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

Optimal. Leaf size=195 \[ -\frac{6 \left (a^2 x^2+1\right )^{5/2}}{625 a^5}+\frac{76 \left (a^2 x^2+1\right )^{3/2}}{1125 a^5}-\frac{298 \sqrt{a^2 x^2+1}}{375 a^5}-\frac{3 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{25 a}-\frac{8 x^3 \sinh ^{-1}(a x)}{75 a^2}+\frac{4 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{25 a^5}+\frac{16 x \sinh ^{-1}(a x)}{25 a^4}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3+\frac{6}{125} x^5 \sinh ^{-1}(a x) \]

[Out]

(-298*Sqrt[1 + a^2*x^2])/(375*a^5) + (76*(1 + a^2*x^2)^(3/2))/(1125*a^5) - (6*(1 + a^2*x^2)^(5/2))/(625*a^5) +
(16*x*ArcSinh[a*x])/(25*a^4) - (8*x^3*ArcSinh[a*x])/(75*a^2) + (6*x^5*ArcSinh[a*x])/125 - (8*Sqrt[1 + a^2*x^2
]*ArcSinh[a*x]^2)/(25*a^5) + (4*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[1 + a^2*x^2]*ArcS
inh[a*x]^2)/(25*a) + (x^5*ArcSinh[a*x]^3)/5

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Rubi [A]  time = 0.36596, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5661, 5758, 5717, 5653, 261, 266, 43} \[ -\frac{6 \left (a^2 x^2+1\right )^{5/2}}{625 a^5}+\frac{76 \left (a^2 x^2+1\right )^{3/2}}{1125 a^5}-\frac{298 \sqrt{a^2 x^2+1}}{375 a^5}-\frac{3 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{25 a}-\frac{8 x^3 \sinh ^{-1}(a x)}{75 a^2}+\frac{4 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{25 a^5}+\frac{16 x \sinh ^{-1}(a x)}{25 a^4}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3+\frac{6}{125} x^5 \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-298*Sqrt[1 + a^2*x^2])/(375*a^5) + (76*(1 + a^2*x^2)^(3/2))/(1125*a^5) - (6*(1 + a^2*x^2)^(5/2))/(625*a^5) +
(16*x*ArcSinh[a*x])/(25*a^4) - (8*x^3*ArcSinh[a*x])/(75*a^2) + (6*x^5*ArcSinh[a*x])/125 - (8*Sqrt[1 + a^2*x^2
]*ArcSinh[a*x]^2)/(25*a^5) + (4*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[1 + a^2*x^2]*ArcS
inh[a*x]^2)/(25*a) + (x^5*ArcSinh[a*x]^3)/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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^4 \sinh ^{-1}(a x)^3 \, dx &=\frac{1}{5} x^5 \sinh ^{-1}(a x)^3-\frac{1}{5} (3 a) \int \frac{x^5 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3+\frac{6}{25} \int x^4 \sinh ^{-1}(a x) \, dx+\frac{12 \int \frac{x^3 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{25 a}\\ &=\frac{6}{125} x^5 \sinh ^{-1}(a x)+\frac{4 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3-\frac{8 \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{25 a^3}-\frac{8 \int x^2 \sinh ^{-1}(a x) \, dx}{25 a^2}-\frac{1}{125} (6 a) \int \frac{x^5}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{8 x^3 \sinh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \sinh ^{-1}(a x)-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^5}+\frac{4 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3+\frac{16 \int \sinh ^{-1}(a x) \, dx}{25 a^4}+\frac{8 \int \frac{x^3}{\sqrt{1+a^2 x^2}} \, dx}{75 a}-\frac{1}{125} (3 a) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac{16 x \sinh ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \sinh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \sinh ^{-1}(a x)-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^5}+\frac{4 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3-\frac{16 \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx}{25 a^3}+\frac{4 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )}{75 a}-\frac{1}{125} (3 a) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{1+a^2 x}}-\frac{2 \sqrt{1+a^2 x}}{a^4}+\frac{\left (1+a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{86 \sqrt{1+a^2 x^2}}{125 a^5}+\frac{4 \left (1+a^2 x^2\right )^{3/2}}{125 a^5}-\frac{6 \left (1+a^2 x^2\right )^{5/2}}{625 a^5}+\frac{16 x \sinh ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \sinh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \sinh ^{-1}(a x)-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^5}+\frac{4 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3+\frac{4 \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{1+a^2 x}}+\frac{\sqrt{1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{75 a}\\ &=-\frac{298 \sqrt{1+a^2 x^2}}{375 a^5}+\frac{76 \left (1+a^2 x^2\right )^{3/2}}{1125 a^5}-\frac{6 \left (1+a^2 x^2\right )^{5/2}}{625 a^5}+\frac{16 x \sinh ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \sinh ^{-1}(a x)}{75 a^2}+\frac{6}{125} x^5 \sinh ^{-1}(a x)-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^5}+\frac{4 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0689866, size = 120, normalized size = 0.62 \[ \frac{-2 \sqrt{a^2 x^2+1} \left (27 a^4 x^4-136 a^2 x^2+2072\right )+1125 a^5 x^5 \sinh ^{-1}(a x)^3+30 a x \left (9 a^4 x^4-20 a^2 x^2+120\right ) \sinh ^{-1}(a x)-225 \sqrt{a^2 x^2+1} \left (3 a^4 x^4-4 a^2 x^2+8\right ) \sinh ^{-1}(a x)^2}{5625 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + a^2*x^2]*(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)*ArcSinh[a*x] -
225*Sqrt[1 + a^2*x^2]*(8 - 4*a^2*x^2 + 3*a^4*x^4)*ArcSinh[a*x]^2 + 1125*a^5*x^5*ArcSinh[a*x]^3)/(5625*a^5)

