3.1.59 \(\int \frac {x^3}{\cosh ^{-1}(a x)^3} \, dx\) [59]

Optimal. Leaf size=87 \[ -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {3 x^2}{2 a^2 \cosh ^{-1}(a x)}-\frac {2 x^4}{\cosh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{a^4} \]

[Out]

3/2*x^2/a^2/arccosh(a*x)-2*x^4/arccosh(a*x)+1/2*Shi(2*arccosh(a*x))/a^4+Shi(4*arccosh(a*x))/a^4-1/2*x^3*(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 = 87, normalized size of antiderivative = 1.00, number of steps used = 12, 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, 12} \begin {gather*} \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{a^4}+\frac {3 x^2}{2 a^2 \cosh ^{-1}(a x)}-\frac {2 x^4}{\cosh ^{-1}(a x)}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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

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Mathematica [A]
time = 0.13, size = 75, normalized size = 0.86 \begin {gather*} \frac {-\frac {a^2 x^2 \left (a x \sqrt {-1+a x} \sqrt {1+a x}+\left (-3+4 a^2 x^2\right ) \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^2}+\text {Shi}\left (2 \cosh ^{-1}(a x)\right )+2 \text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 2.08, size = 82, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicSineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \,\mathrm {arccosh}\left (a x \right )}+\hyperbolicSineIntegral \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{4}}\)	\(82\)
default	\(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicSineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \,\mathrm {arccosh}\left (a x \right )}+\hyperbolicSineIntegral \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{4}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(-1/8/arccosh(a*x)^2*sinh(2*arccosh(a*x))-1/4/arccosh(a*x)*cosh(2*arccosh(a*x))+1/2*Shi(2*arccosh(a*x))-
1/16/arccosh(a*x)^2*sinh(4*arccosh(a*x))-1/4/arccosh(a*x)*cosh(4*arccosh(a*x))+Shi(4*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^3/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^10 - 3*a^6*x^8 + 3*a^4*x^6 - a^2*x^4 + (a^5*x^7 - a^3*x^5)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^
6*x^8 - 5*a^4*x^6 + 2*a^2*x^4)*(a*x + 1)*(a*x - 1) + (3*a^7*x^9 - 7*a^5*x^7 + 5*a^3*x^5 - a*x^3)*sqrt(a*x + 1)
*sqrt(a*x - 1) + (4*a^8*x^10 - 12*a^6*x^8 + 12*a^4*x^6 - 4*a^2*x^4 + 2*(2*a^5*x^7 - 3*a^3*x^5 + a*x^3)*(a*x +
1)^(3/2)*(a*x - 1)^(3/2) + 3*(4*a^6*x^8 - 8*a^4*x^6 + 5*a^2*x^4 - x^2)*(a*x + 1)*(a*x - 1) + (12*a^7*x^9 - 30*
a^5*x^7 + 25*a^3*x^5 - 7*a*x^3)*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*(16*a^10*x^11 - 64*a^8*x^9 + 96*a^6*x^7 - 64*a^4*x^5 + 4*(4*a^6*x^7 - 3*a^4*x^5)*(a*x + 1)^2
*(a*x - 1)^2 + 16*a^2*x^3 + (64*a^7*x^8 - 100*a^5*x^6 + 42*a^3*x^4 - 3*a*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)
+ 6*(16*a^8*x^9 - 38*a^6*x^7 + 30*a^4*x^5 - 9*a^2*x^3 + x)*(a*x + 1)*(a*x - 1) + (64*a^9*x^10 - 204*a^7*x^8 +
234*a^5*x^6 - 115*a^3*x^4 + 21*a*x^2)*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^3/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^3/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^{3}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/acosh(a*x)**3, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3/acosh(a*x)^3, x)

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