3.1.55 \(\int \frac {1}{\cosh ^{-1}(a x)^2} \, dx\) [55]

Optimal. Leaf size=39 \[ -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5880, 5953, 3382} \begin {gather*} \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^(-2),x]

[Out]

-((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/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 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)^2} \, dx &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+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}}{a \cosh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 60, normalized size = 1.54 \begin {gather*} \frac {1-a x+\sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^(-2),x]

[Out]

(1 - a*x + Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]])/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*A
rcCosh[a*x])

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Maple [A]
time = 1.82, size = 33, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\mathrm {arccosh}\left (a x \right )}+\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{a}\)	\(33\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\mathrm {arccosh}\left (a x \right )}+\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{a}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

[Out]

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

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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(1/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^(-2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**2,x)

[Out]

Integral(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(1/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/acosh(a*x)^2,x)

[Out]

int(1/acosh(a*x)^2, x)

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