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

Optimal. Leaf size=105 \[ \frac{2 \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^2}+\frac{1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac{x^2}{3 \cosh ^{-1}(a x)^2}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

[Out]

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

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Rubi [A]  time = 0.396025, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5668, 5775, 5666, 3301, 5676} \[ \frac{2 \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^2}+\frac{1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac{x^2}{3 \cosh ^{-1}(a x)^2}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x/ArcCosh[a*x]^4,x]

[Out]

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

Rule 5668

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

Rule 5775

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

Rule 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

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]

Rubi steps

\begin{align*} \int \frac{x}{\cosh ^{-1}(a x)^4} \, dx &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac{\int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3} \, dx}{3 a}+\frac{1}{3} (2 a) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac{x^2}{3 \cosh ^{-1}(a x)^2}+\frac{2}{3} \int \frac{x}{\cosh ^{-1}(a x)^2} \, dx\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac{x^2}{3 \cosh ^{-1}(a x)^2}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac{x^2}{3 \cosh ^{-1}(a x)^2}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)}+\frac{2 \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.29365, size = 131, normalized size = 1.25 \[ \frac{\frac{-2 a^3 x^3+\left (4 a x-4 a^3 x^3\right ) \cosh ^{-1}(a x)^2-\sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2-1\right ) \cosh ^{-1}(a x)+2 a x}{\cosh ^{-1}(a x)^3}+4 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{6 a^2 \sqrt{a x-1} \sqrt{a x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/ArcCosh[a*x]^4,x]

[Out]

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

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Maple [A]  time = 0.026, size = 60, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{6\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}-{\frac{\cosh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{6\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{3\,{\rm arccosh} \left (ax\right )}}+{\frac{2\,{\it Chi} \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arccosh(a*x)^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arccosh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^12*x^12 - 10*a^10*x^10 + 20*a^8*x^8 - 20*a^6*x^6 + 10*a^4*x^4 + 2*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2
)*(a*x - 1)^(5/2) + 2*(5*a^8*x^8 - 9*a^6*x^6 + 4*a^4*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^9*x^9 - 13*a^7*x^7
+ 11*a^5*x^5 - 3*a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a^2*x^2 + 4*(5*a^10*x^10 - 17*a^8*x^8 + 21*a^6*x
^6 - 11*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*(a*x - 1) + (4*a^12*x^12 - 20*a^10*x^10 + 40*a^8*x^8 - 40*a^6*x^6 + 20*
a^4*x^4 + 4*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (20*a^8*x^8 - 36*a^6*x^6 + 16*a^4*x^4 + 3*a^
2*x^2 - 3)*(a*x + 1)^2*(a*x - 1)^2 + (40*a^9*x^9 - 104*a^7*x^7 + 88*a^5*x^5 - 21*a^3*x^3 - 3*a*x)*(a*x + 1)^(3
/2)*(a*x - 1)^(3/2) - 4*a^2*x^2 + (40*a^10*x^10 - 136*a^8*x^8 + 168*a^6*x^6 - 91*a^4*x^4 + 22*a^2*x^2 - 3)*(a*
x + 1)*(a*x - 1) + (20*a^11*x^11 - 84*a^9*x^9 + 136*a^7*x^7 - 107*a^5*x^5 + 42*a^3*x^3 - 7*a*x)*sqrt(a*x + 1)*
sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*a^11*x^11 - 21*a^9*x^9 + 34*a^7*x^7 - 26*a^5*x^
5 + 9*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + (2*a^12*x^12 - 10*a^10*x^10 + 20*a^8*x^8 - 20*a^6*x^6 + 10*
a^4*x^4 + 2*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (10*a^8*x^8 - 18*a^6*x^6 + 9*a^4*x^4 - a^2*x
^2)*(a*x + 1)^2*(a*x - 1)^2 + (20*a^9*x^9 - 52*a^7*x^7 + 47*a^5*x^5 - 17*a^3*x^3 + 2*a*x)*(a*x + 1)^(3/2)*(a*x
- 1)^(3/2) - 2*a^2*x^2 + (20*a^10*x^10 - 68*a^8*x^8 + 87*a^6*x^6 - 51*a^4*x^4 + 13*a^2*x^2 - 1)*(a*x + 1)*(a*
x - 1) + (10*a^11*x^11 - 42*a^9*x^9 + 69*a^7*x^7 - 55*a^5*x^5 + 21*a^3*x^3 - 3*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1
))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^12*x^10 - 5*a^10*x^8 + (a*x + 1)^(5/2)*(a*x - 1)^(5/2)*a^7*x^5
+ 10*a^8*x^6 - 10*a^6*x^4 + 5*a^4*x^2 + 5*(a^8*x^6 - a^6*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 10*(a^9*x^7 - 2*a^7*x^
5 + a^5*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 10*(a^10*x^8 - 3*a^8*x^6 + 3*a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x
- 1) + 5*(a^11*x^9 - 4*a^9*x^7 + 6*a^7*x^5 - 4*a^5*x^3 + a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a^2)*log(a*x + s
qrt(a*x + 1)*sqrt(a*x - 1))^3) + integrate(1/6*(8*a^13*x^13 - 48*a^11*x^11 + 8*(a*x + 1)^3*(a*x - 1)^3*a^7*x^7
+ 120*a^9*x^9 - 160*a^7*x^7 + 120*a^5*x^5 + (48*a^8*x^8 - 48*a^6*x^6 + 4*a^4*x^4 - 12*a^2*x^2 + 15)*(a*x + 1)
^(5/2)*(a*x - 1)^(5/2) - 48*a^3*x^3 + 8*(15*a^9*x^9 - 30*a^7*x^7 + 17*a^5*x^5 - 5*a^3*x^3 + 3*a*x)*(a*x + 1)^2
*(a*x - 1)^2 + 2*(80*a^10*x^10 - 240*a^8*x^8 + 252*a^6*x^6 - 104*a^4*x^4 + 3*a^2*x^2 + 9)*(a*x + 1)^(3/2)*(a*x
- 1)^(3/2) + 8*(15*a^11*x^11 - 60*a^9*x^9 + 92*a^7*x^7 - 63*a^5*x^5 + 15*a^3*x^3 + a*x)*(a*x + 1)*(a*x - 1) +
(48*a^12*x^12 - 240*a^10*x^10 + 484*a^8*x^8 - 484*a^6*x^6 + 243*a^4*x^4 - 58*a^2*x^2 + 7)*sqrt(a*x + 1)*sqrt(
a*x - 1) + 8*a*x)/((a^13*x^12 - 6*a^11*x^10 + (a*x + 1)^3*(a*x - 1)^3*a^7*x^6 + 15*a^9*x^8 - 20*a^7*x^6 + 15*a
^5*x^4 + 6*(a^8*x^7 - a^6*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 15*(a^9*x^8 - 2*a^7*x^6 + a^5*x^4)*(a*x + 1)^
2*(a*x - 1)^2 - 6*a^3*x^2 + 20*(a^10*x^9 - 3*a^8*x^7 + 3*a^6*x^5 - a^4*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) +
15*(a^11*x^10 - 4*a^9*x^8 + 6*a^7*x^6 - 4*a^5*x^4 + a^3*x^2)*(a*x + 1)*(a*x - 1) + 6*(a^12*x^11 - 5*a^10*x^9 +
10*a^8*x^7 - 10*a^6*x^5 + 5*a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sqrt(a*
x - 1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x/arccosh(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{acosh}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/acosh(a*x)**4,x)

[Out]

Integral(x/acosh(a*x)**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x/arccosh(a*x)^4, x)