\(\int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx\) [177]

Optimal result

Integrand size = 21, antiderivative size = 86 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 x^2}{16 a^3}-\frac {x^4}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{4 a^2}+\frac {3 \text {arcsinh}(a x)^2}{16 a^5} \]

[Out]

3/16*x^2/a^3-1/16*x^4/a+3/16*arcsinh(a*x)^2/a^5-3/8*x*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^4+1/4*x^3*arcsinh(a*x)*
(a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5812, 5783, 30} \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 \text {arcsinh}(a x)^2}{16 a^5}+\frac {3 x^2}{16 a^3}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {3 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{8 a^4}-\frac {x^4}{16 a} \]

[In]

Int[(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(3*x^2)/(16*a^3) - x^4/(16*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[
a*x])/(4*a^2) + (3*ArcSinh[a*x]^2)/(16*a^5)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && 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 {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{4 a^2}-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}-\frac {\int x^3 \, dx}{4 a} \\ & = -\frac {x^4}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{4 a^2}+\frac {3 \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^4}+\frac {3 \int x \, dx}{8 a^3} \\ & = \frac {3 x^2}{16 a^3}-\frac {x^4}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{4 a^2}+\frac {3 \text {arcsinh}(a x)^2}{16 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 a^2 x^2-a^4 x^4+2 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)+3 \text {arcsinh}(a x)^2}{16 a^5} \]

[In]

Integrate[(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(3*a^2*x^2 - a^4*x^4 + 2*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x] + 3*ArcSinh[a*x]^2)/(16*a^5)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {4 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}-a^{4} x^{4}-6 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +3 a^{2} x^{2}+3 \operatorname {arcsinh}\left (a x \right )^{2}+3}{16 a^{5}}\)	\(74\)

[In]

int(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/16*(4*a^3*x^3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)-a^4*x^4-6*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x+3*a^2*x^2+3*arcsin
h(a*x)^2+3)/a^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a^{4} x^{4} - 3 \, a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{16 \, a^{5}} \]

[In]

integrate(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/16*(a^4*x^4 - 3*a^2*x^2 - 2*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)) - 3*log(a*x
+ sqrt(a^2*x^2 + 1))^2)/a^5

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{4}}{16 a} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{2}}{16 a^{3}} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{8 a^{4}} + \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*asinh(a*x)/(a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((-x**4/(16*a) + x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)/(4*a**2) + 3*x**2/(16*a**3) - 3*x*sqrt(a**2*x**2
+ 1)*asinh(a*x)/(8*a**4) + 3*asinh(a*x)**2/(16*a**5), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {1}{16} \, {\left (\frac {x^{4}}{a^{2}} - \frac {3 \, x^{2}}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )^{2}}{a^{6}}\right )} a + \frac {1}{8} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} x^{3}}{a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} \operatorname {arsinh}\left (a x\right ) \]

[In]

integrate(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-1/16*(x^4/a^2 - 3*x^2/a^4 + 3*arcsinh(a*x)^2/a^6)*a + 1/8*(2*sqrt(a^2*x^2 + 1)*x^3/a^2 - 3*sqrt(a^2*x^2 + 1)*
x/a^4 + 3*arcsinh(a*x)/a^5)*arcsinh(a*x)

Giac [F]

\[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]

[In]

integrate(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4*arcsinh(a*x)/sqrt(a^2*x^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^4\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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