\(\int \text {arcsinh}(a x)^3 \, dx\) [26]

Optimal result

Integrand size = 6, antiderivative size = 58 \[ \int \text {arcsinh}(a x)^3 \, dx=-\frac {6 \sqrt {1+a^2 x^2}}{a}+6 x \text {arcsinh}(a x)-\frac {3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a}+x \text {arcsinh}(a x)^3 \]

[Out]

6*x*arcsinh(a*x)+x*arcsinh(a*x)^3-6*(a^2*x^2+1)^(1/2)/a-3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5772, 5798, 267} \[ \int \text {arcsinh}(a x)^3 \, dx=-\frac {3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a}-\frac {6 \sqrt {a^2 x^2+1}}{a}+x \text {arcsinh}(a x)^3+6 x \text {arcsinh}(a x) \]

[In]

Int[ArcSinh[a*x]^3,x]

[Out]

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

Rule 267

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

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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]

Rubi steps \begin{align*} \text {integral}& = x \text {arcsinh}(a x)^3-(3 a) \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a}+x \text {arcsinh}(a x)^3+6 \int \text {arcsinh}(a x) \, dx \\ & = 6 x \text {arcsinh}(a x)-\frac {3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a}+x \text {arcsinh}(a x)^3-(6 a) \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {6 \sqrt {1+a^2 x^2}}{a}+6 x \text {arcsinh}(a x)-\frac {3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a}+x \text {arcsinh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \text {arcsinh}(a x)^3 \, dx=-\frac {6 \sqrt {1+a^2 x^2}}{a}+6 x \text {arcsinh}(a x)-\frac {3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a}+x \text {arcsinh}(a x)^3 \]

[In]

Integrate[ArcSinh[a*x]^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {a x \operatorname {arcsinh}\left (a x \right )^{3}-3 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+6 a x \,\operatorname {arcsinh}\left (a x \right )-6 \sqrt {a^{2} x^{2}+1}}{a}\)	\(55\)
default	\(\frac {a x \operatorname {arcsinh}\left (a x \right )^{3}-3 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+6 a x \,\operatorname {arcsinh}\left (a x \right )-6 \sqrt {a^{2} x^{2}+1}}{a}\)	\(55\)

[In]

int(arcsinh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x*arcsinh(a*x)^3-3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+6*a*x*arcsinh(a*x)-6*(a^2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.55 \[ \int \text {arcsinh}(a x)^3 \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 6 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, \sqrt {a^{2} x^{2} + 1}}{a} \]

[In]

integrate(arcsinh(a*x)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \text {arcsinh}(a x)^3 \, dx=\begin {cases} x \operatorname {asinh}^{3}{\left (a x \right )} + 6 x \operatorname {asinh}{\left (a x \right )} - \frac {3 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{a} - \frac {6 \sqrt {a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(asinh(a*x)**3,x)

[Out]

Piecewise((x*asinh(a*x)**3 + 6*x*asinh(a*x) - 3*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/a - 6*sqrt(a**2*x**2 + 1)/a,
Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \text {arcsinh}(a x)^3 \, dx=x \operatorname {arsinh}\left (a x\right )^{3} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{2}}{a} + \frac {6 \, {\left (a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}\right )}}{a} \]

[In]

integrate(arcsinh(a*x)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.69 \[ \int \text {arcsinh}(a x)^3 \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 3 \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{2}} - \frac {2 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a}\right )}}{a}\right )} \]

[In]

integrate(arcsinh(a*x)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \text {arcsinh}(a x)^3 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]

[In]

int(asinh(a*x)^3,x)

[Out]

int(asinh(a*x)^3, x)