3.1.51 \(\int \frac {x^4}{\cosh ^{-1}(a x)^2} \, dx\) [51]

Optimal. Leaf size=73 \[ -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5} \]

[Out]

1/8*Chi(arccosh(a*x))/a^5+9/16*Chi(3*arccosh(a*x))/a^5+5/16*Chi(5*arccosh(a*x))/a^5-x^4*(a*x-1)^(1/2)*(a*x+1)^
(1/2)/a/arccosh(a*x)

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Rubi [A]
time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5885, 3382} \begin {gather*} \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^2,x]

[Out]

-((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/(8*a^5) + (9*CoshIntegral[
3*ArcCosh[a*x]])/(16*a^5) + (5*CoshIntegral[5*ArcCosh[a*x]])/(16*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]

Rubi steps

\begin {align*} \int \frac {x^4}{\cosh ^{-1}(a x)^2} \, dx &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}-\frac {\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 )}{a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {5 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 101, normalized size = 1.38 \begin {gather*} \frac {-16 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-16 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}+2 \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )+9 \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )+5 \cosh ^{-1}(a x) \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5 \cosh ^{-1}(a x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcCosh[a*x]^2,x]

[Out]

(-16*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 16*a^5*x^5*Sqrt[(-1 + a*x)/(1 + a*x)] + 2*ArcCosh[a*x]*CoshIntegral[
ArcCosh[a*x]] + 9*ArcCosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] + 5*ArcCosh[a*x]*CoshIntegral[5*ArcCosh[a*x]])/(16
*a^5*ArcCosh[a*x])

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Maple [A]
time = 2.39, size = 83, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {5 \hyperbolicCosineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(83\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {5 \hyperbolicCosineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(-1/8/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/8*Chi(arccosh(a*x))-3/16/arccosh(a*x)*sinh(3*arccosh(a*
x))+9/16*Chi(3*arccosh(a*x))-1/16/arccosh(a*x)*sinh(5*arccosh(a*x))+5/16*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)^2,x, algorithm="maxima")

[Out]

-(a^3*x^7 - a*x^5 + (a^2*x^6 - x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x
- a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) + integrate((5*a^5*x^8 - 10*a^3*x^6 + 5*a*x^4 + (5*a^3*x^6 - 3*a
*x^4)*(a*x + 1)*(a*x - 1) + (10*a^4*x^7 - 13*a^2*x^5 + 4*x^3)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^5*x^4 + (a*x +
1)*(a*x - 1)*a^3*x^2 - 2*a^3*x^2 + 2*(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))), 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)^2,x, algorithm="fricas")

[Out]

integral(x^4/arccosh(a*x)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**2,x)

[Out]

Integral(x**4/acosh(a*x)**2, 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)^2,x, algorithm="giac")

[Out]

integrate(x^4/arccosh(a*x)^2, 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 )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/acosh(a*x)^2,x)

[Out]

int(x^4/acosh(a*x)^2, x)

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