3.135 \(\int \frac{x^4 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{3 x^2 \sqrt{a x-1}}{16 a^3 \sqrt{1-a x}}-\frac{x^3 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{8 a^4}+\frac{3 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{16 a^5 \sqrt{1-a x}}-\frac{x^4 \sqrt{a x-1}}{16 a \sqrt{1-a x}} \]

[Out]

(-3*x^2*Sqrt[-1 + a*x])/(16*a^3*Sqrt[1 - a*x]) - (x^4*Sqrt[-1 + a*x])/(16*a*Sqrt[1 - a*x]) - (3*x*Sqrt[1 - a^2
*x^2]*ArcCosh[a*x])/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(4*a^2) + (3*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)
/(16*a^5*Sqrt[1 - a*x])

________________________________________________________________________________________

Rubi [A]  time = 0.500843, antiderivative size = 206, normalized size of antiderivative = 1.42, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5798, 5759, 5676, 30} \[ -\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{16 a \sqrt{1-a^2 x^2}}-\frac{3 x^2 \sqrt{a x-1} \sqrt{a x+1}}{16 a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (a x+1) \cosh ^{-1}(a x)}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (a x+1) \cosh ^{-1}(a x)}{8 a^4 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{16 a^5 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

(-3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(16*a^3*Sqrt[1 - a^2*x^2]) - (x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(16*a*Sq
rt[1 - a^2*x^2]) - (3*x*(1 - a*x)*(1 + a*x)*ArcCosh[a*x])/(8*a^4*Sqrt[1 - a^2*x^2]) - (x^3*(1 - a*x)*(1 + a*x)
*ArcCosh[a*x])/(4*a^2*Sqrt[1 - a^2*x^2]) + (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(16*a^5*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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -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^4 \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^4 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^3 \, dx}{4 a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{16 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int x \, dx}{8 a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{16 a^3 \sqrt{1-a^2 x^2}}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{16 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a^5 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.254521, size = 93, normalized size = 0.64 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (-16 \cosh \left (2 \cosh ^{-1}(a x)\right )-\cosh \left (4 \cosh ^{-1}(a x)\right )+4 \cosh ^{-1}(a x) \left (6 \cosh ^{-1}(a x)+8 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )\right )}{128 a^5 \sqrt{-(a x-1) (a x+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-16*Cosh[2*ArcCosh[a*x]] - Cosh[4*ArcCosh[a*x]] + 4*ArcCosh[a*x]*(6*Arc
Cosh[a*x] + 8*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]])))/(128*a^5*Sqrt[-((-1 + a*x)*(1 + a*x))])

________________________________________________________________________________________

Maple [B]  time = 0.299, size = 456, normalized size = 3.1 \begin{align*} -{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{-1+4\,{\rm arccosh} \left (ax\right )}{256\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 8\,{x}^{5}{a}^{5}-12\,{x}^{3}{a}^{3}+8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+4\,ax-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{-1+2\,{\rm arccosh} \left (ax\right )}{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax+2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{1+2\,{\rm arccosh} \left (ax\right )}{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{1+4\,{\rm arccosh} \left (ax\right )}{256\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 8\,{x}^{5}{a}^{5}-12\,{x}^{3}{a}^{3}-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+4\,ax+8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*arccosh(a*x)^2-1/256*(-a^2*x^2+1)^(1/2)*(
8*x^5*a^5-12*x^3*a^3+8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+4*a*x-8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2+(a*x-1)
^(1/2)*(a*x+1)^(1/2))*(-1+4*arccosh(a*x))/a^5/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(2*x^3*a^3-2*a*x+2*(a*x+1)^(
1/2)*(a*x-1)^(1/2)*x^2*a^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(-1+2*arccosh(a*x))/a^5/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^
(1/2)*(2*x^3*a^3-2*a*x-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+2*arccosh(a*x))/a
^5/(a^2*x^2-1)-1/256*(-a^2*x^2+1)^(1/2)*(8*x^5*a^5-12*x^3*a^3-8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+4*a*x+8*(a
*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+4*arccosh(a*x))/a^5/(a^2*x^2-1)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{4} \operatorname{arcosh}\left (a x\right )}{a^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^4*arccosh(a*x)/(a^2*x^2 - 1), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \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**4*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**4*acosh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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