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

Optimal. Leaf size=276 \[ -\frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}-\frac{x^5 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{9 a}-\frac{x^5 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{54 a}-\frac{5 x^4 \sinh ^{-1}(a x)^2}{48 a^2}+\frac{5 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{36 a^3}+\frac{65 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{864 a^3}+\frac{5 x^2 \sinh ^{-1}(a x)^2}{16 a^4}-\frac{5 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{24 a^5}-\frac{245 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{576 a^5}+\frac{5 \sinh ^{-1}(a x)^4}{96 a^6}+\frac{245 \sinh ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4+\frac{1}{18} x^6 \sinh ^{-1}(a x)^2+\frac{x^6}{324} \]

[Out]

(245*x^2)/(1152*a^4) - (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(576*a^5) + (65*
x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(864*a^3) - (x^5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(54*a) + (245*ArcSinh[a*x
]^2)/(1152*a^6) + (5*x^2*ArcSinh[a*x]^2)/(16*a^4) - (5*x^4*ArcSinh[a*x]^2)/(48*a^2) + (x^6*ArcSinh[a*x]^2)/18
- (5*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(36*a^3) - (x^5*S
qrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(9*a) + (5*ArcSinh[a*x]^4)/(96*a^6) + (x^6*ArcSinh[a*x]^4)/6

________________________________________________________________________________________

Rubi [A]  time = 0.855782, antiderivative size = 276, 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 = {5661, 5758, 5675, 30} \[ -\frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}-\frac{x^5 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{9 a}-\frac{x^5 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{54 a}-\frac{5 x^4 \sinh ^{-1}(a x)^2}{48 a^2}+\frac{5 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{36 a^3}+\frac{65 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{864 a^3}+\frac{5 x^2 \sinh ^{-1}(a x)^2}{16 a^4}-\frac{5 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{24 a^5}-\frac{245 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{576 a^5}+\frac{5 \sinh ^{-1}(a x)^4}{96 a^6}+\frac{245 \sinh ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4+\frac{1}{18} x^6 \sinh ^{-1}(a x)^2+\frac{x^6}{324} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcSinh[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^2*x^2]*ArcSinh[a*x])/(576*a^5) + (65*
x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(864*a^3) - (x^5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(54*a) + (245*ArcSinh[a*x
]^2)/(1152*a^6) + (5*x^2*ArcSinh[a*x]^2)/(16*a^4) - (5*x^4*ArcSinh[a*x]^2)/(48*a^2) + (x^6*ArcSinh[a*x]^2)/18
- (5*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(36*a^3) - (x^5*S
qrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(9*a) + (5*ArcSinh[a*x]^4)/(96*a^6) + (x^6*ArcSinh[a*x]^4)/6

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 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 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 \sinh ^{-1}(a x)^4 \, dx &=\frac{1}{6} x^6 \sinh ^{-1}(a x)^4-\frac{1}{3} (2 a) \int \frac{x^6 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4+\frac{1}{3} \int x^5 \sinh ^{-1}(a x)^2 \, dx+\frac{5 \int \frac{x^4 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{9 a}\\ &=\frac{1}{18} x^6 \sinh ^{-1}(a x)^2+\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4-\frac{5 \int \frac{x^2 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{12 a^3}-\frac{5 \int x^3 \sinh ^{-1}(a x)^2 \, dx}{12 a^2}-\frac{1}{9} a \int \frac{x^6 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{54 a}-\frac{5 x^4 \sinh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \sinh ^{-1}(a x)^2-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4+\frac{\int x^5 \, dx}{54}+\frac{5 \int \frac{\sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{24 a^5}+\frac{5 \int x \sinh ^{-1}(a x)^2 \, dx}{8 a^4}+\frac{5 \int \frac{x^4 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{54 a}+\frac{5 \int \frac{x^4 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{24 a}\\ &=\frac{x^6}{324}+\frac{65 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{54 a}+\frac{5 x^2 \sinh ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \sinh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \sinh ^{-1}(a x)^2-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac{5 \sinh ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4-\frac{5 \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{72 a^3}-\frac{5 \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{32 a^3}-\frac{5 \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, 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^2 x^2} \sinh ^{-1}(a x)}{576 a^5}+\frac{65 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{54 a}+\frac{5 x^2 \sinh ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \sinh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \sinh ^{-1}(a x)^2-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac{5 \sinh ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4+\frac{5 \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{144 a^5}+\frac{5 \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{64 a^5}+\frac{5 \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, 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^2 x^2} \sinh ^{-1}(a x)}{576 a^5}+\frac{65 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{54 a}+\frac{245 \sinh ^{-1}(a x)^2}{1152 a^6}+\frac{5 x^2 \sinh ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \sinh ^{-1}(a x)^2}{48 a^2}+\frac{1}{18} x^6 \sinh ^{-1}(a x)^2-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{9 a}+\frac{5 \sinh ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \sinh ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.0902957, size = 165, normalized size = 0.6 \[ \frac{a^2 x^2 \left (32 a^4 x^4-195 a^2 x^2+2205\right )+108 \left (16 a^6 x^6+5\right ) \sinh ^{-1}(a x)^4-144 a x \sqrt{a^2 x^2+1} \left (8 a^4 x^4-10 a^2 x^2+15\right ) \sinh ^{-1}(a x)^3+9 \left (64 a^6 x^6-120 a^4 x^4+360 a^2 x^2+245\right ) \sinh ^{-1}(a x)^2-6 a x \sqrt{a^2 x^2+1} \left (32 a^4 x^4-130 a^2 x^2+735\right ) \sinh ^{-1}(a x)}{10368 a^6} \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

