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

Optimal. Leaf size=44 \[ -\frac{x \sqrt{a^2 x^2+1}}{4 a}+\frac{\sinh ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0156758, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5661, 321, 215} \[ -\frac{x \sqrt{a^2 x^2+1}}{4 a}+\frac{\sinh ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.009513, size = 40, normalized size = 0.91 \[ \frac{\left (2 a^2 x^2+1\right ) \sinh ^{-1}(a x)-a x \sqrt{a^2 x^2+1}}{4 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) }{2}}-{\frac{ax}{4}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{{\it Arcsinh} \left ( ax \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.19328, size = 69, normalized size = 1.57 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arsinh}\left (a x\right ) - \frac{1}{4} \, a{\left (\frac{\sqrt{a^{2} x^{2} + 1} x}{a^{2}} - \frac{\operatorname{arsinh}\left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76822, size = 109, normalized size = 2.48 \begin{align*} -\frac{\sqrt{a^{2} x^{2} + 1} a x -{\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.236765, size = 37, normalized size = 0.84 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asinh}{\left (a x \right )}}{2} - \frac{x \sqrt{a^{2} x^{2} + 1}}{4 a} + \frac{\operatorname{asinh}{\left (a x \right )}}{4 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),x)

[Out]

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

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Giac [A]  time = 1.28725, size = 92, normalized size = 2.09 \begin{align*} \frac{1}{2} \, x^{2} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{1}{4} \, a{\left (\frac{\sqrt{a^{2} x^{2} + 1} x}{a^{2}} + \frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}{\left | a \right |}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(a*x + sqrt(a^2*x^2 + 1)) - 1/4*a*(sqrt(a^2*x^2 + 1)*x/a^2 + log(-x*abs(a) + sqrt(a^2*x^2 + 1))/(a^
2*abs(a)))