3.1.52 \(\int \frac {x^3}{\cosh ^{-1}(a x)^2} \, dx\) [52]

Optimal. Leaf size=61 \[ -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4} \]

[Out]

1/2*Chi(2*arccosh(a*x))/a^4+1/2*Chi(4*arccosh(a*x))/a^4-x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)

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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, 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 (2 \cosh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/ArcCosh[a*x]^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.18, size = 58, normalized size = 0.95 \begin {gather*} \frac {-\frac {2 a^3 x^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{\cosh ^{-1}(a x)}+\text {Chi}\left (2 \cosh ^{-1}(a x)\right )+\text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcCosh[a*x]^2,x]

[Out]

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

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Maple [A]
time = 2.08, size = 54, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{2}}{a^{4}}\)	\(54\)
default	\(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{2}}{a^{4}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

[Out]

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

[Out]

integral(x^3/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^{3}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/acosh(a*x)**2,x)

[Out]

Integral(x**3/acosh(a*x)**2, 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)^2,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 = -0.02 \begin {gather*} \int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/acosh(a*x)^2,x)

[Out]

int(x^3/acosh(a*x)^2, x)

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