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

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

[Out]

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

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Rubi [A]  time = 0.50, 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 30

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

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 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 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]

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*}

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Mathematica [A]  time = 0.26, 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))])

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{4} \operatorname {arcosh}\left (a x\right )}{a^{2} x^{2} - 1}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]

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)

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maple [B]  time = 0.80, size = 456, normalized size = 3.14 \[ -\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2}}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-1+4 \,\mathrm {arccosh}\left (a x \right )\right )}{256 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-1+2 \,\mathrm {arccosh}\left (a x \right )\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (1+2 \,\mathrm {arccosh}\left (a x \right )\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (1+4 \,\mathrm {arccosh}\left (a x \right )\right )}{256 a^{5} \left (a^{2} x^{2}-1\right )} \]

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^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(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^2*x^2*(
a*x-1)^(1/2)*(a*x+1)^(1/2)-(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^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(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^
2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+4*arccosh(a*x))/a^5/(a^2*x^2-1)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

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)

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