3.32 \(\int x^5 \cosh ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=306 \[ \frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}+\frac{5 x^4 \cosh ^{-1}(a x)^2}{48 a^2}-\frac{5 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{65 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{864 a^3}+\frac{5 x^2 \cosh ^{-1}(a x)^2}{16 a^4}-\frac{5 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{24 a^5}-\frac{245 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{576 a^5}-\frac{5 \cosh ^{-1}(a x)^4}{96 a^6}-\frac{245 \cosh ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4+\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{x^5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{9 a}-\frac{x^5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{54 a}+\frac{x^6}{324} \]

[Out]

(245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(576*
a^5) - (65*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(864*a^3) - (x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCos
h[a*x])/(54*a) - (245*ArcCosh[a*x]^2)/(1152*a^6) + (5*x^2*ArcCosh[a*x]^2)/(16*a^4) + (5*x^4*ArcCosh[a*x]^2)/(4
8*a^2) + (x^6*ArcCosh[a*x]^2)/18 - (5*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(24*a^5) - (5*x^3*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(36*a^3) - (x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) - (5*
ArcCosh[a*x]^4)/(96*a^6) + (x^6*ArcCosh[a*x]^4)/6

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Rubi [A]  time = 2.19252, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5662, 5759, 5676, 30} \[ \frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}+\frac{5 x^4 \cosh ^{-1}(a x)^2}{48 a^2}-\frac{5 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{65 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{864 a^3}+\frac{5 x^2 \cosh ^{-1}(a x)^2}{16 a^4}-\frac{5 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{24 a^5}-\frac{245 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{576 a^5}-\frac{5 \cosh ^{-1}(a x)^4}{96 a^6}-\frac{245 \cosh ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4+\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{x^5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{9 a}-\frac{x^5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{54 a}+\frac{x^6}{324} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCosh[a*x]^4,x]

[Out]

(245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(576*
a^5) - (65*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(864*a^3) - (x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCos
h[a*x])/(54*a) - (245*ArcCosh[a*x]^2)/(1152*a^6) + (5*x^2*ArcCosh[a*x]^2)/(16*a^4) + (5*x^4*ArcCosh[a*x]^2)/(4
8*a^2) + (x^6*ArcCosh[a*x]^2)/18 - (5*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(24*a^5) - (5*x^3*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(36*a^3) - (x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) - (5*
ArcCosh[a*x]^4)/(96*a^6) + (x^6*ArcCosh[a*x]^4)/6

