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

Optimal. Leaf size=244 \[ -\frac{1088 x^3}{16875 a^2}-\frac{4 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{25 a}-\frac{24 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{625 a}-\frac{16 x^3 \sinh ^{-1}(a x)^2}{75 a^2}+\frac{16 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{75 a^3}+\frac{1088 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{5625 a^3}-\frac{32 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{75 a^5}-\frac{16576 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{5625 a^5}+\frac{16576 x}{5625 a^4}+\frac{32 x \sinh ^{-1}(a x)^2}{25 a^4}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4+\frac{12}{125} x^5 \sinh ^{-1}(a x)^2+\frac{24 x^5}{3125} \]

[Out]

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

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Rubi [A]  time = 0.657166, antiderivative size = 244, 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 = {5661, 5758, 5717, 5653, 8, 30} \[ -\frac{1088 x^3}{16875 a^2}-\frac{4 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{25 a}-\frac{24 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{625 a}-\frac{16 x^3 \sinh ^{-1}(a x)^2}{75 a^2}+\frac{16 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{75 a^3}+\frac{1088 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{5625 a^3}-\frac{32 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{75 a^5}-\frac{16576 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{5625 a^5}+\frac{16576 x}{5625 a^4}+\frac{32 x \sinh ^{-1}(a x)^2}{25 a^4}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4+\frac{12}{125} x^5 \sinh ^{-1}(a x)^2+\frac{24 x^5}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], 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 \sinh ^{-1}(a x)^4 \, dx &=\frac{1}{5} x^5 \sinh ^{-1}(a x)^4-\frac{1}{5} (4 a) \int \frac{x^5 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{4 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4+\frac{12}{25} \int x^4 \sinh ^{-1}(a x)^2 \, dx+\frac{16 \int \frac{x^3 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{25 a}\\ &=\frac{12}{125} x^5 \sinh ^{-1}(a x)^2+\frac{16 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4-\frac{32 \int \frac{x \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{75 a^3}-\frac{16 \int x^2 \sinh ^{-1}(a x)^2 \, dx}{25 a^2}-\frac{1}{125} (24 a) \int \frac{x^5 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{24 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{625 a}-\frac{16 x^3 \sinh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \sinh ^{-1}(a x)^2-\frac{32 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^5}+\frac{16 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4+\frac{24 \int x^4 \, dx}{625}+\frac{32 \int \sinh ^{-1}(a x)^2 \, dx}{25 a^4}+\frac{96 \int \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{625 a}+\frac{32 \int \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{75 a}\\ &=\frac{24 x^5}{3125}+\frac{1088 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{5625 a^3}-\frac{24 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{625 a}+\frac{32 x \sinh ^{-1}(a x)^2}{25 a^4}-\frac{16 x^3 \sinh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \sinh ^{-1}(a x)^2-\frac{32 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^5}+\frac{16 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4-\frac{64 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{625 a^3}-\frac{64 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{225 a^3}-\frac{64 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, 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^2 x^2} \sinh ^{-1}(a x)}{5625 a^5}+\frac{1088 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{5625 a^3}-\frac{24 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{625 a}+\frac{32 x \sinh ^{-1}(a x)^2}{25 a^4}-\frac{16 x^3 \sinh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \sinh ^{-1}(a x)^2-\frac{32 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^5}+\frac{16 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \sinh ^{-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^2 x^2} \sinh ^{-1}(a x)}{5625 a^5}+\frac{1088 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{5625 a^3}-\frac{24 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{625 a}+\frac{32 x \sinh ^{-1}(a x)^2}{25 a^4}-\frac{16 x^3 \sinh ^{-1}(a x)^2}{75 a^2}+\frac{12}{125} x^5 \sinh ^{-1}(a x)^2-\frac{32 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^5}+\frac{16 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{75 a^3}-\frac{4 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.081171, size = 148, normalized size = 0.61 \[ \frac{8 a x \left (81 a^4 x^4-680 a^2 x^2+31080\right )+16875 a^5 x^5 \sinh ^{-1}(a x)^4+900 a x \left (9 a^4 x^4-20 a^2 x^2+120\right ) \sinh ^{-1}(a x)^2-4500 \sqrt{a^2 x^2+1} \left (3 a^4 x^4-4 a^2 x^2+8\right ) \sinh ^{-1}(a x)^3-120 \sqrt{a^2 x^2+1} \left (27 a^4 x^4-136 a^2 x^2+2072\right ) \sinh ^{-1}(a x)}{84375 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 272, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{3}{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4} \left ({a}^{2}{x}^{2}+1 \right ) }{5}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}ax \left ({a}^{2}{x}^{2}+1 \right ) }{5}}+{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}ax}{5}}-{\frac{4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{2}{x}^{2}}{25} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{28\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{2}{x}^{2}}{75}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{32\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{75}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{12\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}{125}}+{\frac{596\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax}{375}}-{\frac{152\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax \left ({a}^{2}{x}^{2}+1 \right ) }{375}}-{\frac{24\,{\it Arcsinh} \left ( ax \right ){a}^{2}{x}^{2}}{625} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{1304\,{\it Arcsinh} \left ( ax \right ){a}^{2}{x}^{2}}{5625}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{16576\,{\it Arcsinh} \left ( ax \right ) }{5625}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{24\,ax \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}{3125}}+{\frac{254728\,ax}{84375}}-{\frac{6736\,ax \left ({a}^{2}{x}^{2}+1 \right ) }{84375}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/5*a^3*x^3*arcsinh(a*x)^4*(a^2*x^2+1)-1/5*arcsinh(a*x)^4*a*x*(a^2*x^2+1)+1/5*arcsinh(a*x)^4*a*x-4/25*a
rcsinh(a*x)^3*a^2*x^2*(a^2*x^2+1)^(3/2)+28/75*a^2*x^2*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)-32/75*arcsinh(a*x)^3*(a
^2*x^2+1)^(1/2)+12/125*arcsinh(a*x)^2*a*x*(a^2*x^2+1)^2+596/375*arcsinh(a*x)^2*a*x-152/375*arcsinh(a*x)^2*a*x*
(a^2*x^2+1)-24/625*arcsinh(a*x)*a^2*x^2*(a^2*x^2+1)^(3/2)+1304/5625*arcsinh(a*x)*a^2*x^2*(a^2*x^2+1)^(1/2)-165
76/5625*arcsinh(a*x)*(a^2*x^2+1)^(1/2)+24/3125*a*x*(a^2*x^2+1)^2+254728/84375*a*x-6736/84375*a*x*(a^2*x^2+1))

