\(\int \text {arcsinh}(a x)^4 \, dx\) [37]

Optimal result

Integrand size = 6, antiderivative size = 67 \[ \int \text {arcsinh}(a x)^4 \, dx=24 x-\frac {24 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+12 x \text {arcsinh}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+x \text {arcsinh}(a x)^4 \]

[Out]

24*x+12*x*arcsinh(a*x)^2+x*arcsinh(a*x)^4-24*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a-4*arcsinh(a*x)^3*(a^2*x^2+1)^(1/
2)/a

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5772, 5798, 8} \[ \int \text {arcsinh}(a x)^4 \, dx=-\frac {4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a}-\frac {24 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a}+x \text {arcsinh}(a x)^4+12 x \text {arcsinh}(a x)^2+24 x \]

[In]

Int[ArcSinh[a*x]^4,x]

[Out]

24*x - (24*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a + 12*x*ArcSinh[a*x]^2 - (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a +
x*ArcSinh[a*x]^4

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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)^4-(4 a) \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+x \text {arcsinh}(a x)^4+12 \int \text {arcsinh}(a x)^2 \, dx \\ & = 12 x \text {arcsinh}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+x \text {arcsinh}(a x)^4-(24 a) \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {24 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+12 x \text {arcsinh}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+x \text {arcsinh}(a x)^4+24 \int 1 \, dx \\ & = 24 x-\frac {24 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+12 x \text {arcsinh}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+x \text {arcsinh}(a x)^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \text {arcsinh}(a x)^4 \, dx=24 x-\frac {24 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+12 x \text {arcsinh}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+x \text {arcsinh}(a x)^4 \]

[In]

Integrate[ArcSinh[a*x]^4,x]

[Out]

24*x - (24*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a + 12*x*ArcSinh[a*x]^2 - (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a +
x*ArcSinh[a*x]^4

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {a x \operatorname {arcsinh}\left (a x \right )^{4}-4 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}+12 a x \operatorname {arcsinh}\left (a x \right )^{2}-24 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}+24 a x}{a}\)	\(65\)
default	\(\frac {a x \operatorname {arcsinh}\left (a x \right )^{4}-4 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}+12 a x \operatorname {arcsinh}\left (a x \right )^{2}-24 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}+24 a x}{a}\)	\(65\)

[In]

int(arcsinh(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x*arcsinh(a*x)^4-4*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)+12*a*x*arcsinh(a*x)^2-24*arcsinh(a*x)*(a^2*x^2+1)^(
1/2)+24*a*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.67 \[ \int \text {arcsinh}(a x)^4 \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 12 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 4 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 24 \, a x - 24 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a} \]

[In]

integrate(arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

(a*x*log(a*x + sqrt(a^2*x^2 + 1))^4 + 12*a*x*log(a*x + sqrt(a^2*x^2 + 1))^2 - 4*sqrt(a^2*x^2 + 1)*log(a*x + sq
rt(a^2*x^2 + 1))^3 + 24*a*x - 24*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)))/a

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \text {arcsinh}(a x)^4 \, dx=\begin {cases} x \operatorname {asinh}^{4}{\left (a x \right )} + 12 x \operatorname {asinh}^{2}{\left (a x \right )} + 24 x - \frac {4 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a} - \frac {24 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(asinh(a*x)**4,x)

[Out]

Piecewise((x*asinh(a*x)**4 + 12*x*asinh(a*x)**2 + 24*x - 4*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/a - 24*sqrt(a**2*
x**2 + 1)*asinh(a*x)/a, Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \text {arcsinh}(a x)^4 \, dx=x \operatorname {arsinh}\left (a x\right )^{4} - \frac {4 \, \sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{3}}{a} + 12 \, {\left (\frac {x \operatorname {arsinh}\left (a x\right )^{2}}{a} + \frac {2 \, {\left (x - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a}\right )}}{a}\right )} a \]

[In]

integrate(arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

x*arcsinh(a*x)^4 - 4*sqrt(a^2*x^2 + 1)*arcsinh(a*x)^3/a + 12*(x*arcsinh(a*x)^2/a + 2*(x - sqrt(a^2*x^2 + 1)*ar
csinh(a*x)/a)/a)*a

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \text {arcsinh}(a x)^4 \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 4 \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{2}} - \frac {3 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}}\right )}\right )}}{a}\right )} a \]

[In]

integrate(arcsinh(a*x)^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \text {arcsinh}(a x)^4 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]

[In]

int(asinh(a*x)^4,x)

[Out]

int(asinh(a*x)^4, x)