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

Optimal. Leaf size=68 \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}+\frac {1}{2 a^2 \cosh ^{-1}(a x)}-\frac {x^2}{\cosh ^{-1}(a x)}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.39, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5668, 5775, 5670, 5448, 12, 3298, 5676} \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}+\frac {1}{2 a^2 \cosh ^{-1}(a x)}-\frac {x^2}{\cosh ^{-1}(a x)}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x/ArcCosh[a*x]^3,x]

[Out]

-(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + 1/(2*a^2*ArcCosh[a*x]) - x^2/ArcCosh[a*x] + SinhInteg
ral[2*ArcCosh[a*x]]/a^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3298

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

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

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 5670

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 67, normalized size = 0.99 \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}+\frac {1-2 a^2 x^2}{2 a^2 \cosh ^{-1}(a x)}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/ArcCosh[a*x]^3,x]

[Out]

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

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\operatorname {arcosh}\left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x/arccosh(a*x)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x/arccosh(a*x)^3, x)

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maple [A]  time = 0.03, size = 43, normalized size = 0.63 \[ \frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2 \,\mathrm {arccosh}\left (a x \right )}+\Shi \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arccosh(a*x)^3,x)

[Out]

1/a^2*(-1/4*sinh(2*arccosh(a*x))/arccosh(a*x)^2-1/2/arccosh(a*x)*cosh(2*arccosh(a*x))+Shi(2*arccosh(a*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{8} x^{8} - 3 \, a^{6} x^{6} + 3 \, a^{4} x^{4} + {\left (a^{5} x^{5} - a^{3} x^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} - a^{2} x^{2} + {\left (3 \, a^{6} x^{6} - 5 \, a^{4} x^{4} + 2 \, a^{2} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (3 \, a^{7} x^{7} - 7 \, a^{5} x^{5} + 5 \, a^{3} x^{3} - a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + {\left (2 \, a^{8} x^{8} - 6 \, a^{6} x^{6} + 6 \, a^{4} x^{4} + 2 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} - 2 \, a^{2} x^{2} + {\left (6 \, a^{6} x^{6} - 10 \, a^{4} x^{4} + 5 \, a^{2} x^{2} - 1\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (6 \, a^{7} x^{7} - 14 \, a^{5} x^{5} + 11 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{2 \, {\left (a^{8} x^{6} + {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} a^{5} x^{3} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} + 3 \, {\left (a^{6} x^{4} - a^{4} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 3 \, {\left (a^{7} x^{5} - 2 \, a^{5} x^{3} + a^{3} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a^{2}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}} + \int \frac {4 \, a^{9} x^{9} + 4 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} a^{5} x^{5} - 16 \, a^{7} x^{7} + 24 \, a^{5} x^{5} - 16 \, a^{3} x^{3} + {\left (16 \, a^{6} x^{6} - 16 \, a^{4} x^{4} + 3\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + 24 \, {\left (a^{7} x^{7} - 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (16 \, a^{8} x^{8} - 48 \, a^{6} x^{6} + 48 \, a^{4} x^{4} - 19 \, a^{2} x^{2} + 3\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 4 \, a x}{2 \, {\left (a^{9} x^{8} + {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} a^{5} x^{4} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + 4 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{7} x^{6} - 2 \, a^{5} x^{4} + a^{3} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 4 \, {\left (a^{8} x^{7} - 3 \, a^{6} x^{5} + 3 \, a^{4} x^{3} - a^{2} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^8 - 3*a^6*x^6 + 3*a^4*x^4 + (a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - a^2*x^2 + (3*a^6
*x^6 - 5*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*(a*x - 1) + (3*a^7*x^7 - 7*a^5*x^5 + 5*a^3*x^3 - a*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) + (2*a^8*x^8 - 6*a^6*x^6 + 6*a^4*x^4 + 2*(a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a
^2*x^2 + (6*a^6*x^6 - 10*a^4*x^4 + 5*a^2*x^2 - 1)*(a*x + 1)*(a*x - 1) + (6*a^7*x^7 - 14*a^5*x^5 + 11*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^8*x^6 + (a*x + 1)^(3/2)*(a*x
- 1)^(3/2)*a^5*x^3 - 3*a^6*x^4 + 3*a^4*x^2 + 3*(a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x - 1) + 3*(a^7*x^5 - 2*a^5*x^
3 + a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2) + integrate(1/2*(4*a^9
*x^9 + 4*(a*x + 1)^2*(a*x - 1)^2*a^5*x^5 - 16*a^7*x^7 + 24*a^5*x^5 - 16*a^3*x^3 + (16*a^6*x^6 - 16*a^4*x^4 + 3
)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 24*(a^7*x^7 - 2*a^5*x^5 + a^3*x^3)*(a*x + 1)*(a*x - 1) + (16*a^8*x^8 - 48*
a^6*x^6 + 48*a^4*x^4 - 19*a^2*x^2 + 3)*sqrt(a*x + 1)*sqrt(a*x - 1) + 4*a*x)/((a^9*x^8 + (a*x + 1)^2*(a*x - 1)^
2*a^5*x^4 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + 4*(a^6*x^5 - a^4*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^7
*x^6 - 2*a^5*x^4 + a^3*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^8*x^7 - 3*a^6*x^5 + 3*a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/acosh(a*x)^3,x)

[Out]

int(x/acosh(a*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/acosh(a*x)**3,x)

[Out]

Integral(x/acosh(a*x)**3, x)

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