3.282 \(\int \frac{x^4 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=153 \[ \frac{x^3 \sqrt{a^2 x^2+1}}{32 a^2}-\frac{15 x \sqrt{a^2 x^2+1}}{64 a^4}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}-\frac{x^4 \sinh ^{-1}(a x)}{8 a} \]

[Out]

(-15*x*Sqrt[1 + a^2*x^2])/(64*a^4) + (x^3*Sqrt[1 + a^2*x^2])/(32*a^2) + (15*ArcSinh[a*x])/(64*a^5) + (3*x^2*Ar
cSinh[a*x])/(8*a^3) - (x^4*ArcSinh[a*x])/(8*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(8*a^4) + (x^3*Sqrt[1
+ a^2*x^2]*ArcSinh[a*x]^2)/(4*a^2) + ArcSinh[a*x]^3/(8*a^5)

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Rubi [A]  time = 0.28968, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5758, 5675, 5661, 321, 215} \[ \frac{x^3 \sqrt{a^2 x^2+1}}{32 a^2}-\frac{15 x \sqrt{a^2 x^2+1}}{64 a^4}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}-\frac{x^4 \sinh ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*x*Sqrt[1 + a^2*x^2])/(64*a^4) + (x^3*Sqrt[1 + a^2*x^2])/(32*a^2) + (15*ArcSinh[a*x])/(64*a^5) + (3*x^2*Ar
cSinh[a*x])/(8*a^3) - (x^4*ArcSinh[a*x])/(8*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(8*a^4) + (x^3*Sqrt[1
+ a^2*x^2]*ArcSinh[a*x]^2)/(4*a^2) + ArcSinh[a*x]^3/(8*a^5)

Rule 5758

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

Rule 5675

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

Rule 5661

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 321

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 215

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]

Rubi steps

\begin{align*} \int \frac{x^4 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}-\frac{3 \int \frac{x^2 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{4 a^2}-\frac{\int x^3 \sinh ^{-1}(a x) \, dx}{2 a}\\ &=-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{1}{8} \int \frac{x^4}{\sqrt{1+a^2 x^2}} \, dx+\frac{3 \int \frac{\sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{8 a^4}+\frac{3 \int x \sinh ^{-1}(a x) \, dx}{4 a^3}\\ &=\frac{x^3 \sqrt{1+a^2 x^2}}{32 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}-\frac{3 \int \frac{x^2}{\sqrt{1+a^2 x^2}} \, dx}{32 a^2}-\frac{3 \int \frac{x^2}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2}\\ &=-\frac{15 x \sqrt{1+a^2 x^2}}{64 a^4}+\frac{x^3 \sqrt{1+a^2 x^2}}{32 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{3 \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{64 a^4}+\frac{3 \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{16 a^4}\\ &=-\frac{15 x \sqrt{1+a^2 x^2}}{64 a^4}+\frac{x^3 \sqrt{1+a^2 x^2}}{32 a^2}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0694461, size = 98, normalized size = 0.64 \[ \frac{a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-15\right )+8 a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)^2+\left (-8 a^4 x^4+24 a^2 x^2+15\right ) \sinh ^{-1}(a x)+8 \sinh ^{-1}(a x)^3}{64 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2) + (15 + 24*a^2*x^2 - 8*a^4*x^4)*ArcSinh[a*x] + 8*a*x*Sqrt[1 + a^2*x^2
]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^2 + 8*ArcSinh[a*x]^3)/(64*a^5)

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Maple [A]  time = 0.059, size = 125, normalized size = 0.8 \begin{align*}{\frac{1}{64\,{a}^{5}} \left ( 16\,{a}^{3}{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-8\,{a}^{4}{x}^{4}{\it Arcsinh} \left ( ax \right ) +2\,{a}^{3}{x}^{3}\sqrt{{a}^{2}{x}^{2}+1}-24\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax\sqrt{{a}^{2}{x}^{2}+1}+24\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) +8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}-15\,ax\sqrt{{a}^{2}{x}^{2}+1}+15\,{\it Arcsinh} \left ( ax \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(16*a^3*x^3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-8*a^4*x^4*arcsinh(a*x)+2*a^3*x^3*(a^2*x^2+1)^(1/2)-24*arcsin
h(a*x)^2*a*x*(a^2*x^2+1)^(1/2)+24*a^2*x^2*arcsinh(a*x)+8*arcsinh(a*x)^3-15*a*x*(a^2*x^2+1)^(1/2)+15*arcsinh(a*
x))/a^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arsinh}\left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.12243, size = 297, normalized size = 1.94 \begin{align*} \frac{8 \,{\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} + 8 \, \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{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 ) +{\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt{a^{2} x^{2} + 1}}{64 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(8*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2 + 8*log(a*x + sqrt(a^2*x^2 + 1))^
3 - (8*a^4*x^4 - 24*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 + 1)) + (2*a^3*x^3 - 15*a*x)*sqrt(a^2*x^2 + 1))/a^5

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Sympy [A]  time = 5.75894, size = 146, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{x^{4} \operatorname{asinh}{\left (a x \right )}}{8 a} + \frac{x^{3} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{4 a^{2}} + \frac{x^{3} \sqrt{a^{2} x^{2} + 1}}{32 a^{2}} + \frac{3 x^{2} \operatorname{asinh}{\left (a x \right )}}{8 a^{3}} - \frac{3 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{8 a^{4}} - \frac{15 x \sqrt{a^{2} x^{2} + 1}}{64 a^{4}} + \frac{\operatorname{asinh}^{3}{\left (a x \right )}}{8 a^{5}} + \frac{15 \operatorname{asinh}{\left (a x \right )}}{64 a^{5}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**4*asinh(a*x)/(8*a) + x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(4*a**2) + x**3*sqrt(a**2*x**2 + 1)
/(32*a**2) + 3*x**2*asinh(a*x)/(8*a**3) - 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(8*a**4) - 15*x*sqrt(a**2*x**2
+ 1)/(64*a**4) + asinh(a*x)**3/(8*a**5) + 15*asinh(a*x)/(64*a**5), Ne(a, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arsinh}\left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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