frac{16 x \operatorname{acosh}{\left (a x \right )}}{25 a^{4}} - \frac{8 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{25 a^{5}} - \frac{4144 \sqrt{a^{2} x^{2} - 1}}{5625 a^{5}} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} x^{5}}{40} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acosh(a*x)**3,x)

[Out]

Piecewise((x**5*acosh(a*x)**3/5 + 6*x**5*acosh(a*x)/125 - 3*x**4*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(25*a) - 6*
x**4*sqrt(a**2*x**2 - 1)/(625*a) + 8*x**3*acosh(a*x)/(75*a**2) - 4*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(25*
a**3) - 272*x**2*sqrt(a**2*x**2 - 1)/(5625*a**3) + 16*x*acosh(a*x)/(25*a**4) - 8*sqrt(a**2*x**2 - 1)*acosh(a*x
)**2/(25*a**5) - 4144*sqrt(a**2*x**2 - 1)/(5625*a**5), Ne(a, 0)), (-I*pi**3*x**5/40, True))

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Giac [A]  time = 1.60727, size = 243, normalized size = 1.05 \begin{align*} \frac{1}{5} \, x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - \frac{1}{5625} \, a{\left (\frac{225 \,{\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 )^{2}}{a^{6}} - \frac{2 \,{\left (15 \,{\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 ) - \frac{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}}{a}\right )}}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^3,x, algorithm="giac")

[Out]

1/5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^3 - 1/5625*a*(225*(3*(a^2*x^2 - 1)^(5/2) + 10*(a^2*x^2 - 1)^(3/2) + 15*sq
rt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))^2/a^6 - 2*(15*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*log(a*x + sqrt(a^
2*x^2 - 1)) - (27*(a^2*x^2 - 1)^(5/2) + 190*(a^2*x^2 - 1)^(3/2) + 2235*sqrt(a^2*x^2 - 1))/a)/a^5)