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

Optimal. Leaf size=70 \[ \frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^4}+\frac{2 x}{3 a^3}-\frac{x^3}{9 a} \]

[Out]

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

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Rubi [A]  time = 0.111001, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5758, 5717, 8, 30} \[ \frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^4}+\frac{2 x}{3 a^3}-\frac{x^3}{9 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/(3*a^3) - x^3/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/
(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 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 \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^2}-\frac{2 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{3 a^2}-\frac{\int x^2 \, dx}{3 a}\\ &=-\frac{x^3}{9 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^2}+\frac{2 \int 1 \, dx}{3 a^3}\\ &=\frac{2 x}{3 a^3}-\frac{x^3}{9 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0395997, size = 48, normalized size = 0.69 \[ \frac{-a^3 x^3+3 \left (a^2 x^2-2\right ) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+6 a x}{9 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 82, normalized size = 1.2 \begin{align*}{\frac{1}{9\,{a}^{4}} \left ( 3\,{a}^{4}{x}^{4}{\it Arcsinh} \left ( ax \right ) -3\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) -{a}^{3}{x}^{3}\sqrt{{a}^{2}{x}^{2}+1}-6\,{\it Arcsinh} \left ( ax \right ) +6\,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)/(a^2*x^2+1)^(1/2),x)

[Out]

1/9/a^4/(a^2*x^2+1)^(1/2)*(3*a^4*x^4*arcsinh(a*x)-3*a^2*x^2*arcsinh(a*x)-a^3*x^3*(a^2*x^2+1)^(1/2)-6*arcsinh(a
*x)+6*a*x*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.1451, size = 80, normalized size = 1.14 \begin{align*} -\frac{1}{9} \, a{\left (\frac{x^{3}}{a^{2}} - \frac{6 \, x}{a^{4}}\right )} + \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.53258, size = 126, normalized size = 1.8 \begin{align*} -\frac{a^{3} x^{3} - 3 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 6 \, a x}{9 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.52718, size = 65, normalized size = 0.93 \begin{align*} \begin{cases} - \frac{x^{3}}{9 a} + \frac{x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{3 a^{2}} + \frac{2 x}{3 a^{3}} - \frac{2 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{3 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)/(a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.33975, size = 85, normalized size = 1.21 \begin{align*} -\frac{a^{2} x^{3} - 6 \, x}{9 \, a^{3}} + \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 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(a^2*x^3 - 6*x)/a^3 + 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))/a^4