3.1.58 \(\int \frac {x^4}{\cosh ^{-1}(a x)^3} \, dx\) [58]

Optimal. Leaf size=102 \[ -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5} \]

[Out]

2*x^3/a^2/arccosh(a*x)-5/2*x^5/arccosh(a*x)+1/16*Shi(arccosh(a*x))/a^5+27/32*Shi(3*arccosh(a*x))/a^5+25/32*Shi
(5*arccosh(a*x))/a^5-1/2*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^2

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Rubi [A]
time = 0.43, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5886, 5951, 5887, 5556, 3379} \begin {gather*} \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^3,x]

[Out]

-1/2*(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) + (2*x^3)/(a^2*ArcCosh[a*x]) - (5*x^5)/(2*ArcCosh[a
*x]) + SinhIntegral[ArcCosh[a*x]]/(16*a^5) + (27*SinhIntegral[3*ArcCosh[a*x]])/(32*a^5) + (25*SinhIntegral[5*A
rcCosh[a*x]])/(32*a^5)

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^4}{\cosh ^{-1}(a x)^3} \, dx &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}-\frac {2 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (5 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {25}{2} \int \frac {x^4}{\cosh ^{-1}(a x)} \, dx-\frac {6 \int \frac {x^2}{\cosh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {6 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {6 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 x}+\frac {3 \sinh (3 x)}{16 x}+\frac {\sinh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {25 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac {25 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {75 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 107, normalized size = 1.05 \begin {gather*} \frac {-16 a^4 x^4 \sqrt {-1+a x} \sqrt {1+a x}+64 a^3 x^3 \cosh ^{-1}(a x)-80 a^5 x^5 \cosh ^{-1}(a x)+2 \cosh ^{-1}(a x)^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )+27 \cosh ^{-1}(a x)^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )+25 \cosh ^{-1}(a x)^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5 \cosh ^{-1}(a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcCosh[a*x]^3,x]

[Out]

(-16*a^4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + 64*a^3*x^3*ArcCosh[a*x] - 80*a^5*x^5*ArcCosh[a*x] + 2*ArcCosh[a*x]
^2*SinhIntegral[ArcCosh[a*x]] + 27*ArcCosh[a*x]^2*SinhIntegral[3*ArcCosh[a*x]] + 25*ArcCosh[a*x]^2*SinhIntegra
l[5*ArcCosh[a*x]])/(32*a^5*ArcCosh[a*x]^2)

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Maple [A]
time = 2.32, size = 123, normalized size = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{16 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {a x}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{16}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {9 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {27 \hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {5 \cosh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {25 \hyperbolicSineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32}}{a^{5}}\)	\(123\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{16 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {a x}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{16}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {9 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {27 \hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {5 \cosh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {25 \hyperbolicSineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32}}{a^{5}}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(-1/16/arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/16/arccosh(a*x)*a*x+1/16*Shi(arccosh(a*x))-3/32/arcc
osh(a*x)^2*sinh(3*arccosh(a*x))-9/32/arccosh(a*x)*cosh(3*arccosh(a*x))+27/32*Shi(3*arccosh(a*x))-1/32/arccosh(
a*x)^2*sinh(5*arccosh(a*x))-5/32/arccosh(a*x)*cosh(5*arccosh(a*x))+25/32*Shi(5*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^4/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^11 - 3*a^6*x^9 + 3*a^4*x^7 - a^2*x^5 + (a^5*x^8 - a^3*x^6)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^
6*x^9 - 5*a^4*x^7 + 2*a^2*x^5)*(a*x + 1)*(a*x - 1) + (3*a^7*x^10 - 7*a^5*x^8 + 5*a^3*x^6 - a*x^4)*sqrt(a*x + 1
)*sqrt(a*x - 1) + (5*a^8*x^11 - 15*a^6*x^9 + 15*a^4*x^7 - 5*a^2*x^5 + (5*a^5*x^8 - 8*a^3*x^6 + 3*a*x^4)*(a*x +
1)^(3/2)*(a*x - 1)^(3/2) + (15*a^6*x^9 - 31*a^4*x^7 + 20*a^2*x^5 - 4*x^3)*(a*x + 1)*(a*x - 1) + (15*a^7*x^10
- 38*a^5*x^8 + 32*a^3*x^6 - 9*a*x^4)*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*(25*a^10*x^12 - 100*a^8*x^10 + 150*a^6*x^8 - 100*a^4*x^6 + 25*a^2*x^4 + (25*a^6*x^8 - 2
4*a^4*x^6 + 3*a^2*x^4)*(a*x + 1)^2*(a*x - 1)^2 + (100*a^7*x^9 - 172*a^5*x^7 + 87*a^3*x^5 - 12*a*x^3)*(a*x + 1)
^(3/2)*(a*x - 1)^(3/2) + 3*(50*a^8*x^10 - 124*a^6*x^8 + 105*a^4*x^6 - 35*a^2*x^4 + 4*x^2)*(a*x + 1)*(a*x - 1)
+ (100*a^9*x^11 - 324*a^7*x^9 + 381*a^5*x^7 - 193*a^3*x^5 + 36*a*x^3)*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^4/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4/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^{4}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**3,x)

[Out]

Integral(x**4/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^4/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4/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^4}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/acosh(a*x)^3,x)

[Out]

int(x^4/acosh(a*x)^3, x)

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