\(\int \frac {a+b \text {arccosh}(c x)}{x (d-c^2 d x^2)^2} \, dx\) [42]

Optimal result

Integrand size = 25, antiderivative size = 116 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5936, 5916, 5569, 4267, 2317, 2438, 39} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {2 \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b c x}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^2),x]

[Out]

-1/2*(b*c*x)/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(2*d^2*(1 - c^2*x^2)) + (2*(a + b*ArcCo
sh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^2 + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^2) - (b*PolyLog[2, E^(2*A
rcCosh[c*x])])/(2*d^2)

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5916

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

Rule 5936

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

Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d} \\ & = -\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^2}-\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(244\) vs. \(2(116)=232\).

Time = 0.61 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.10 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {-b \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \text {arccosh}(c x)}{1-c x}+\frac {b \text {arccosh}(c x)}{1+c x}+4 b \text {arccosh}(c x)^2+4 b \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-4 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-4 b \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )-2 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-4 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-4 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.19

method	result	size
parts	\(\frac {a \left (\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}+\ln \left (x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(254\)
derivativedivides	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(256\)
default	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(256\)

[In]

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

[Out]

a/d^2*(1/4/(c*x+1)-1/2*ln(c*x+1)+ln(x)-1/4/(c*x-1)-1/2*ln(c*x-1))+b/d^2*(-1/2*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x
-c^2*x^2+arccosh(c*x)+1)/(c^2*x^2-1)+arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+1/2*polylog(2,-(c*
x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)-arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-polylog(2,c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))-arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2
)))

Fricas [F]

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

[In]

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

[Out]

integral((b*arccosh(c*x) + a)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a/(c**4*x**5 - 2*c**2*x**3 + x), x) + Integral(b*acosh(c*x)/(c**4*x**5 - 2*c**2*x**3 + x), x))/d**2

Maxima [F]

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

[In]

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

[Out]

-1/2*a*(1/(c^2*d^2*x^2 - d^2) + log(c*x + 1)/d^2 + log(c*x - 1)/d^2 - 2*log(x)/d^2) + b*integrate(log(c*x + sq
rt(c*x + 1)*sqrt(c*x - 1))/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*acosh(c*x))/(x*(d - c^2*d*x^2)^2),x)

[Out]

int((a + b*acosh(c*x))/(x*(d - c^2*d*x^2)^2), x)