\(\int \frac {a+b \text {arccosh}(c x)}{x (d-c^2 d x^2)} \, dx\) [33]

Optimal result

Integrand size = 25, antiderivative size = 61 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx=\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5916, 5569, 4267, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx=\frac {2 \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d} \]

[In]

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

[Out]

(2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d) - (b*PolyLo
g[2, E^(2*ArcCosh[c*x])])/(2*d)

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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(61)=122\).

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.11 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx=-\frac {-b \text {arccosh}(c x)^2-2 b \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+2 b \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+2 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-2 a \log (x)+a \log \left (1-c^2 x^2\right )+b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-2 b \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )+2 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.13

method	result	size
parts	\(-\frac {a \left (\frac {\ln \left (c x +1\right )}{2}-\ln \left (x \right )+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(191\)
derivativedivides	\(-\frac {a \left (-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(193\)
default	\(-\frac {a \left (-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(193\)

[In]

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

[Out]

-a/d*(1/2*ln(c*x+1)-ln(x)+1/2*ln(c*x-1))-b/d*(-arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)-1/2*poly
log(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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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