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

Optimal. Leaf size=58 \[ -\frac{6 \sqrt{a^2 x^2+1}}{a}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 x \sinh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0808533, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5653, 5717, 261} \[ -\frac{6 \sqrt{a^2 x^2+1}}{a}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 x \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]

Rubi steps

\begin{align*} \int \sinh ^{-1}(a x)^3 \, dx &=x \sinh ^{-1}(a x)^3-(3 a) \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 \int \sinh ^{-1}(a x) \, dx\\ &=6 x \sinh ^{-1}(a x)-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3-(6 a) \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{6 \sqrt{1+a^2 x^2}}{a}+6 x \sinh ^{-1}(a x)-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0160733, size = 58, normalized size = 1. \[ -\frac{6 \sqrt{a^2 x^2+1}}{a}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 x \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 55, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax-3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}+6\,ax{\it Arcsinh} \left ( ax \right ) -6\,\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.3084, size = 77, normalized size = 1.33 \begin{align*} x \operatorname{arsinh}\left (a x\right )^{3} - \frac{3 \, \sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{2}}{a} + \frac{6 \,{\left (a x \operatorname{arsinh}\left (a x\right ) - \sqrt{a^{2} x^{2} + 1}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.0196, size = 205, normalized size = 3.53 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} + 6 \, a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 3 \, \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 6 \, \sqrt{a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.517625, size = 54, normalized size = 0.93 \begin{align*} \begin{cases} x \operatorname{asinh}^{3}{\left (a x \right )} + 6 x \operatorname{asinh}{\left (a x \right )} - \frac{3 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{a} - \frac{6 \sqrt{a^{2} x^{2} + 1}}{a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3,x)

[Out]

Piecewise((x*asinh(a*x)**3 + 6*x*asinh(a*x) - 3*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/a - 6*sqrt(a**2*x**2 + 1)/a,
Ne(a, 0)), (0, True))

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Giac [A]  time = 1.43003, size = 132, normalized size = 2.28 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 3 \, a{\left (\frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{a^{2}} - \frac{2 \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{\sqrt{a^{2} x^{2} + 1}}{a}\right )}}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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