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Maxima [A]  time = 1.23611, size = 271, normalized size = 1.11 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arsinh}\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{arsinh}\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{arsinh}\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{arsinh}\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*arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

1/5*x^5*arcsinh(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*arcsinh(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)*arcsinh(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)*arcsinh(a*x)^2/a^5)*a

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Fricas [A]  time = 2.14352, size = 466, normalized size = 1.91 \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*arcsinh(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

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Sympy [A]  time = 12.4645, size = 241, normalized size = 0.99 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{asinh}^{4}{\left (a x \right )}}{5} + \frac{12 x^{5} \operatorname{asinh}^{2}{\left (a x \right )}}{125} + \frac{24 x^{5}}{3125} - \frac{4 x^{4} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{25 a} - \frac{24 x^{4} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{625 a} - \frac{16 x^{3} \operatorname{asinh}^{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{asinh}^{3}{\left (a x \right )}}{75 a^{3}} + \frac{1088 x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{5625 a^{3}} + \frac{32 x \operatorname{asinh}^{2}{\left (a x \right )}}{25 a^{4}} + \frac{16576 x}{5625 a^{4}} - \frac{32 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{75 a^{5}} - \frac{16576 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{5625 a^{5}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.81725, size = 293, normalized size = 1.2 \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*arcsinh(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)