3.1.65 \(\int \frac {x^4}{\cosh ^{-1}(a x)^4} \, dx\) [65]

Optimal. Leaf size=170 \[ -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{48 a^5}+\frac {27 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {125 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5} \]

[Out]

2/3*x^3/a^2/arccosh(a*x)^2-5/6*x^5/arccosh(a*x)^2+1/48*Chi(arccosh(a*x))/a^5+27/32*Chi(3*arccosh(a*x))/a^5+125
/96*Chi(5*arccosh(a*x))/a^5-1/3*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^3+2*x^2*(a*x-1)^(1/2)*(a*x+1)^(
1/2)/a^3/arccosh(a*x)-25/6*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)

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Rubi [A]
time = 0.41, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5886, 5951, 5885, 3382} \begin {gather*} \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{48 a^5}+\frac {27 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {125 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5}+\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a^3 \cosh ^{-1}(a x)}+\frac {2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {5 x^5}{6 \cosh ^{-1}(a x)^2}-\frac {25 x^4 \sqrt {a x-1} \sqrt {a x+1}}{6 a \cosh ^{-1}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) + (2*x^3)/(3*a^2*ArcCosh[a*x]^2) - (5*x^5)/(6*ArcCo
sh[a*x]^2) + (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (25*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(
6*a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(48*a^5) + (27*CoshIntegral[3*ArcCosh[a*x]])/(32*a^5) + (125*Co
shIntegral[5*ArcCosh[a*x]])/(96*a^5)

Rule 3382

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 5885

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^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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 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)^4} \, dx &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {4 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx}{3 a}+\frac {1}{3} (5 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac {25}{6} \int \frac {x^4}{\cosh ^{-1}(a x)^2} \, dx-\frac {2 \int \frac {x^2}{\cosh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}-\frac {3 \cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {25 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{8 x}-\frac {9 \cosh (3 x)}{16 x}-\frac {5 \cosh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^5}+\frac {125 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac {75 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{48 a^5}+\frac {27 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {125 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(356\) vs. \(2(170)=340\).
time = 0.28, size = 356, normalized size = 2.09 \begin {gather*} \frac {\sqrt {-1+a x} \left (32 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-32 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}}+64 a^3 x^3 \sqrt {-1+a x} \sqrt {\frac {-1+a x}{1+a x}} \sqrt {1+a x} \cosh ^{-1}(a x)-80 a^5 x^5 \sqrt {-1+a x} \sqrt {\frac {-1+a x}{1+a x}} \sqrt {1+a x} \cosh ^{-1}(a x)-192 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)^2+592 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)^2-400 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)^2+2 (-1+a x) \cosh ^{-1}(a x)^3 \text {Chi}\left (\cosh ^{-1}(a x)\right )+81 (-1+a x) \cosh ^{-1}(a x)^3 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )-125 \cosh ^{-1}(a x)^3 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )+125 a x \cosh ^{-1}(a x)^3 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )\right )}{96 a^5 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-1 + a*x]*(32*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 32*a^6*x^6*Sqrt[(-1 + a*x)/(1 + a*x)] + 64*a^3*x^3*Sq
rt[-1 + a*x]*Sqrt[(-1 + a*x)/(1 + a*x)]*Sqrt[1 + a*x]*ArcCosh[a*x] - 80*a^5*x^5*Sqrt[-1 + a*x]*Sqrt[(-1 + a*x)
/(1 + a*x)]*Sqrt[1 + a*x]*ArcCosh[a*x] - 192*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 + 592*a^4*x^4*S
qrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 - 400*a^6*x^6*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 + 2*(-1 + a*x
)*ArcCosh[a*x]^3*CoshIntegral[ArcCosh[a*x]] + 81*(-1 + a*x)*ArcCosh[a*x]^3*CoshIntegral[3*ArcCosh[a*x]] - 125*
ArcCosh[a*x]^3*CoshIntegral[5*ArcCosh[a*x]] + 125*a*x*ArcCosh[a*x]^3*CoshIntegral[5*ArcCosh[a*x]]))/(96*a^5*((
-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^3)

