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

Optimal. Leaf size=52 \[ -\frac {\left (a^2 x^2+1\right )^{3/2}}{9 a^3}+\frac {\sqrt {a^2 x^2+1}}{3 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5661, 266, 43} \[ -\frac {\left (a^2 x^2+1\right )^{3/2}}{9 a^3}+\frac {\sqrt {a^2 x^2+1}}{3 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps

\begin {align*} \int x^2 \sinh ^{-1}(a x) \, dx &=\frac {1}{3} x^3 \sinh ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a x)-\frac {1}{6} a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a x)-\frac {1}{6} a \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {\sqrt {1+a^2 x^2}}{3 a^3}-\frac {\left (1+a^2 x^2\right )^{3/2}}{9 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.79 \[ \frac {1}{9} \left (\frac {\left (2-a^2 x^2\right ) \sqrt {a^2 x^2+1}}{a^3}+3 x^3 \sinh ^{-1}(a x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 52, normalized size = 1.00 \[ \frac {3 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )}}{9 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.02, size = 50, normalized size = 0.96 \[ \frac {\frac {a^{3} x^{3} \arcsinh \left (a x \right )}{3}-\frac {a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{9}+\frac {2 \sqrt {a^{2} x^{2}+1}}{9}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 48, normalized size = 0.92 \[ \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right ) - \frac {1}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\mathrm {asinh}\left (a\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*asinh(a*x), x)

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sympy [A]  time = 0.44, size = 48, normalized size = 0.92 \[ \begin {cases} \frac {x^{3} \operatorname {asinh}{\left (a x \right )}}{3} - \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{9 a} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*asinh(a*x),x)

[Out]

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

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