\(\int x^3 \text {arcsinh}(a x)^3 \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 163 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {45 \text {arcsinh}(a x)}{256 a^4}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3 \]

[Out]

-45/256*arcsinh(a*x)/a^4-9/32*x^2*arcsinh(a*x)/a^2+3/32*x^4*arcsinh(a*x)-3/32*arcsinh(a*x)^3/a^4+1/4*x^4*arcsi
nh(a*x)^3+45/256*x*(a^2*x^2+1)^(1/2)/a^3-3/128*x^3*(a^2*x^2+1)^(1/2)/a+9/32*x*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)
/a^3-3/16*x^3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 5812, 5783, 327, 221} \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}-\frac {45 \text {arcsinh}(a x)}{256 a^4}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}-\frac {3 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{16 a}-\frac {3 x^3 \sqrt {a^2 x^2+1}}{128 a}+\frac {9 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{32 a^3}+\frac {45 x \sqrt {a^2 x^2+1}}{256 a^3}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3+\frac {3}{32} x^4 \text {arcsinh}(a x) \]

[In]

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

[Out]

(45*x*Sqrt[1 + a^2*x^2])/(256*a^3) - (3*x^3*Sqrt[1 + a^2*x^2])/(128*a) - (45*ArcSinh[a*x])/(256*a^4) - (9*x^2*
ArcSinh[a*x])/(32*a^2) + (3*x^4*ArcSinh[a*x])/32 + (9*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(32*a^3) - (3*x^3*Sq
rt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(16*a) - (3*ArcSinh[a*x]^3)/(32*a^4) + (x^4*ArcSinh[a*x]^3)/4

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}{4} x^4 \text {arcsinh}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3+\frac {3}{8} \int x^3 \text {arcsinh}(a x) \, dx+\frac {9 \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a} \\ & = \frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3-\frac {9 \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{32 a^3}-\frac {9 \int x \text {arcsinh}(a x) \, dx}{16 a^2}-\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3+\frac {9 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{128 a}+\frac {9 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{32 a} \\ & = \frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3-\frac {9 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{256 a^3}-\frac {9 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3} \\ & = \frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {45 \text {arcsinh}(a x)}{256 a^4}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.67 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\frac {3 a x \left (15-2 a^2 x^2\right ) \sqrt {1+a^2 x^2}+3 \left (-15-24 a^2 x^2+8 a^4 x^4\right ) \text {arcsinh}(a x)-24 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)^2+8 \left (-3+8 a^4 x^4\right ) \text {arcsinh}(a x)^3}{256 a^4} \]

[In]

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

[Out]

(3*a*x*(15 - 2*a^2*x^2)*Sqrt[1 + a^2*x^2] + 3*(-15 - 24*a^2*x^2 + 8*a^4*x^4)*ArcSinh[a*x] - 24*a*x*Sqrt[1 + a^
2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcSinh[a*x]^3)/(256*a^4)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16}+\frac {9 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{32}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{128}+\frac {45 a x \sqrt {a^{2} x^{2}+1}}{256}+\frac {27 \,\operatorname {arcsinh}\left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{32}}{a^{4}}\)	\(141\)
default	\(\frac {\frac {a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16}+\frac {9 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{32}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{128}+\frac {45 a x \sqrt {a^{2} x^{2}+1}}{256}+\frac {27 \,\operatorname {arcsinh}\left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{32}}{a^{4}}\)	\(141\)

[In]

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

[Out]

1/a^4*(1/4*a^4*x^4*arcsinh(a*x)^3-3/16*a^3*x^3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+9/32*arcsinh(a*x)^2*(a^2*x^2+1
)^(1/2)*a*x-3/32*arcsinh(a*x)^3+3/32*a^4*x^4*arcsinh(a*x)-3/128*a^3*x^3*(a^2*x^2+1)^(1/2)+45/256*a*x*(a^2*x^2+
1)^(1/2)+27/256*arcsinh(a*x)-9/32*(a^2*x^2+1)*arcsinh(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 24 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{256 \, a^{4}} \]

[In]

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

[Out]

1/256*(8*(8*a^4*x^4 - 3)*log(a*x + sqrt(a^2*x^2 + 1))^3 - 24*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + s
qrt(a^2*x^2 + 1))^2 + 3*(8*a^4*x^4 - 24*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 + 1)) - 3*(2*a^3*x^3 - 15*a*x)*sq
rt(a^2*x^2 + 1))/a^4

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {x^{4} \operatorname {asinh}^{3}{\left (a x \right )}}{4} + \frac {3 x^{4} \operatorname {asinh}{\left (a x \right )}}{32} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1}}{128 a} - \frac {9 x^{2} \operatorname {asinh}{\left (a x \right )}}{32 a^{2}} + \frac {9 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{32 a^{3}} + \frac {45 x \sqrt {a^{2} x^{2} + 1}}{256 a^{3}} - \frac {3 \operatorname {asinh}^{3}{\left (a x \right )}}{32 a^{4}} - \frac {45 \operatorname {asinh}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**4*asinh(a*x)**3/4 + 3*x**4*asinh(a*x)/32 - 3*x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(16*a) - 3*x
**3*sqrt(a**2*x**2 + 1)/(128*a) - 9*x**2*asinh(a*x)/(32*a**2) + 9*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(32*a**3
) + 45*x*sqrt(a**2*x**2 + 1)/(256*a**3) - 3*asinh(a*x)**3/(32*a**4) - 45*asinh(a*x)/(256*a**4), Ne(a, 0)), (0,
True))

Maxima [F]

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

[In]

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

[Out]

1/4*x^4*log(a*x + sqrt(a^2*x^2 + 1))^3 - integrate(3/4*(a^3*x^6 + sqrt(a^2*x^2 + 1)*a^2*x^5 + a*x^4)*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(-2)]

Exception generated. \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^3*asinh(a*x)^3, x)