3.344 \(\int \frac{x^2 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 a^2}+\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac{\sinh ^{-1}(a x)^4}{8 a^3}-\frac{3 \sinh ^{-1}(a x)^2}{8 a^3}-\frac{3 x^2}{8 a}-\frac{3 x^2 \sinh ^{-1}(a x)^2}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.224513, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5758, 5675, 5661, 30} \[ \frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 a^2}+\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac{\sinh ^{-1}(a x)^4}{8 a^3}-\frac{3 \sinh ^{-1}(a x)^2}{8 a^3}-\frac{3 x^2}{8 a}-\frac{3 x^2 \sinh ^{-1}(a x)^2}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]

[Out]

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

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0671444, size = 83, normalized size = 0.79 \[ -\frac{3 a^2 x^2-4 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+\left (6 a^2 x^2+3\right ) \sinh ^{-1}(a x)^2-6 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^4}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]

[Out]

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

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Maple [A]  time = 0.052, size = 84, normalized size = 0.8 \begin{align*} -{\frac{1}{8\,{a}^{3}} \left ( -4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax\sqrt{{a}^{2}{x}^{2}+1}+6\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}+ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}-6\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax+3\,{a}^{2}{x}^{2}+3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+3 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{arsinh}\left (a x\right )^{3}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2*arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)

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Fricas [A]  time = 2.11824, size = 293, normalized size = 2.79 \begin{align*} \frac{4 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 3 \, a^{2} x^{2} - \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{4} + 6 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 3 \,{\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{8 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/8*(4*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1))^3 - 3*a^2*x^2 - log(a*x + sqrt(a^2*x^2 + 1))^4 + 6*s
qrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1)) - 3*(2*a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2)/a^3

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Sympy [A]  time = 3.07738, size = 100, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{3 x^{2} \operatorname{asinh}^{2}{\left (a x \right )}}{4 a} - \frac{3 x^{2}}{8 a} + \frac{x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{2 a^{2}} + \frac{3 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{4 a^{2}} - \frac{\operatorname{asinh}^{4}{\left (a x \right )}}{8 a^{3}} - \frac{3 \operatorname{asinh}^{2}{\left (a x \right )}}{8 a^{3}} & \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**2*asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{arsinh}\left (a x\right )^{3}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^2*arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)