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 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^5 \cosh ^{-1}(a x)^4 \, dx &=\frac{1}{6} x^6 \cosh ^{-1}(a x)^4-\frac{1}{3} (2 a) \int \frac{x^6 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4+\frac{1}{3} \int x^5 \cosh ^{-1}(a x)^2 \, dx-\frac{5 \int \frac{x^4 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a}\\ &=\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{5 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4-\frac{5 \int \frac{x^2 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{12 a^3}+\frac{5 \int x^3 \cosh ^{-1}(a x)^2 \, dx}{12 a^2}-\frac{1}{9} a \int \frac{x^6 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{54 a}+\frac{5 x^4 \cosh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4+\frac{\int x^5 \, dx}{54}-\frac{5 \int \frac{\cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{24 a^5}+\frac{5 \int x \cosh ^{-1}(a x)^2 \, dx}{8 a^4}-\frac{5 \int \frac{x^4 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{54 a}-\frac{5 \int \frac{x^4 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{24 a}\\ &=\frac{x^6}{324}-\frac{65 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{54 a}+\frac{5 x^2 \cosh ^{-1}(a x)^2}{16 a^4}+\frac{5 x^4 \cosh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}-\frac{5 \cosh ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4-\frac{5 \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{72 a^3}-\frac{5 \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 a^3}-\frac{5 \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a^3}+\frac{5 \int x^3 \, dx}{216 a^2}+\frac{5 \int x^3 \, dx}{96 a^2}\\ &=\frac{65 x^4}{3456 a^2}+\frac{x^6}{324}-\frac{245 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{576 a^5}-\frac{65 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{54 a}+\frac{5 x^2 \cosh ^{-1}(a x)^2}{16 a^4}+\frac{5 x^4 \cosh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}-\frac{5 \cosh ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4-\frac{5 \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{144 a^5}-\frac{5 \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{64 a^5}-\frac{5 \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 a^5}+\frac{5 \int x \, dx}{144 a^4}+\frac{5 \int x \, dx}{64 a^4}+\frac{5 \int x \, dx}{16 a^4}\\ &=\frac{245 x^2}{1152 a^4}+\frac{65 x^4}{3456 a^2}+\frac{x^6}{324}-\frac{245 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{576 a^5}-\frac{65 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{54 a}-\frac{245 \cosh ^{-1}(a x)^2}{1152 a^6}+\frac{5 x^2 \cosh ^{-1}(a x)^2}{16 a^4}+\frac{5 x^4 \cosh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \cosh ^{-1}(a x)^2-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{9 a}-\frac{5 \cosh ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \cosh ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.152674, size = 175, normalized size = 0.57 \[ \frac{a^2 x^2 \left (32 a^4 x^4+195 a^2 x^2+2205\right )+108 \left (16 a^6 x^6-5\right ) \cosh ^{-1}(a x)^4-144 a x \sqrt{a x-1} \sqrt{a x+1} \left (8 a^4 x^4+10 a^2 x^2+15\right ) \cosh ^{-1}(a x)^3+9 \left (64 a^6 x^6+120 a^4 x^4+360 a^2 x^2-245\right ) \cosh ^{-1}(a x)^2-6 a x \sqrt{a x-1} \sqrt{a x+1} \left (32 a^4 x^4+130 a^2 x^2+735\right ) \cosh ^{-1}(a x)}{10368 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCosh[a*x]^4,x]

[Out]

(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) - 6*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(735 + 130*a^2*x^2 + 32*a^4*x^
4)*ArcCosh[a*x] + 9*(-245 + 360*a^2*x^2 + 120*a^4*x^4 + 64*a^6*x^6)*ArcCosh[a*x]^2 - 144*a*x*Sqrt[-1 + a*x]*Sq
rt[1 + a*x]*(15 + 10*a^2*x^2 + 8*a^4*x^4)*ArcCosh[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcCosh[a*x]^4)/(10368*a^6)

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Maple [A]  time = 0.052, size = 344, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{6}} \left ({\frac{{a}^{4}{x}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{4} \left ( ax-1 \right ) \left ( ax+1 \right ) }{6}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{6}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}{a}^{2}{x}^{2}}{6}}-{\frac{{x}^{5}{a}^{5} \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{9}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{5\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3}}{36}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{5\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax}{24}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{5\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{96}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{4}{x}^{4}}{18}}+{\frac{23\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{144}}+{\frac{17\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{2}{x}^{2}}{36}}-{\frac{{a}^{5}{x}^{5}{\rm arccosh} \left (ax\right )}{54}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{65\,{a}^{3}{x}^{3}{\rm arccosh} \left (ax\right )}{864}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{245\,ax{\rm arccosh} \left (ax\right )}{576}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{245\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{1152}}+{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{4}{x}^{4}}{324}}+{\frac{ \left ( 227\,ax-227 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{10368}}+{\frac{19\,{a}^{2}{x}^{2}}{81}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccosh(a*x)^4,x)

[Out]

