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]

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

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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 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 \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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.43, size = 48, normalized size = 1.09 \[ -\frac {\sqrt {a^{2} x^{2} + 1} a x - {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{4 \, a^{2}} \]

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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giac [A]  time = 0.14, size = 68, normalized size = 1.55 \[ \frac {1}{2} \, x^{2} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} + \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2} {\left | a \right |}}\right )} \]

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)))

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maple [A]  time = 0.02, size = 39, normalized size = 0.89 \[ \frac {\frac {a^{2} x^{2} \arcsinh \left (a x \right )}{2}-\frac {a x \sqrt {a^{2} x^{2}+1}}{4}+\frac {\arcsinh \left (a x \right )}{4}}{a^{2}} \]

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 = 0.31, size = 39, normalized size = 0.89 \[ \frac {1}{2} \, x^{2} \operatorname {arsinh}\left (a x\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )} \]

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*x)/a^3)

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mupad [B]  time = 1.65, size = 36, normalized size = 0.82 \[ x\,\mathrm {asinh}\left (a\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,a^2\,x}\right )-\frac {x\,\sqrt {a^2\,x^2+1}}{4\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.20, size = 37, normalized size = 0.84 \[ \begin {cases} \frac {x^{2} \operatorname {asinh}{\left (a x \right )}}{2} - \frac {x \sqrt {a^{2} x^{2} + 1}}{4 a} + \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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