\(\int \frac {\text {arccosh}(a x)}{(c-a^2 c x^2)^3} \, dx\) [57]

Optimal result

Integrand size = 18, antiderivative size = 164 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3} \]

[Out]

1/12/a/c^3/(a*x-1)^(3/2)/(a*x+1)^(3/2)+1/4*x*arccosh(a*x)/c^3/(-a^2*x^2+1)^2+3/8*x*arccosh(a*x)/c^3/(-a^2*x^2+
1)+3/4*arccosh(a*x)*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+3/8*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1
/2))/a/c^3-3/8*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/8/a/c^3/(a*x-1)^(1/2)/(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5901, 5903, 4267, 2317, 2438, 75} \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {3}{8 a c^3 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}} \]

[In]

Int[ArcCosh[a*x]/(c - a^2*c*x^2)^3,x]

[Out]

1/(12*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - 3/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x])/(4
*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x])/(8*c^3*(1 - a^2*x^2)) + (3*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(4
*a*c^3) + (3*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) - (3*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3)

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*} \text {integral}& = \frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {a \int \frac {x}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c} \\ & = \frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {(3 a) \int \frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{8 c^3}+\frac {3 \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx}{8 c^2} \\ & = \frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}-\frac {3 \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arccosh}(a x))}{8 a c^3} \\ & = \frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{8 a c^3}-\frac {3 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{8 a c^3} \\ & = \frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{8 a c^3} \\ & = \frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.79 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.36 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-\frac {2 (-2+a x) \sqrt {1+a x}}{(-1+a x)^{3/2}}+\frac {2 \sqrt {-1+a x} (2+a x)}{(1+a x)^{3/2}}+\frac {6 \text {arccosh}(a x)}{(-1+a x)^2}-\frac {6 \text {arccosh}(a x)}{(1+a x)^2}+18 \left (-\frac {1}{\sqrt {\frac {-1+a x}{1+a x}}}+\frac {\text {arccosh}(a x)}{1-a x}\right )+18 \left (\sqrt {\frac {-1+a x}{1+a x}}-\frac {\text {arccosh}(a x)}{1+a x}\right )+9 \text {arccosh}(a x) \left (\text {arccosh}(a x)-4 \log \left (1-e^{\text {arccosh}(a x)}\right )\right )-9 \text {arccosh}(a x) \left (\text {arccosh}(a x)-4 \log \left (1+e^{\text {arccosh}(a x)}\right )\right )+36 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-36 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{96 a c^3} \]

[In]

Integrate[ArcCosh[a*x]/(c - a^2*c*x^2)^3,x]

[Out]

((-2*(-2 + a*x)*Sqrt[1 + a*x])/(-1 + a*x)^(3/2) + (2*Sqrt[-1 + a*x]*(2 + a*x))/(1 + a*x)^(3/2) + (6*ArcCosh[a*
x])/(-1 + a*x)^2 - (6*ArcCosh[a*x])/(1 + a*x)^2 + 18*(-(1/Sqrt[(-1 + a*x)/(1 + a*x)]) + ArcCosh[a*x]/(1 - a*x)
) + 18*(Sqrt[(-1 + a*x)/(1 + a*x)] - ArcCosh[a*x]/(1 + a*x)) + 9*ArcCosh[a*x]*(ArcCosh[a*x] - 4*Log[1 - E^ArcC
osh[a*x]]) - 9*ArcCosh[a*x]*(ArcCosh[a*x] - 4*Log[1 + E^ArcCosh[a*x]]) + 36*PolyLog[2, -E^ArcCosh[a*x]] - 36*P
olyLog[2, E^ArcCosh[a*x]])/(96*a*c^3)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )+9 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \,\operatorname {arccosh}\left (a x \right )-11 \sqrt {a x -1}\, \sqrt {a x +1}}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}}{a}\)	\(205\)
default	\(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )+9 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \,\operatorname {arccosh}\left (a x \right )-11 \sqrt {a x -1}\, \sqrt {a x +1}}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}}{a}\)	\(205\)

[In]

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

[Out]

1/a*(-1/24*(9*a^3*x^3*arccosh(a*x)+9*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-15*a*x*arccosh(a*x)-11*(a*x-1)^(1/2)*
(a*x+1)^(1/2))/(a^4*x^4-2*a^2*x^2+1)/c^3-3/8/c^3*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/8/c^3*po
lylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*po
lylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2)))

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]

[In]

integrate(arccosh(a*x)/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

integral(-arccosh(a*x)/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {\int \frac {\operatorname {acosh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]

[In]

integrate(acosh(a*x)/(-a**2*c*x**2+c)**3,x)

[Out]

-Integral(acosh(a*x)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)/c**3

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]

[In]

integrate(arccosh(a*x)/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/64*(10*a^3*x^3 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(
a*x - 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 14*a*x + 4*(6*a^3*x^3 - 10*a*x - 3*(a^4*x^4 - 2*a^2*x^
2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)) - 7*(a^
4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) + 3/16*(log(a*x - 1)*log(1/2*a*x +
1/2) + dilog(-1/2*a*x + 1/2))/(a*c^3) - 7/64*log(a*x + 1)/(a*c^3) + integrate(-1/16*(6*a^3*x^3 - 10*a*x - 3*(a
^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))/(a^7*c^3*x^7 - 3*a^5*c^3*x^5
+ 3*a^3*c^3*x^3 - a*c^3*x + (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt(a*x + 1)*sqrt(a*x - 1)),
x)

Giac [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]

[In]

integrate(arccosh(a*x)/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

integrate(-arccosh(a*x)/(a^2*c*x^2 - c)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^3} \,d x \]

[In]

int(acosh(a*x)/(c - a^2*c*x^2)^3,x)

[Out]

int(acosh(a*x)/(c - a^2*c*x^2)^3, x)