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Maple [A]  time = 0.04, size = 222, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{3}{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3} \left ({a}^{2}{x}^{2}+1 \right ) }{5}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax \left ({a}^{2}{x}^{2}+1 \right ) }{5}}+{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax}{5}}-{\frac{3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}}{25} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{7\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}}{25}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{25}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{6\,{\it Arcsinh} \left ( ax \right ) ax \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}{125}}+{\frac{298\,ax{\it Arcsinh} \left ( ax \right ) }{375}}-{\frac{76\,{\it Arcsinh} \left ( ax \right ) ax \left ({a}^{2}{x}^{2}+1 \right ) }{375}}-{\frac{6\,{a}^{2}{x}^{2}}{625} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{326\,{a}^{2}{x}^{2}}{5625}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{4144}{5625}\sqrt{{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/5*a^3*x^3*arcsinh(a*x)^3*(a^2*x^2+1)-1/5*arcsinh(a*x)^3*a*x*(a^2*x^2+1)+1/5*arcsinh(a*x)^3*a*x-3/25*a
rcsinh(a*x)^2*a^2*x^2*(a^2*x^2+1)^(3/2)+7/25*a^2*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-8/25*arcsinh(a*x)^2*(a^2
*x^2+1)^(1/2)+6/125*arcsinh(a*x)*a*x*(a^2*x^2+1)^2+298/375*a*x*arcsinh(a*x)-76/375*arcsinh(a*x)*a*x*(a^2*x^2+1
)-6/625*a^2*x^2*(a^2*x^2+1)^(3/2)+326/5625*a^2*x^2*(a^2*x^2+1)^(1/2)-4144/5625*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.0839, size = 223, normalized size = 1.14 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arsinh}\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{arsinh}\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{arsinh}\left (a x\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.16287, size = 359, normalized size = 1.84 \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*arcsinh(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 = 7.48059, size = 196, normalized size = 1.01 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{asinh}^{3}{\left (a x \right )}}{5} + \frac{6 x^{5} \operatorname{asinh}{\left (a x \right )}}{125} - \frac{3 x^{4} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{25 a} - \frac{6 x^{4} \sqrt{a^{2} x^{2} + 1}}{625 a} - \frac{8 x^{3} \operatorname{asinh}{\left (a x \right )}}{75 a^{2}} + \frac{4 x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{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{asinh}{\left (a x \right )}}{25 a^{4}} - \frac{8 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{25 a^{5}} - \frac{4144 \sqrt{a^{2} x^{2} + 1}}{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)**3,x)

[Out]

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

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Giac [A]  time = 1.59894, size = 243, normalized size = 1.25 \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*arcsinh(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)