3.1.60 \(\int \frac {x^2}{\cosh ^{-1}(a x)^3} \, dx\) [60]

Optimal. Leaf size=85 \[ -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}+\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3} \]

[Out]

x/a^2/arccosh(a*x)-3/2*x^3/arccosh(a*x)+1/8*Shi(arccosh(a*x))/a^3+9/8*Shi(3*arccosh(a*x))/a^3-1/2*x^2*(a*x-1)^
(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^2

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Rubi [A]
time = 0.33, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5886, 5951, 5887, 5556, 3379, 5881} \begin {gather*} \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/ArcCosh[a*x]^3,x]

[Out]

-1/2*(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) + x/(a^2*ArcCosh[a*x]) - (3*x^3)/(2*ArcCosh[a*x]) +
SinhIntegral[ArcCosh[a*x]]/(8*a^3) + (9*SinhIntegral[3*ArcCosh[a*x]])/(8*a^3)

Rule 3379

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 5556

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 5881

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

Rule 5886

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 5887

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

Rule 5951

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*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], 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] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {x^2}{\cosh ^{-1}(a x)^3} \, dx &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}-\frac {\int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}+\frac {9}{2} \int \frac {x^2}{\cosh ^{-1}(a x)} \, dx-\frac {\int \frac {1}{\cosh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}+\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 69, normalized size = 0.81 \begin {gather*} \frac {-\frac {4 a x \left (a x \sqrt {-1+a x} \sqrt {1+a x}+\left (-2+3 a^2 x^2\right ) \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^2}+\text {Shi}\left (\cosh ^{-1}(a x)\right )+9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcCosh[a*x]^3,x]

[Out]

((-4*a*x*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + (-2 + 3*a^2*x^2)*ArcCosh[a*x]))/ArcCosh[a*x]^2 + SinhIntegral[Arc
Cosh[a*x]] + 9*SinhIntegral[3*ArcCosh[a*x]])/(8*a^3)

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Maple [A]
time = 1.85, size = 84, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {a x}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{8}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\)	\(84\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {a x}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{8}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^3*(-1/8/arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/8/arccosh(a*x)*a*x+1/8*Shi(arccosh(a*x))-1/8/arccosh(
a*x)^2*sinh(3*arccosh(a*x))-3/8/arccosh(a*x)*cosh(3*arccosh(a*x))+9/8*Shi(3*arccosh(a*x)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^8*x^9 - 3*a^6*x^7 + 3*a^4*x^5 - a^2*x^3 + (a^5*x^6 - a^3*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^6
*x^7 - 5*a^4*x^5 + 2*a^2*x^3)*(a*x + 1)*(a*x - 1) + (3*a^7*x^8 - 7*a^5*x^6 + 5*a^3*x^4 - a*x^2)*sqrt(a*x + 1)*
sqrt(a*x - 1) + (3*a^8*x^9 - 9*a^6*x^7 + 9*a^4*x^5 - 3*a^2*x^3 + (3*a^5*x^6 - 4*a^3*x^4 + a*x^2)*(a*x + 1)^(3/
2)*(a*x - 1)^(3/2) + (9*a^6*x^7 - 17*a^4*x^5 + 10*a^2*x^3 - 2*x)*(a*x + 1)*(a*x - 1) + (9*a^7*x^8 - 22*a^5*x^6
+ 18*a^3*x^4 - 5*a*x^2)*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) + int
egrate(1/2*(9*a^10*x^10 - 36*a^8*x^8 + 54*a^6*x^6 - 36*a^4*x^4 + (9*a^6*x^6 - 4*a^4*x^4 - a^2*x^2)*(a*x + 1)^2
*(a*x - 1)^2 + (36*a^7*x^7 - 48*a^5*x^5 + 13*a^3*x^3 + 2*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 9*a^2*x^2 + (5
4*a^8*x^8 - 120*a^6*x^6 + 83*a^4*x^4 - 19*a^2*x^2 + 2)*(a*x + 1)*(a*x - 1) + (36*a^9*x^9 - 112*a^7*x^7 + 123*a
^5*x^5 - 57*a^3*x^3 + 10*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^10*x^8 + (a*x + 1)^2*(a*x - 1)^2*a^6*x^4 - 4*a^
8*x^6 + 6*a^6*x^4 - 4*a^4*x^2 + 4*(a^7*x^5 - a^5*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^8*x^6 - 2*a^6*x^4
+ a^4*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^9*x^7 - 3*a^7*x^5 + 3*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))), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arccosh(a*x)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/acosh(a*x)**3,x)

[Out]

Integral(x**2/acosh(a*x)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/acosh(a*x)^3,x)

[Out]

int(x^2/acosh(a*x)^3, x)

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