1/a^6*(1/6*a^4*x^4*arccosh(a*x)^4*(a*x-1)*(a*x+1)+1/6*arccosh(a*x)^4*(a*x-1)*(a*x+1)*a^2*x^2+1/6*arccosh(a*x)^
4*a^2*x^2-1/9*a^5*x^5*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-5/36*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/
2)*a^3*x^3-5/24*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-5/96*arccosh(a*x)^4+1/18*arccosh(a*x)^2*(a*x-1)
*(a*x+1)*a^4*x^4+23/144*arccosh(a*x)^2*(a*x-1)*(a*x+1)*a^2*x^2+17/36*arccosh(a*x)^2*a^2*x^2-1/54*arccosh(a*x)*
(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^5*x^5-65/864*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^3*x^3-245/576*arccosh(a*
x)*a*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)-245/1152*arccosh(a*x)^2+1/324*(a*x-1)*(a*x+1)*a^4*x^4+227/10368*(a*x-1)*(a*
x+1)*a^2*x^2+19/81*a^2*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, x^{6} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{4} - \int \frac{2 \,{\left (a^{3} x^{8} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x^{7} - a x^{6}\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3}}{3 \,{\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^5*arccosh(a*x)^4,x, algorithm="maxima")

[Out]

1/6*x^6*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4 - integrate(2/3*(a^3*x^8 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^
7 - a*x^6)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(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 = 2.39829, size = 497, normalized size = 1.62 \begin{align*} \frac{32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \,{\left (16 \, a^{6} x^{6} - 5\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{4} - 144 \,{\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + 2205 \, a^{2} x^{2} + 9 \,{\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{10368 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccosh(a*x)^4,x, algorithm="fricas")

[Out]

1/10368*(32*a^6*x^6 + 195*a^4*x^4 + 108*(16*a^6*x^6 - 5)*log(a*x + sqrt(a^2*x^2 - 1))^4 - 144*(8*a^5*x^5 + 10*
a^3*x^3 + 15*a*x)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^3 + 2205*a^2*x^2 + 9*(64*a^6*x^6 + 120*a^4*x^
4 + 360*a^2*x^2 - 245)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 6*(32*a^5*x^5 + 130*a^3*x^3 + 735*a*x)*sqrt(a^2*x^2 -
1)*log(a*x + sqrt(a^2*x^2 - 1)))/a^6

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Sympy [A]  time = 25.0856, size = 275, normalized size = 0.9 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acosh}^{4}{\left (a x \right )}}{6} + \frac{x^{6} \operatorname{acosh}^{2}{\left (a x \right )}}{18} + \frac{x^{6}}{324} - \frac{x^{5} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{9 a} - \frac{x^{5} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{54 a} + \frac{5 x^{4} \operatorname{acosh}^{2}{\left (a x \right )}}{48 a^{2}} + \frac{65 x^{4}}{3456 a^{2}} - \frac{5 x^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{36 a^{3}} - \frac{65 x^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{864 a^{3}} + \frac{5 x^{2} \operatorname{acosh}^{2}{\left (a x \right )}}{16 a^{4}} + \frac{245 x^{2}}{1152 a^{4}} - \frac{5 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{24 a^{5}} - \frac{245 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{576 a^{5}} - \frac{5 \operatorname{acosh}^{4}{\left (a x \right )}}{96 a^{6}} - \frac{245 \operatorname{acosh}^{2}{\left (a x \right )}}{1152 a^{6}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{6}}{96} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acosh(a*x)**4,x)

[Out]

Piecewise((x**6*acosh(a*x)**4/6 + x**6*acosh(a*x)**2/18 + x**6/324 - x**5*sqrt(a**2*x**2 - 1)*acosh(a*x)**3/(9
*a) - x**5*sqrt(a**2*x**2 - 1)*acosh(a*x)/(54*a) + 5*x**4*acosh(a*x)**2/(48*a**2) + 65*x**4/(3456*a**2) - 5*x*
*3*sqrt(a**2*x**2 - 1)*acosh(a*x)**3/(36*a**3) - 65*x**3*sqrt(a**2*x**2 - 1)*acosh(a*x)/(864*a**3) + 5*x**2*ac
osh(a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) - 5*x*sqrt(a**2*x**2 - 1)*acosh(a*x)**3/(24*a**5) - 245*x*sqrt(a*
*2*x**2 - 1)*acosh(a*x)/(576*a**5) - 5*acosh(a*x)**4/(96*a**6) - 245*acosh(a*x)**2/(1152*a**6), Ne(a, 0)), (pi
**4*x**6/96, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{arcosh}\left (a x\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^5*arccosh(a*x)^4, x)