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

Optimal. Leaf size=110 \[ -\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{a^2}-\frac{6 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{a^2}-\frac{6 x \sqrt{a x-1}}{a \sqrt{1-a x}}-\frac{3 x \sqrt{a x-1} \cosh ^{-1}(a x)^2}{a \sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.392537, antiderivative size = 153, normalized size of antiderivative = 1.39, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5798, 5718, 5654, 8} \[ -\frac{6 x \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{6 (1-a x) (a x+1) \cosh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[1 - a^2*x^2]) - (6*(1 - a*x)*(1 + a*x)*ArcCosh[a*x])/(a^2*Sqrt[1 -
a^2*x^2]) - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) - ((1 - a*x)*(1 + a*x)*Ar
cCosh[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 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 5654

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

Rule 8

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

Rubi steps

\begin{align*} \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \cosh ^{-1}(a x)^2 \, dx}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}+\frac{\left (6 \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}{\sqrt{1-a^2 x^2}}\\ &=-\frac{6 (1-a x) (1+a x) \cosh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}-\frac{\left (6 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{6 x \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{1-a^2 x^2}}-\frac{6 (1-a x) (1+a x) \cosh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0992229, size = 101, normalized size = 0.92 \[ \frac{\sqrt{1-a^2 x^2} \left (6 a x-\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3+3 a x \cosh ^{-1}(a x)^2-6 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)\right )}{a^2 \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.121, size = 155, normalized size = 1.4 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )-6}{2\,{a}^{2} \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{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+6}{2\,{a}^{2} \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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-a^2*x^2+1)^(1/2)*((a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+a^2*x^2-1)*(arccosh(a*x)^3-3*arccosh(a*x)^2+6*arccosh
(a*x)-6)/a^2/(a^2*x^2-1)-1/2*(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-1)*(arccosh(a*x)^3+3*
arccosh(a*x)^2+6*arccosh(a*x)+6)/a^2/(a^2*x^2-1)

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Maxima [C]  time = 1.14622, size = 88, normalized size = 0.8 \begin{align*} \frac{3 i \, x \operatorname{arcosh}\left (a x\right )^{2}}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )^{3}}{a^{2}} - \frac{3 \,{\left (-2 i \, x + \frac{2 i \, \sqrt{a^{2} x^{2} - 1} \operatorname{arcosh}\left (a x\right )}{a}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.17433, size = 348, normalized size = 3.16 \begin{align*} \frac{3 \, \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} +{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + 6 \, \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} a x - 6 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{acosh}^{3}{\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*acosh(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [C]  time = 1.28001, size = 139, normalized size = 1.26 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3}}{a^{2}} - \frac{3 i \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 2 \, a{\left (\frac{x}{a} - \frac{\sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a^{2}}\right )}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^3/a^2 - 3*I*(x*log(a*x + sqrt(a^2*x^2 - 1))^2 + 2*a*(x/a - sq
rt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))/a^2))/a