\(\int x \text {arcsinh}(a x)^3 \, dx\) [25]

Optimal result

Integrand size = 8, antiderivative size = 97 \[ \int x \text {arcsinh}(a x)^3 \, dx=-\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3 \text {arcsinh}(a x)}{8 a^2}+\frac {3}{4} x^2 \text {arcsinh}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a}+\frac {\text {arcsinh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3 \]

[Out]

3/8*arcsinh(a*x)/a^2+3/4*x^2*arcsinh(a*x)+1/4*arcsinh(a*x)^3/a^2+1/2*x^2*arcsinh(a*x)^3-3/8*x*(a^2*x^2+1)^(1/2
)/a-3/4*x*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5776, 5812, 5783, 327, 221} \[ \int x \text {arcsinh}(a x)^3 \, dx=-\frac {3 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{4 a}+\frac {\text {arcsinh}(a x)^3}{4 a^2}+\frac {3 \text {arcsinh}(a x)}{8 a^2}-\frac {3 x \sqrt {a^2 x^2+1}}{8 a}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3+\frac {3}{4} x^2 \text {arcsinh}(a x) \]

[In]

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

[Out]

(-3*x*Sqrt[1 + a^2*x^2])/(8*a) + (3*ArcSinh[a*x])/(8*a^2) + (3*x^2*ArcSinh[a*x])/4 - (3*x*Sqrt[1 + a^2*x^2]*Ar
cSinh[a*x]^2)/(4*a) + ArcSinh[a*x]^3/(4*a^2) + (x^2*ArcSinh[a*x]^3)/2

Rule 221

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 327

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 5776

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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5812

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)
))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3+\frac {3}{2} \int x \text {arcsinh}(a x) \, dx+\frac {3 \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{4 a} \\ & = \frac {3}{4} x^2 \text {arcsinh}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a}+\frac {\text {arcsinh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3}{4} x^2 \text {arcsinh}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a}+\frac {\text {arcsinh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3+\frac {3 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a} \\ & = -\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3 \text {arcsinh}(a x)}{8 a^2}+\frac {3}{4} x^2 \text {arcsinh}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a}+\frac {\text {arcsinh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int x \text {arcsinh}(a x)^3 \, dx=\frac {-3 a x \sqrt {1+a^2 x^2}+\left (3+6 a^2 x^2\right ) \text {arcsinh}(a x)-6 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2+\left (2+4 a^2 x^2\right ) \text {arcsinh}(a x)^3}{8 a^2} \]

[In]

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

[Out]

(-3*a*x*Sqrt[1 + a^2*x^2] + (3 + 6*a^2*x^2)*ArcSinh[a*x] - 6*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 + (2 + 4*a^2
*x^2)*ArcSinh[a*x]^3)/(8*a^2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{3} \left (a^{2} x^{2}+1\right )}{2}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{4}-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{4}-\frac {3 a x \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \,\operatorname {arcsinh}\left (a x \right )}{8}}{a^{2}}\)	\(88\)
default	\(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{3} \left (a^{2} x^{2}+1\right )}{2}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{4}-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{4}-\frac {3 a x \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \,\operatorname {arcsinh}\left (a x \right )}{8}}{a^{2}}\)	\(88\)

[In]

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

[Out]

1/a^2*(1/2*arcsinh(a*x)^3*(a^2*x^2+1)-3/4*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)*a*x-1/4*arcsinh(a*x)^3+3/4*(a^2*x^2
+1)*arcsinh(a*x)-3/8*a*x*(a^2*x^2+1)^(1/2)-3/8*arcsinh(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.15 \[ \int x \text {arcsinh}(a x)^3 \, dx=-\frac {6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} + 1} a x - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{8 \, a^{2}} \]

[In]

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

[Out]

-1/8*(6*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1))^2 - 2*(2*a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^
3 + 3*sqrt(a^2*x^2 + 1)*a*x - 3*(2*a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)))/a^2

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int x \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {x^{2} \operatorname {asinh}^{3}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}{\left (a x \right )}}{4} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {a^{2} x^{2} + 1}}{8 a} + \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asinh}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**2*asinh(a*x)**3/2 + 3*x**2*asinh(a*x)/4 - 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(4*a) - 3*x*sqrt
(a**2*x**2 + 1)/(8*a) + asinh(a*x)**3/(4*a**2) + 3*asinh(a*x)/(8*a**2), Ne(a, 0)), (0, True))

Maxima [F]

\[ \int x \text {arcsinh}(a x)^3 \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

1/2*x^2*log(a*x + sqrt(a^2*x^2 + 1))^3 - integrate(3/2*(a^3*x^4 + sqrt(a^2*x^2 + 1)*a^2*x^3 + a*x^2)*log(a*x +
sqrt(a^2*x^2 + 1))^2/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)

Giac [F]

\[ \int x \text {arcsinh}(a x)^3 \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integrate(x*arcsinh(a*x)^3, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*asinh(a*x)^3, x)