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

Optimal. Leaf size=49 \[ \frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a^2}-\frac{\sinh ^{-1}(a x)^2}{4 a^3}-\frac{x^2}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.103116, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5758, 5675, 30} \[ \frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a^2}-\frac{\sinh ^{-1}(a x)^2}{4 a^3}-\frac{x^2}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -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^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{2 a^2}-\frac{\int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2}-\frac{\int x \, dx}{2 a}\\ &=-\frac{x^2}{4 a}+\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{2 a^2}-\frac{\sinh ^{-1}(a x)^2}{4 a^3}\\ \end{align*}

Mathematica [A]  time = 0.038447, size = 42, normalized size = 0.86 \[ -\frac{a^2 x^2-2 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^2}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 40, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{a}^{3}} \left ( -2\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax+{a}^{2}{x}^{2}+ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15993, size = 100, normalized size = 2.04 \begin{align*} -\frac{1}{4} \, a{\left (\frac{x^{2}}{a^{2}} - \frac{\operatorname{arsinh}\left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )^{2}}{a^{4}}\right )} + \frac{1}{2} \,{\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 )} \operatorname{arsinh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.54241, size = 146, normalized size = 2.98 \begin{align*} -\frac{a^{2} x^{2} - 2 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.934297, size = 42, normalized size = 0.86 \begin{align*} \begin{cases} - \frac{x^{2}}{4 a} + \frac{x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{2 a^{2}} - \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{4 a^{3}} & \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**2*asinh(a*x)/(a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((-x**2/(4*a) + x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(2*a**2) - asinh(a*x)**2/(4*a**3), Ne(a, 0)), (0, Tr
ue))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arcsinh(a*x)/sqrt(a^2*x^2 + 1), x)