3.1.69 \(\int \frac {1}{\cosh ^{-1}(a x)^4} \, dx\) [69]
Optimal. Leaf size=86 \[ -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {x}{6 \cosh ^{-1}(a x)^2}-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{6 a} \]
[Out]
-1/6*x/arccosh(a*x)^2+1/6*Chi(arccosh(a*x))/a-1/3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^3-1/6*(a*x-1)^(1/
2)*(a*x+1)^(1/2)/a/arccosh(a*x)
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5880, 5951,
5953, 3382} \begin {gather*} \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{6 a}-\frac {x}{6 \cosh ^{-1}(a x)^2}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{6 a \cosh ^{-1}(a x)}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[ArcCosh[a*x]^(-4),x]
[Out]
-1/3*(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) - x/(6*ArcCosh[a*x]^2) - (Sqrt[-1 + a*x]*Sqrt[1 + a*x])
/(6*a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(6*a)
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 5880
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c
*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[
-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
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]
Rule 5953
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{-1}(a x)^4} \, dx &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {1}{3} a \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {x}{6 \cosh ^{-1}(a x)^2}+\frac {1}{6} \int \frac {1}{\cosh ^{-1}(a x)^2} \, dx\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {x}{6 \cosh ^{-1}(a x)^2}-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}+\frac {1}{6} a \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)} \, dx\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {x}{6 \cosh ^{-1}(a x)^2}-\frac {\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 )}{6 a}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {x}{6 \cosh ^{-1}(a x)^2}-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{6 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{6 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 114, normalized size = 1.33 \begin {gather*} \frac {\frac {2-2 a^2 x^2-a x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)+\left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2}{\cosh ^{-1}(a x)^3}+\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )}{6 a \sqrt {-1+a x} \sqrt {1+a x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcCosh[a*x]^(-4),x]
[Out]
((2 - 2*a^2*x^2 - a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + (1 - a^2*x^2)*ArcCosh[a*x]^2)/ArcCosh[a*x]^3
+ Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*CoshIntegral[ArcCosh[a*x]])/(6*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
________________________________________________________________________________________
Maple [A]
time = 1.83, size = 67, normalized size = 0.78
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{3 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {a x}{6 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{6 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{6}}{a}\) |
\(67\) |
| default |
\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{3 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {a x}{6 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{6 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{6}}{a}\) |
\(67\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/arccosh(a*x)^4,x,method=_RETURNVERBOSE)
[Out]
1/a*(-1/3/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/6/arccosh(a*x)^2*a*x-1/6/arccosh(a*x)*(a*x-1)^(1/2)*(a*
x+1)^(1/2)+1/6*Chi(arccosh(a*x)))
________________________________________________________________________________________
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(1/arccosh(a*x)^4,x, algorithm="maxima")
[Out]
-1/6*(2*a^11*x^11 - 10*a^9*x^9 + 20*a^7*x^7 - 20*a^5*x^5 + 2*(a^6*x^6 - a^4*x^4)*(a*x + 1)^(5/2)*(a*x - 1)^(5/
