3.1.56 \(\int \frac {\cosh ^{-1}(a x)}{(c-a^2 c x^2)^2} \, dx\) [56]

Optimal. Leaf size=109 \[ -\frac {1}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {\text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5901, 5903, 4267, 2317, 2438, 75} \begin {gather*} \frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {\text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c - a^2*c*x^2)^2,x]

[Out]

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

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 5901

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(
1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {a \int \frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{2 c^2}+\frac {\int \frac {\cosh ^{-1}(a x)}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac {1}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {\text {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {1}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}-\frac {\text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {1}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}\\ &=-\frac {1}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {\text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 120, normalized size = 1.10 \begin {gather*} \frac {-\frac {2 \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\cosh ^{-1}(a x) \left (a x+\left (-1+a^2 x^2\right ) \log \left (1-e^{\cosh ^{-1}(a x)}\right )+\left (1-a^2 x^2\right ) \log \left (1+e^{\cosh ^{-1}(a x)}\right )\right )\right )}{-1+a^2 x^2}+2 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{4 a c^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x]*(a*x + (-1 + a^2*x^2)*Log[1 - E^ArcCosh[a*x]] + (1 -
a^2*x^2)*Log[1 + E^ArcCosh[a*x]])))/(-1 + a^2*x^2) + 2*PolyLog[2, -E^ArcCosh[a*x]] - 2*PolyLog[2, E^ArcCosh[a
*x]])/(4*a*c^2)

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Maple [A]
time = 3.95, size = 161, normalized size = 1.48

method	result	size
derivativedivides	\(\frac {-\frac {a x \,\mathrm {arccosh}\left (a x \right )+\sqrt {a x -1}\, \sqrt {a x +1}}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}}{a}\)	\(161\)
default	\(\frac {-\frac {a x \,\mathrm {arccosh}\left (a x \right )+\sqrt {a x -1}\, \sqrt {a x +1}}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}}{a}\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/16*((a^2*x^2 - 1)*log(a*x + 1)^2 + 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^2
+ 4*a*x + 4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*
x - 1)) - 2*(a^2*x^2 - 1)*log(a*x - 1))/(a^3*c^2*x^2 - a*c^2) + 1/4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-
1/2*a*x + 1/2))/(a*c^2) - 1/8*log(a*x + 1)/(a*c^2) + integrate(-1/4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2
*x^2 - 1)*log(a*x - 1))/(a^5*c^2*x^5 - 2*a^3*c^2*x^3 + a*c^2*x + (a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*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)/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\operatorname {acosh}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2

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

[Out]

integrate(arccosh(a*x)/(a^2*c*x^2 - c)^2, 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 )}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)/(c - a^2*c*x^2)^2,x)

[Out]

int(acosh(a*x)/(c - a^2*c*x^2)^2, x)

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