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

Optimal. Leaf size=64 \[ \frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{6 x}{a}-\frac{3 x \sinh ^{-1}(a x)^2}{a} \]

[Out]

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

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Rubi [A]  time = 0.10974, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5717, 5653, 8} \[ \frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{6 x}{a}-\frac{3 x \sinh ^{-1}(a x)^2}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0327067, size = 58, normalized size = 0.91 \[ \frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)-6 a x-3 a x \sinh ^{-1}(a x)^2}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 90, normalized size = 1.4 \begin{align*}{\frac{1}{{a}^{2}} \left ( \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}+ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}-3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax\sqrt{{a}^{2}{x}^{2}+1}+6\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) +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*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.15992, size = 82, normalized size = 1.28 \begin{align*} -\frac{3 \, x \operatorname{arsinh}\left (a x\right )^{2}}{a} + \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{3}}{a^{2}} - \frac{6 \,{\left (x - \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{a}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.10925, size = 209, normalized size = 3.27 \begin{align*} -\frac{3 \, a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} + 6 \, a x - 6 \, \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.47613, size = 61, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{3 x \operatorname{asinh}^{2}{\left (a x \right )}}{a} - \frac{6 x}{a} + \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{a^{2}} + \frac{6 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{a^{2}} & \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*asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.48015, size = 136, normalized size = 2.12 \begin{align*} \frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3}}{a^{2}} - \frac{3 \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 2 \, a{\left (\frac{x}{a} - \frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}}\right )}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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