3.136 \(\int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{3 a^4}-\frac{2 x \sqrt{a x-1}}{3 a^3 \sqrt{1-a x}}-\frac{x^3 \sqrt{a x-1}}{9 a \sqrt{1-a x}} \]

[Out]

(-2*x*Sqrt[-1 + a*x])/(3*a^3*Sqrt[1 - a*x]) - (x^3*Sqrt[-1 + a*x])/(9*a*Sqrt[1 - a*x]) - (2*Sqrt[1 - a^2*x^2]*
ArcCosh[a*x])/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(3*a^2)

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Rubi [A]  time = 0.392274, antiderivative size = 158, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {5798, 5759, 5718, 8, 30} \[ -\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{9 a \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{3 a^3 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{3 a^4 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

(-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^3*Sqrt[1 - a^2*x^2]) - (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a*Sqrt[1
- a^2*x^2]) - (2*(1 - a*x)*(1 + a*x)*ArcCosh[a*x])/(3*a^4*Sqrt[1 - a^2*x^2]) - (x^2*(1 - a*x)*(1 + a*x)*ArcCo
sh[a*x])/(3*a^2*Sqrt[1 - a^2*x^2])

Rule 5798

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

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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^3 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^2 \, dx}{3 a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x}}{9 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{3 a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x}}{9 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.122945, size = 74, normalized size = 0.67 \[ -\frac{a x \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+6\right )-3 \left (a^4 x^4+a^2 x^2-2\right ) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

-(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(6 + a^2*x^2) - 3*(-2 + a^2*x^2 + a^4*x^4)*ArcCosh[a*x])/(9*a^4*Sqrt[1 - a^
2*x^2])

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Maple [B]  time = 0.207, size = 311, normalized size = 2.8 \begin{align*} -{\frac{-1+3\,{\rm arccosh} \left (ax\right )}{72\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}+4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}-3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) }-{\frac{-3+3\,{\rm arccosh} \left (ax\right )}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \sqrt{ax+1}\sqrt{ax-1}ax+{a}^{2}{x}^{2}-1 \right ) }-{\frac{3+3\,{\rm arccosh} \left (ax\right )}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }-{\frac{1+3\,{\rm arccosh} \left (ax\right )}{72\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}-4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}+3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(-a^2*x^2+1)^(1/2)*(4*x^4*a^4-5*a^2*x^2+4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3*(a*x+1)^(1/2)*(a*x-1)^(1
/2)*a*x+1)*(-1+3*arccosh(a*x))/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*((a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+a^2*x^2
-1)*(-1+arccosh(a*x))/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-1)*(1+ar
ccosh(a*x))/a^4/(a^2*x^2-1)-1/72*(-a^2*x^2+1)^(1/2)*(4*x^4*a^4-5*a^2*x^2-4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)
+3*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+1)*(1+3*arccosh(a*x))/a^4/(a^2*x^2-1)

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Maxima [C]  time = 1.92775, size = 84, normalized size = 0.76 \begin{align*} \frac{1}{9} \, a{\left (\frac{i \, x^{3}}{a^{2}} + \frac{6 i \, x}{a^{4}}\right )} - \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{arcosh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*(I*x^3/a^2 + 6*I*x/a^4) - 1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccosh(a*x)

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Fricas [A]  time = 2.1322, size = 209, normalized size = 1.9 \begin{align*} -\frac{3 \,{\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1}}{9 \,{\left (a^{6} x^{2} - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{acosh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**3*acosh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)

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Giac [C]  time = 1.22301, size = 89, normalized size = 0.81 \begin{align*} \frac{-i \, a^{2} x^{3} - 6 i \, x}{9 \, a^{3}} + \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 )}{3 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(-I*a^2*x^3 - 6*I*x)/a^3 + 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))/
a^4