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Maple [A]
time = 2.33, size = 175, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {a x}{48 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{48 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{48}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {3 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {9 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {27 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{48 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {5 \cosh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{96 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {25 \sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{96 \,\mathrm {arccosh}\left (a x \right )}+\frac {125 \hyperbolicCosineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{96}}{a^{5}}\)	\(175\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {a x}{48 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{48 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{48}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {3 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {9 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {27 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{48 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {5 \cosh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{96 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {25 \sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{96 \,\mathrm {arccosh}\left (a x \right )}+\frac {125 \hyperbolicCosineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{96}}{a^{5}}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(-1/24/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/48/arccosh(a*x)^2*a*x-1/48/arccosh(a*x)*(a*x-1)^(1/2
)*(a*x+1)^(1/2)+1/48*Chi(arccosh(a*x))-1/16/arccosh(a*x)^3*sinh(3*arccosh(a*x))-3/32/arccosh(a*x)^2*cosh(3*arc
cosh(a*x))-9/32/arccosh(a*x)*sinh(3*arccosh(a*x))+27/32*Chi(3*arccosh(a*x))-1/48/arccosh(a*x)^3*sinh(5*arccosh
(a*x))-5/96/arccosh(a*x)^2*cosh(5*arccosh(a*x))-25/96/arccosh(a*x)*sinh(5*arccosh(a*x))+125/96*Chi(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)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^13*x^15 - 10*a^11*x^13 + 20*a^9*x^11 - 20*a^7*x^9 + 10*a^5*x^7 - 2*a^3*x^5 + 2*(a^8*x^10 - a^6*x^8)*
(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 2*(5*a^9*x^11 - 9*a^7*x^9 + 4*a^5*x^7)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x
^12 - 13*a^8*x^10 + 11*a^6*x^8 - 3*a^4*x^6)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^11*x^13 - 17*a^9*x^11 + 2
1*a^7*x^9 - 11*a^5*x^7 + 2*a^3*x^5)*(a*x + 1)*(a*x - 1) + (25*a^13*x^15 - 125*a^11*x^13 + 250*a^9*x^11 - 250*a
^7*x^9 + 125*a^5*x^7 - 25*a^3*x^5 + (25*a^8*x^10 - 49*a^6*x^8 + 27*a^4*x^6 - 3*a^2*x^4)*(a*x + 1)^(5/2)*(a*x -
1)^(5/2) + (125*a^9*x^11 - 321*a^7*x^9 + 286*a^5*x^7 - 102*a^3*x^5 + 12*a*x^3)*(a*x + 1)^2*(a*x - 1)^2 + (250
*a^10*x^12 - 794*a^8*x^10 + 946*a^6*x^8 - 519*a^4*x^6 + 129*a^2*x^4 - 12*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)
+ 2*(125*a^11*x^13 - 473*a^9*x^11 + 696*a^7*x^9 - 497*a^5*x^7 + 173*a^3*x^5 - 24*a*x^3)*(a*x + 1)*(a*x - 1) +