2) + 10*a^3*x^3 + 2*(5*a^7*x^7 - 9*a^5*x^5 + 4*a^3*x^3)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^8*x^8 - 13*a^6*x^6 +
11*a^4*x^4 - 3*a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^9*x^9 - 17*a^7*x^7 + 21*a^5*x^5 - 11*a^3*x^3
+ 2*a*x)*(a*x + 1)*(a*x - 1) + (a^11*x^11 - 5*a^9*x^9 + 10*a^7*x^7 - 10*a^5*x^5 + (a^6*x^6 - a^4*x^4 + 3*a^2*x
^2 - 3)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 5*a^3*x^3 + (5*a^7*x^7 - 9*a^5*x^5 + 10*a^3*x^3 - 6*a*x)*(a*x + 1)^2
*(a*x - 1)^2 + (10*a^8*x^8 - 26*a^6*x^6 + 22*a^4*x^4 - 3*a^2*x^2 - 3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(5*a
^9*x^9 - 17*a^7*x^7 + 18*a^5*x^5 - 5*a^3*x^3 - a*x)*(a*x + 1)*(a*x - 1) + (5*a^10*x^10 - 21*a^8*x^8 + 31*a^6*x
^6 - 20*a^4*x^4 + 6*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 +
2*(5*a^10*x^10 - 21*a^8*x^8 + 34*a^6*x^6 - 26*a^4*x^4 + 9*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - 2*a*x +
(a^11*x^11 - 5*a^9*x^9 + 10*a^7*x^7 - 10*a^5*x^5 + (a^6*x^6 - a^2*x^2)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 5*a^3
*x^3 + (5*a^7*x^7 - 5*a^5*x^5 - 2*a^3*x^3 + 2*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (10*a^8*x^8 - 20*a^6*x^6 + 10*a^4
*x^4 + a^2*x^2 - 1)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(5*a^9*x^9 - 15*a^7*x^7 + 16*a^5*x^5 - 7*a^3*x^3 + a*x
)*(a*x + 1)*(a*x - 1) + (5*a^10*x^10 - 20*a^8*x^8 + 31*a^6*x^6 - 23*a^4*x^4 + 8*a^2*x^2 - 1)*sqrt(a*x + 1)*sqr
t(a*x - 1) - a*x)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^11*x^10 - 5*a^9*x^8 + (a*x + 1)^(5/2)*(a*x - 1)^
(5/2)*a^6*x^5 + 10*a^7*x^6 - 10*a^5*x^4 + 5*(a^7*x^6 - a^5*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 5*a^3*x^2 + 10*(a^8*
x^7 - 2*a^6*x^5 + a^4*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 10*(a^9*x^8 - 3*a^7*x^6 + 3*a^5*x^4 - a^3*x^2)*(a
*x + 1)*(a*x - 1) + 5*(a^10*x^9 - 4*a^8*x^7 + 6*a^6*x^5 - 4*a^4*x^3 + a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a)*
log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3) + integrate(1/6*(a^12*x^12 - 6*a^10*x^10 + 15*a^8*x^8 - 20*a^6*x^6 +
15*a^4*x^4 + (a^6*x^6 + a^4*x^4 - 9*a^2*x^2 + 15)*(a*x + 1)^3*(a*x - 1)^3 + (6*a^7*x^7 - a^5*x^5 - 31*a^3*x^3
+ 33*a*x)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (15*a^8*x^8 - 20*a^6*x^6 - 19*a^4*x^4 + 3*a^2*x^2 + 21)*(a*x + 1)
^2*(a*x - 1)^2 + (20*a^9*x^9 - 50*a^7*x^7 + 54*a^5*x^5 - 59*a^3*x^3 + 35*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)
- 6*a^2*x^2 + (15*a^10*x^10 - 55*a^8*x^8 + 101*a^6*x^6 - 90*a^4*x^4 + 22*a^2*x^2 + 7)*(a*x + 1)*(a*x - 1) + (6
*a^11*x^11 - 29*a^9*x^9 + 65*a^7*x^7 - 66*a^5*x^5 + 23*a^3*x^3 + a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)/((a^12*
x^12 - 6*a^10*x^10 + (a*x + 1)^3*(a*x - 1)^3*a^6*x^6 + 15*a^8*x^8 - 20*a^6*x^6 + 15*a^4*x^4 + 6*(a^7*x^7 - a^5
*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 15*(a^8*x^8 - 2*a^6*x^6 + a^4*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 20*(a^9*x
^9 - 3*a^7*x^7 + 3*a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 6*a^2*x^2 + 15*(a^10*x^10 - 4*a^8*x^8
+ 6*a^6*x^6 - 4*a^4*x^4 + a^2*x^2)*(a*x + 1)*(a*x - 1) + 6*(a^11*x^11 - 5*a^9*x^9 + 10*a^7*x^7 - 10*a^5*x^5 +
5*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)
________________________________________________________________________________________
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(1/arccosh(a*x)^4,x, algorithm="fricas")
[Out]
integral(arccosh(a*x)^(-4), x)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/acosh(a*x)**4,x)
[Out]
Integral(acosh(a*x)**(-4), x)
________________________________________________________________________________________
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(1/arccosh(a*x)^4,x, algorithm="giac")
[Out]
integrate(arccosh(a*x)^(-4), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/acosh(a*x)^4,x)
[Out]
int(1/acosh(a*x)^4, x)
________________________________________________________________________________________