3.2.67 \(\int \frac {\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx\) [167]

Optimal. Leaf size=98 \[ \frac {2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c} \]

[Out]

2*arccosh(a*x)^2*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c+2*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x
+1)^(1/2))/a/c-2*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c-2*polylog(3,-a*x-(a*x-1)^(1/2)*(a
*x+1)^(1/2))/a/c+2*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c

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Rubi [A]
time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5903, 4267, 2611, 2320, 6724} \begin {gather*} \frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^2/(c - a^2*c*x^2),x]

[Out]

(2*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c) + (2*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c) - (2*Arc
Cosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c) - (2*PolyLog[3, -E^ArcCosh[a*x]])/(a*c) + (2*PolyLog[3, E^ArcCosh[
a*x]])/(a*c)

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5903

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-(c*d)^(-1), Subst[Int[
(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx &=-\frac {\text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}-\frac {2 \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \text {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}+\frac {2 \text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {2 \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 95, normalized size = 0.97 \begin {gather*} \frac {-\cosh ^{-1}(a x)^2 \log \left (1-e^{\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^2 \log \left (1+e^{\cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(ArcCosh[a*x]^2*Log[1 - E^ArcCosh[a*x]]) + ArcCosh[a*x]^2*Log[1 + E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2
, -E^ArcCosh[a*x]] - 2*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]] - 2*PolyLog[3, -E^ArcCosh[a*x]] + 2*PolyLog[3,
E^ArcCosh[a*x]])/(a*c)

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Maple [A]
time = 2.31, size = 187, normalized size = 1.91

method	result	size
derivativedivides	\(\frac {-\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {2 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {2 \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {2 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {2 \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}}{a}\)	\(187\)
default	\(\frac {-\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {2 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {2 \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {2 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {2 \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}}{a}\)	\(187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^2/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

1/a*(-1/c*arccosh(a*x)^2*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-2/c*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a
*x+1)^(1/2))+2/c*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+1/c*arccosh(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^
(1/2))+2/c*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-2/c*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(
1/2)))

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

[Out]

1/2*(log(a*x + 1) - log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a*c) - integrate(((a*x*log(a*x + 1
) - a*x*log(a*x - 1))*sqrt(a*x + 1)*sqrt(a*x - 1) + (a^2*x^2 - 1)*log(a*x + 1) - (a^2*x^2 - 1)*log(a*x - 1))*l
og(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/(a^3*c*x^3 - a*c*x + (a^2*c*x^2 - c)*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(arccosh(a*x)^2/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(-arccosh(a*x)^2/(a^2*c*x^2 - c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**2/(-a**2*c*x**2+c),x)

[Out]

-Integral(acosh(a*x)**2/(a**2*x**2 - 1), x)/c

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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(arccosh(a*x)^2/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(-arccosh(a*x)^2/(a^2*c*x^2 - c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^2/(c - a^2*c*x^2),x)

[Out]

int(acosh(a*x)^2/(c - a^2*c*x^2), x)

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