(125*a^12*x^14 - 549*a^10*x^12 + 955*a^8*x^10 - 824*a^6*x^8 + 354*a^4*x^6 - 61*a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x
- 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*a^12*x^14 - 21*a^10*x^12 + 34*a^8*x^10 - 26*a^6*x^8 + 9
*a^4*x^6 - a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1) + (5*a^13*x^15 - 25*a^11*x^13 + 50*a^9*x^11 - 50*a^7*x^9 + 25*
a^5*x^7 - 5*a^3*x^5 + (5*a^8*x^10 - 8*a^6*x^8 + 3*a^4*x^6)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (25*a^9*x^11 - 57
*a^7*x^9 + 42*a^5*x^7 - 10*a^3*x^5)*(a*x + 1)^2*(a*x - 1)^2 + (50*a^10*x^12 - 148*a^8*x^10 + 158*a^6*x^8 - 71*
a^4*x^6 + 11*a^2*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(25*a^11*x^13 - 91*a^9*x^11 + 126*a^7*x^9 - 81*a^5*x
^7 + 23*a^3*x^5 - 2*a*x^3)*(a*x + 1)*(a*x - 1) + (25*a^12*x^14 - 108*a^10*x^12 + 183*a^8*x^10 - 151*a^6*x^8 +
60*a^4*x^6 - 9*a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^13*x^10 - 5*a
^11*x^8 + (a*x + 1)^(5/2)*(a*x - 1)^(5/2)*a^8*x^5 + 10*a^9*x^6 - 10*a^7*x^4 + 5*a^5*x^2 + 5*(a^9*x^6 - a^7*x^4
)*(a*x + 1)^2*(a*x - 1)^2 + 10*(a^10*x^7 - 2*a^8*x^5 + a^6*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 10*(a^11*x^8
- 3*a^9*x^6 + 3*a^7*x^4 - a^5*x^2)*(a*x + 1)*(a*x - 1) - a^3 + 5*(a^12*x^9 - 4*a^10*x^7 + 6*a^8*x^5 - 4*a^6*x
^3 + a^4*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3) + integrate(1/6*(125*a^15*x
^16 - 750*a^13*x^14 + 1875*a^11*x^12 - 2500*a^9*x^10 + 1875*a^7*x^8 - 750*a^5*x^6 + (125*a^9*x^10 - 147*a^7*x^
8 + 27*a^5*x^6 + 3*a^3*x^4)*(a*x + 1)^3*(a*x - 1)^3 + 125*a^3*x^4 + (750*a^10*x^11 - 1485*a^8*x^9 + 901*a^6*x^
7 - 147*a^4*x^5 - 12*a^2*x^3)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (1875*a^11*x^12 - 5220*a^9*x^10 + 5209*a^7*x^8
- 2185*a^5*x^6 + 321*a^3*x^4)*(a*x + 1)^2*(a*x - 1)^2 + (2500*a^12*x^13 - 8970*a^10*x^11 + 12366*a^8*x^9 - 81
43*a^6*x^7 + 2583*a^4*x^5 - 360*a^2*x^3 + 24*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (1875*a^13*x^14 - 8235*a^11*
x^12 + 14449*a^9*x^10 - 12834*a^7*x^8 + 6030*a^5*x^6 - 1429*a^3*x^4 + 144*a*x^2)*(a*x + 1)*(a*x - 1) + (750*a^
14*x^15 - 3897*a^12*x^13 + 8293*a^10*x^11 - 9226*a^8*x^9 + 5655*a^6*x^7 - 1819*a^4*x^5 + 244*a^2*x^3)*sqrt(a*x
+ 1)*sqrt(a*x - 1))/((a^15*x^12 - 6*a^13*x^10 + (a*x + 1)^3*(a*x - 1)^3*a^9*x^6 + 15*a^11*x^8 - 20*a^9*x^6 +
15*a^7*x^4 - 6*a^5*x^2 + 6*(a^10*x^7 - a^8*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 15*(a^11*x^8 - 2*a^9*x^6 + a
^7*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 20*(a^12*x^9 - 3*a^10*x^7 + 3*a^8*x^5 - a^6*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(
3/2) + 15*(a^13*x^10 - 4*a^11*x^8 + 6*a^9*x^6 - 4*a^7*x^4 + a^5*x^2)*(a*x + 1)*(a*x - 1) + a^3 + 6*(a^14*x^11
- 5*a^12*x^9 + 10*a^10*x^7 - 10*a^8*x^5 + 5*a^6*x^3 - a^4*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*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)^4,x, algorithm="fricas")

[Out]

integral(x^4/arccosh(a*x)^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x^4/arccosh(a*x)^4, 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 )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4/acosh(a*x)^4, x)

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