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

Optimal. Leaf size=122 \[ \frac{2 \left (a^2 x^2+1\right )^{3/2}}{27 a^4}-\frac{14 \sqrt{a^2 x^2+1}}{9 a^4}+\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^4}+\frac{4 x \sinh ^{-1}(a x)}{3 a^3}-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a} \]

[Out]

(-14*Sqrt[1 + a^2*x^2])/(9*a^4) + (2*(1 + a^2*x^2)^(3/2))/(27*a^4) + (4*x*ArcSinh[a*x])/(3*a^3) - (2*x^3*ArcSi
nh[a*x])/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^2)

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Rubi [A]  time = 0.215011, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5758, 5717, 5653, 261, 5661, 266, 43} \[ \frac{2 \left (a^2 x^2+1\right )^{3/2}}{27 a^4}-\frac{14 \sqrt{a^2 x^2+1}}{9 a^4}+\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^4}+\frac{4 x \sinh ^{-1}(a x)}{3 a^3}-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14*Sqrt[1 + a^2*x^2])/(9*a^4) + (2*(1 + a^2*x^2)^(3/2))/(27*a^4) + (4*x*ArcSinh[a*x])/(3*a^3) - (2*x^3*ArcSi
nh[a*x])/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^2)

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 5717

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

Rule 5653

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^3 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}-\frac{2 \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{3 a^2}-\frac{2 \int x^2 \sinh ^{-1}(a x) \, dx}{3 a}\\ &=-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}+\frac{2}{9} \int \frac{x^3}{\sqrt{1+a^2 x^2}} \, dx+\frac{4 \int \sinh ^{-1}(a x) \, dx}{3 a^3}\\ &=\frac{4 x \sinh ^{-1}(a x)}{3 a^3}-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )-\frac{4 \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx}{3 a^2}\\ &=-\frac{4 \sqrt{1+a^2 x^2}}{3 a^4}+\frac{4 x \sinh ^{-1}(a x)}{3 a^3}-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}+\frac{1}{9} \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{1+a^2 x}}+\frac{\sqrt{1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{14 \sqrt{1+a^2 x^2}}{9 a^4}+\frac{2 \left (1+a^2 x^2\right )^{3/2}}{27 a^4}+\frac{4 x \sinh ^{-1}(a x)}{3 a^3}-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0611081, size = 79, normalized size = 0.65 \[ \frac{2 \left (a^2 x^2-20\right ) \sqrt{a^2 x^2+1}+9 \left (a^2 x^2-2\right ) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2-6 a x \left (a^2 x^2-6\right ) \sinh ^{-1}(a x)}{27 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2] - 6*a*x*(-6 + a^2*x^2)*ArcSinh[a*x] + 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*
ArcSinh[a*x]^2)/(27*a^4)

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Maple [A]  time = 0.058, size = 113, normalized size = 0.9 \begin{align*}{\frac{1}{27\,{a}^{4}} \left ( 9\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}-9\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}-6\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}{a}^{3}{x}^{3}+2\,{x}^{4}{a}^{4}-38\,{a}^{2}{x}^{2}-18\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+36\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax-40 \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27/a^4/(a^2*x^2+1)^(1/2)*(9*arcsinh(a*x)^2*x^4*a^4-9*arcsinh(a*x)^2*a^2*x^2-6*arcsinh(a*x)*(a^2*x^2+1)^(1/2)
*a^3*x^3+2*x^4*a^4-38*a^2*x^2-18*arcsinh(a*x)^2+36*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x-40)

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Maxima [A]  time = 1.22056, size = 136, normalized size = 1.11 \begin{align*} \frac{1}{3} \,{\left (\frac{\sqrt{a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac{2 \, \sqrt{a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname{arsinh}\left (a x\right )^{2} + \frac{2 \,{\left (\sqrt{a^{2} x^{2} + 1} x^{2} - \frac{20 \, \sqrt{a^{2} x^{2} + 1}}{a^{2}}\right )}}{27 \, a^{2}} - \frac{2 \,{\left (a^{2} x^{3} - 6 \, x\right )} \operatorname{arsinh}\left (a x\right )}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x)^2 + 2/27*(sqrt(a^2*x^2 + 1)*x^2 - 20*sq
rt(a^2*x^2 + 1)/a^2)/a^2 - 2/9*(a^2*x^3 - 6*x)*arcsinh(a*x)/a^3

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Fricas [A]  time = 3.17186, size = 223, normalized size = 1.83 \begin{align*} \frac{9 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 6 \,{\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) + 2 \, \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 20\right )}}{27 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*(a^3*x^3 - 6*a*x)*log(a*x + sqrt(a^
2*x^2 + 1)) + 2*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 20))/a^4

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Sympy [A]  time = 3.07278, size = 121, normalized size = 0.99 \begin{align*} \begin{cases} - \frac{2 x^{3} \operatorname{asinh}{\left (a x \right )}}{9 a} + \frac{x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{3 a^{2}} + \frac{2 x^{2} \sqrt{a^{2} x^{2} + 1}}{27 a^{2}} + \frac{4 x \operatorname{asinh}{\left (a x \right )}}{3 a^{3}} - \frac{2 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{3 a^{4}} - \frac{40 \sqrt{a^{2} x^{2} + 1}}{27 a^{4}} & \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**3*asinh(a*x)**2/(a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((-2*x**3*asinh(a*x)/(9*a) + x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(3*a**2) + 2*x**2*sqrt(a**2*x**2
+ 1)/(27*a**2) + 4*x*asinh(a*x)/(3*a**3) - 2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(3*a**4) - 40*sqrt(a**2*x**2 +
1)/(27*a**4), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.39926, size = 154, normalized size = 1.26 \begin{align*} \frac{{\left ({\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{3 \, a^{4}} - \frac{2 \,{\left (3 \,{\left (a^{2} x^{3} - 6 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 21 \, \sqrt{a^{2} x^{2} + 1}}{a}\right )}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((a^2*x^2 + 1)^(3/2) - 3*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2/a^4 - 2/27*(3*(a^2*x^3 - 6*x)*l
og(a*x + sqrt(a^2*x^2 + 1)) - ((a^2*x^2 + 1)^(3/2) - 21*sqrt(a^2*x^2 + 1))/a)/a^3