Maple [A]  time = 0.16, size = 319, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{6}} \left ({\frac{{a}^{4}{x}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4} \left ({a}^{2}{x}^{2}+1 \right ) }{6}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}{a}^{2}{x}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }{6}}+{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4} \left ({a}^{2}{x}^{2}+1 \right ) }{6}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}}{9} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax}{4} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{11\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax}{24}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{11\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}}{96}}+{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}{18}}-{\frac{31\,{a}^{2}{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}+1 \right ) }{144}}+{\frac{17\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}+1 \right ) }{36}}-{\frac{{\it Arcsinh} \left ( ax \right ) ax}{54} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{97\,{\it Arcsinh} \left ( ax \right ) ax}{864} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{x}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}{324}}-{\frac{259\,{a}^{2}{x}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }{10368}}+{\frac{19\,{a}^{2}{x}^{2}}{81}}+{\frac{19}{81}}-{\frac{299\,{\it Arcsinh} \left ( ax \right ) ax}{576}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{299\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{1152}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^6*(1/6*a^4*x^4*arcsinh(a*x)^4*(a^2*x^2+1)-1/6*arcsinh(a*x)^4*a^2*x^2*(a^2*x^2+1)+1/6*arcsinh(a*x)^4*(a^2*x
^2+1)-1/9*arcsinh(a*x)^3*a^3*x^3*(a^2*x^2+1)^(3/2)+1/4*arcsinh(a*x)^3*a*x*(a^2*x^2+1)^(3/2)-11/24*arcsinh(a*x)
^3*a*x*(a^2*x^2+1)^(1/2)-11/96*arcsinh(a*x)^4+1/18*arcsinh(a*x)^2*a^2*x^2*(a^2*x^2+1)^2-31/144*a^2*x^2*arcsinh
(a*x)^2*(a^2*x^2+1)+17/36*arcsinh(a*x)^2*(a^2*x^2+1)-1/54*arcsinh(a*x)*a*x*(a^2*x^2+1)^(5/2)+97/864*arcsinh(a*
x)*a*x*(a^2*x^2+1)^(3/2)+1/324*a^2*x^2*(a^2*x^2+1)^2-259/10368*a^2*x^2*(a^2*x^2+1)+19/81*a^2*x^2+19/81-299/576
*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x-299/1152*arcsinh(a*x)^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

1/6*x^6*log(a*x + sqrt(a^2*x^2 + 1))^4 - integrate(2/3*(a^3*x^8 + sqrt(a^2*x^2 + 1)*a^2*x^7 + a*x^6)*log(a*x +
sqrt(a^2*x^2 + 1))^3/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)

________________________________________________________________________________________

Fricas [A]  time = 2.14026, size = 497, normalized size = 1.8 \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*arcsinh(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

________________________________________________________________________________________

Sympy [A]  time = 21.5359, size = 269, normalized size = 0.97 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{asinh}^{4}{\left (a x \right )}}{6} + \frac{x^{6} \operatorname{asinh}^{2}{\left (a x \right )}}{18} + \frac{x^{6}}{324} - \frac{x^{5} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{9 a} - \frac{x^{5} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{54 a} - \frac{5 x^{4} \operatorname{asinh}^{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{asinh}^{3}{\left (a x \right )}}{36 a^{3}} + \frac{65 x^{3} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{864 a^{3}} + \frac{5 x^{2} \operatorname{asinh}^{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{asinh}^{3}{\left (a x \right )}}{24 a^{5}} - \frac{245 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{576 a^{5}} + \frac{5 \operatorname{asinh}^{4}{\left (a x \right )}}{96 a^{6}} + \frac{245 \operatorname{asinh}^{2}{\left (a x \right )}}{1152 a^{6}} & \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**5*asinh(a*x)**4,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^5*arcsinh(a*x)^4, x)