3.24 \(\int x^2 \sinh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=132 \[ -\frac{2 \left (a^2 x^2+1\right )^{3/2}}{27 a^3}+\frac{14 \sqrt{a^2 x^2+1}}{9 a^3}-\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a}+\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^3}-\frac{4 x \sinh ^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3+\frac{2}{9} x^3 \sinh ^{-1}(a x) \]

[Out]

(14*Sqrt[1 + a^2*x^2])/(9*a^3) - (2*(1 + a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcSinh[a*x])/(3*a^2) + (2*x^3*ArcSin
h[a*x])/9 + (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a) + (x^3
*ArcSinh[a*x]^3)/3

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Rubi [A]  time = 0.22147, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{2 \left (a^2 x^2+1\right )^{3/2}}{27 a^3}+\frac{14 \sqrt{a^2 x^2+1}}{9 a^3}-\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a}+\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^3}-\frac{4 x \sinh ^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3+\frac{2}{9} x^3 \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSinh[a*x]^3,x]

[Out]

(14*Sqrt[1 + a^2*x^2])/(9*a^3) - (2*(1 + a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcSinh[a*x])/(3*a^2) + (2*x^3*ArcSin
h[a*x])/9 + (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a) + (x^3
*ArcSinh[a*x]^3)/3

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^2 \sinh ^{-1}(a x)^3 \, dx &=\frac{1}{3} x^3 \sinh ^{-1}(a x)^3-a \int \frac{x^3 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3+\frac{2}{3} \int x^2 \sinh ^{-1}(a x) \, dx+\frac{2 \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{3 a}\\ &=\frac{2}{9} x^3 \sinh ^{-1}(a x)+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3-\frac{4 \int \sinh ^{-1}(a x) \, dx}{3 a^2}-\frac{1}{9} (2 a) \int \frac{x^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{4 x \sinh ^{-1}(a x)}{3 a^2}+\frac{2}{9} x^3 \sinh ^{-1}(a x)+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3+\frac{4 \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx}{3 a}-\frac{1}{9} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac{4 \sqrt{1+a^2 x^2}}{3 a^3}-\frac{4 x \sinh ^{-1}(a x)}{3 a^2}+\frac{2}{9} x^3 \sinh ^{-1}(a x)+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3-\frac{1}{9} a \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 )\\ &=\frac{14 \sqrt{1+a^2 x^2}}{9 a^3}-\frac{2 \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-\frac{4 x \sinh ^{-1}(a x)}{3 a^2}+\frac{2}{9} x^3 \sinh ^{-1}(a x)+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0542938, size = 93, normalized size = 0.7 \[ \frac{-2 \left (a^2 x^2-20\right ) \sqrt{a^2 x^2+1}+9 a^3 x^3 \sinh ^{-1}(a x)^3-9 \left (a^2 x^2-2\right ) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2+6 a x \left (a^2 x^2-6\right ) \sinh ^{-1}(a x)}{27 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSinh[a*x]^3,x]

[Out]

(-2*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2] + 6*a*x*(-6 + a^2*x^2)*ArcSinh[a*x] - 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]
*ArcSinh[a*x]^2 + 9*a^3*x^3*ArcSinh[a*x]^3)/(27*a^3)

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Maple [A]  time = 0.028, size = 136, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax}{3}}-{\frac{{a}^{2}{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{3}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{2\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{3}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{2\,{\it Arcsinh} \left ( ax \right ) ax \left ({a}^{2}{x}^{2}+1 \right ) }{9}}-{\frac{14\,ax{\it Arcsinh} \left ( ax \right ) }{9}}-{\frac{2\,{a}^{2}{x}^{2}}{27}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{40}{27}\sqrt{{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsinh(a*x)^3,x)

[Out]

1/a^3*(1/3*arcsinh(a*x)^3*a*x*(a^2*x^2+1)-1/3*arcsinh(a*x)^3*a*x-1/3*a^2*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+
2/3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+2/9*arcsinh(a*x)*a*x*(a^2*x^2+1)-14/9*a*x*arcsinh(a*x)-2/27*a^2*x^2*(a^2*
x^2+1)^(1/2)+40/27*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.14105, size = 157, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arsinh}\left (a x\right )^{3} - \frac{1}{3} \, a{\left (\frac{\sqrt{a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac{2 \, \sqrt{a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname{arsinh}\left (a x\right )^{2} - \frac{2}{27} \, a{\left (\frac{\sqrt{a^{2} x^{2} + 1} x^{2} - \frac{20 \, \sqrt{a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} - \frac{3 \,{\left (a^{2} x^{3} - 6 \, x\right )} \operatorname{arsinh}\left (a x\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^3,x, algorithm="maxima")

[Out]

1/3*x^3*arcsinh(a*x)^3 - 1/3*a*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x)^2 - 2/27*a*(
(sqrt(a^2*x^2 + 1)*x^2 - 20*sqrt(a^2*x^2 + 1)/a^2)/a^2 - 3*(a^2*x^3 - 6*x)*arcsinh(a*x)/a^3)

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Fricas [A]  time = 2.11573, size = 281, normalized size = 2.13 \begin{align*} \frac{9 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 9 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 2 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 20\right )}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^3,x, algorithm="fricas")

[Out]

1/27*(9*a^3*x^3*log(a*x + sqrt(a^2*x^2 + 1))^3 - 9*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1)
)^2 + 6*(a^3*x^3 - 6*a*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 2*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 20))/a^3

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Sympy [A]  time = 2.23995, size = 128, normalized size = 0.97 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asinh}^{3}{\left (a x \right )}}{3} + \frac{2 x^{3} \operatorname{asinh}{\left (a x \right )}}{9} - \frac{x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{3 a} - \frac{2 x^{2} \sqrt{a^{2} x^{2} + 1}}{27 a} - \frac{4 x \operatorname{asinh}{\left (a x \right )}}{3 a^{2}} + \frac{2 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{3 a^{3}} + \frac{40 \sqrt{a^{2} x^{2} + 1}}{27 a^{3}} & \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**2*asinh(a*x)**3,x)

[Out]

Piecewise((x**3*asinh(a*x)**3/3 + 2*x**3*asinh(a*x)/9 - x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(3*a) - 2*x**2*
sqrt(a**2*x**2 + 1)/(27*a) - 4*x*asinh(a*x)/(3*a**2) + 2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(3*a**3) + 40*sqrt(
a**2*x**2 + 1)/(27*a**3), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.56615, size = 190, normalized size = 1.44 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - \frac{1}{27} \, a{\left (\frac{9 \,{\left ({\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{a^{4}} - \frac{2 \,{\left (3 \,{\left (a^{2} x^{3} - 6 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 21 \, \sqrt{a^{2} x^{2} + 1}}{a}\right )}}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^3,x, algorithm="giac")

[Out]

1/3*x^3*log(a*x + sqrt(a^2*x^2 + 1))^3 - 1/27*a*(9*((a^2*x^2 + 1)^(3/2) - 3*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(
a^2*x^2 + 1))^2/a^4 - 2*(3*(a^2*x^3 - 6*x)*log(a*x + sqrt(a^2*x^2 + 1)) - ((a^2*x^2 + 1)^(3/2) - 21*sqrt(a^2*x
^2 + 1))/a)/a^3)