3.343 \(\int \frac{x^3 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=153 \[ \frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a^2}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a^4}-\frac{40 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^4}+\frac{40 x}{9 a^3}+\frac{2 x \sinh ^{-1}(a x)^2}{a^3}-\frac{2 x^3}{27 a}-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a} \]

[Out]

(40*x)/(9*a^3) - (2*x^3)/(27*a) - (40*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4) + (2*x^2*Sqrt[1 + a^2*x^2]*ArcSi
nh[a*x])/(9*a^2) + (2*x*ArcSinh[a*x]^2)/a^3 - (x^3*ArcSinh[a*x]^2)/(3*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3
)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(3*a^2)

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Rubi [A]  time = 0.338049, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5758, 5717, 5653, 8, 5661, 30} \[ \frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a^2}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a^4}-\frac{40 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^4}+\frac{40 x}{9 a^3}+\frac{2 x \sinh ^{-1}(a x)^2}{a^3}-\frac{2 x^3}{27 a}-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]

[Out]

(40*x)/(9*a^3) - (2*x^3)/(27*a) - (40*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4) + (2*x^2*Sqrt[1 + a^2*x^2]*ArcSi
nh[a*x])/(9*a^2) + (2*x*ArcSinh[a*x]^2)/a^3 - (x^3*ArcSinh[a*x]^2)/(3*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3
)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(3*a^2)

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 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^3 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^2}-\frac{2 \int \frac{x \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{3 a^2}-\frac{\int x^2 \sinh ^{-1}(a x)^2 \, dx}{a}\\ &=-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^2}+\frac{2}{3} \int \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx+\frac{2 \int \sinh ^{-1}(a x)^2 \, dx}{a^3}\\ &=\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^2}+\frac{2 x \sinh ^{-1}(a x)^2}{a^3}-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^2}-\frac{4 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{9 a^2}-\frac{4 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{a^2}-\frac{2 \int x^2 \, dx}{9 a}\\ &=-\frac{2 x^3}{27 a}-\frac{40 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^4}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^2}+\frac{2 x \sinh ^{-1}(a x)^2}{a^3}-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^2}+\frac{4 \int 1 \, dx}{9 a^3}+\frac{4 \int 1 \, dx}{a^3}\\ &=\frac{40 x}{9 a^3}-\frac{2 x^3}{27 a}-\frac{40 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^4}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^2}+\frac{2 x \sinh ^{-1}(a x)^2}{a^3}-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.068115, size = 98, normalized size = 0.64 \[ \frac{-2 a x \left (a^2 x^2-60\right )+9 \left (a^2 x^2-2\right ) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3-9 a x \left (a^2 x^2-6\right ) \sinh ^{-1}(a x)^2+6 \left (a^2 x^2-20\right ) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{27 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]

[Out]

(-2*a*x*(-60 + a^2*x^2) + 6*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] - 9*a*x*(-6 + a^2*x^2)*ArcSinh[a*x]
^2 + 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(27*a^4)

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Maple [A]  time = 0.054, size = 164, normalized size = 1.1 \begin{align*}{\frac{1}{27\,{a}^{4}} \left ( 9\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{x}^{4}{a}^{4}-9\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}-9\,{a}^{3}{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}+6\,{a}^{4}{x}^{4}{\it Arcsinh} \left ( ax \right ) -114\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) -2\,{a}^{3}{x}^{3}\sqrt{{a}^{2}{x}^{2}+1}-18\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}+54\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax\sqrt{{a}^{2}{x}^{2}+1}-120\,{\it Arcsinh} \left ( ax \right ) +120\,ax\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x)

[Out]

1/27/a^4/(a^2*x^2+1)^(1/2)*(9*arcsinh(a*x)^3*x^4*a^4-9*arcsinh(a*x)^3*x^2*a^2-9*a^3*x^3*arcsinh(a*x)^2*(a^2*x^
2+1)^(1/2)+6*a^4*x^4*arcsinh(a*x)-114*a^2*x^2*arcsinh(a*x)-2*a^3*x^3*(a^2*x^2+1)^(1/2)-18*arcsinh(a*x)^3+54*ar
csinh(a*x)^2*a*x*(a^2*x^2+1)^(1/2)-120*arcsinh(a*x)+120*a*x*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.16037, size = 171, normalized size = 1.12 \begin{align*} \frac{1}{3} \,{\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 )^{3} + \frac{2}{27} \, a{\left (\frac{3 \,{\left (\sqrt{a^{2} x^{2} + 1} x^{2} - \frac{20 \, \sqrt{a^{2} x^{2} + 1}}{a^{2}}\right )} \operatorname{arsinh}\left (a x\right )}{a^{3}} - \frac{a^{2} x^{3} - 60 \, x}{a^{4}}\right )} - \frac{{\left (a^{2} x^{3} - 6 \, x\right )} \operatorname{arsinh}\left (a x\right )^{2}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/3*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x)^3 + 2/27*a*(3*(sqrt(a^2*x^2 + 1)*x^2 -
20*sqrt(a^2*x^2 + 1)/a^2)*arcsinh(a*x)/a^3 - (a^2*x^3 - 60*x)/a^4) - 1/3*(a^2*x^3 - 6*x)*arcsinh(a*x)^2/a^3

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Fricas [A]  time = 2.14252, size = 296, normalized size = 1.93 \begin{align*} -\frac{2 \, a^{3} x^{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 )^{3} + 9 \,{\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 6 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 20\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 120 \, a x}{27 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/27*(2*a^3*x^3 - 9*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1))^3 + 9*(a^3*x^3 - 6*a*x)*log(
a*x + sqrt(a^2*x^2 + 1))^2 - 6*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 20)*log(a*x + sqrt(a^2*x^2 + 1)) - 120*a*x)/a^4

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Sympy [A]  time = 4.80225, size = 148, normalized size = 0.97 \begin{align*} \begin{cases} - \frac{x^{3} \operatorname{asinh}^{2}{\left (a x \right )}}{3 a} - \frac{2 x^{3}}{27 a} + \frac{x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{3 a^{2}} + \frac{2 x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{9 a^{2}} + \frac{2 x \operatorname{asinh}^{2}{\left (a x \right )}}{a^{3}} + \frac{40 x}{9 a^{3}} - \frac{2 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{3 a^{4}} - \frac{40 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{9 a^{4}} & \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**3*asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.43019, size = 193, normalized size = 1.26 \begin{align*} \frac{{\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 )^{3}}{3 \, a^{4}} - \frac{2 \, a^{2} x^{3} + 9 \,{\left (a^{2} x^{3} - 6 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 120 \, x - \frac{6 \,{\left ({\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 21 \, \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/3*((a^2*x^2 + 1)^(3/2) - 3*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^3/a^4 - 1/27*(2*a^2*x^3 + 9*(a^2*
x^3 - 6*x)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 120*x - 6*((a^2*x^2 + 1)^(3/2) - 21*sqrt(a^2*x^2 + 1))*log(a*x + s
qrt(a^2*x^2 + 1))/a)/a^3