3.14 \(\int x^2 \sinh ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a}+\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^2+\frac{2 x^3}{27} \]

[Out]

(-4*x)/(9*a^2) + (2*x^3)/27 + (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^3) - (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*
x])/(9*a) + (x^3*ArcSinh[a*x]^2)/3

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Rubi [A]  time = 0.122978, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5661, 5758, 5717, 8, 30} \[ -\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a}+\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^2+\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSinh[a*x]^2,x]

[Out]

(-4*x)/(9*a^2) + (2*x^3)/27 + (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^3) - (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*
x])/(9*a) + (x^3*ArcSinh[a*x]^2)/3

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 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 x^2 \sinh ^{-1}(a x)^2 \, dx &=\frac{1}{3} x^3 \sinh ^{-1}(a x)^2-\frac{1}{3} (2 a) \int \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^2+\frac{2 \int x^2 \, dx}{9}+\frac{4 \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{9 a}\\ &=\frac{2 x^3}{27}+\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^3}-\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^2-\frac{4 \int 1 \, dx}{9 a^2}\\ &=-\frac{4 x}{9 a^2}+\frac{2 x^3}{27}+\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^3}-\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.0578393, size = 59, normalized size = 0.74 \[ \frac{1}{27} \left (2 x \left (x^2-\frac{6}{a^2}\right )-\frac{6 \left (a^2 x^2-2\right ) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^3}+9 x^3 \sinh ^{-1}(a x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSinh[a*x]^2,x]

[Out]

(2*x*(-6/a^2 + x^2) - (6*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^3 + 9*x^3*ArcSinh[a*x]^2)/27

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Maple [A]  time = 0.027, size = 92, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax}{3}}-{\frac{2\,{\it Arcsinh} \left ( ax \right ){a}^{2}{x}^{2}}{9}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{4\,{\it Arcsinh} \left ( ax \right ) }{9}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{2\,ax \left ({a}^{2}{x}^{2}+1 \right ) }{27}}-{\frac{14\,ax}{27}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsinh(a*x)^2,x)

[Out]

1/a^3*(1/3*arcsinh(a*x)^2*a*x*(a^2*x^2+1)-1/3*arcsinh(a*x)^2*a*x-2/9*arcsinh(a*x)*a^2*x^2*(a^2*x^2+1)^(1/2)+4/
9*arcsinh(a*x)*(a^2*x^2+1)^(1/2)+2/27*a*x*(a^2*x^2+1)-14/27*a*x)

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Maxima [A]  time = 1.23514, size = 95, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arsinh}\left (a x\right )^{2} - \frac{2}{9} \, a{\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 ) + \frac{2 \,{\left (a^{2} x^{3} - 6 \, x\right )}}{27 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arcsinh(a*x)^2 - 2/9*a*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x) + 2/27*(a^2*
x^3 - 6*x)/a^2

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Fricas [A]  time = 1.8043, size = 188, normalized size = 2.35 \begin{align*} \frac{9 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 2 \, a^{3} x^{3} - 6 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 12 \, a x}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*a^3*x^3*log(a*x + sqrt(a^2*x^2 + 1))^2 + 2*a^3*x^3 - 6*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(
a^2*x^2 + 1)) - 12*a*x)/a^3

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Sympy [A]  time = 1.1273, size = 76, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asinh}^{2}{\left (a x \right )}}{3} + \frac{2 x^{3}}{27} - \frac{2 x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{9 a} - \frac{4 x}{9 a^{2}} + \frac{4 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{9 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)**2,x)

[Out]

Piecewise((x**3*asinh(a*x)**2/3 + 2*x**3/27 - 2*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)/(9*a) - 4*x/(9*a**2) + 4*s
qrt(a**2*x**2 + 1)*asinh(a*x)/(9*a**3), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.43483, size = 120, normalized size = 1.5 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + \frac{2}{27} \, a{\left (\frac{a^{2} x^{3} - 6 \, x}{a^{3}} - \frac{3 \,{\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 )}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(a*x + sqrt(a^2*x^2 + 1))^2 + 2/27*a*((a^2*x^3 - 6*x)/a^3 - 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)