\(\int \frac {a+b \text {arccosh}(c x)}{x^2 (d-c^2 d x^2)} \, dx\) [34]

Optimal result

Integrand size = 25, antiderivative size = 95 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {a+b \text {arccosh}(c x)}{d x}+\frac {b c \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d}+\frac {b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d}-\frac {b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 95, 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 = {5932, 5903, 4267, 2317, 2438, 94, 211} \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=\frac {2 c \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d}-\frac {a+b \text {arccosh}(c x)}{d x}+\frac {b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d}-\frac {b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d}+\frac {b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d} \]

[In]

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

[Out]

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

Rule 94

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 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 5932

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/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 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*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.39 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=\frac {-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b c \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-c (a+b \text {arccosh}(c x)) \log \left (1-e^{\text {arccosh}(c x)}\right )+c (a+b \text {arccosh}(c x)) \log \left (1+e^{\text {arccosh}(c x)}\right )+b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.51

method	result	size
parts	\(-\frac {a \left (\frac {1}{x}-\frac {c \ln \left (c x +1\right )}{2}+\frac {c \ln \left (c x -1\right )}{2}\right )}{d}-\frac {b c \left (\frac {\operatorname {arccosh}\left (c x \right )}{c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(143\)
derivativedivides	\(c \left (-\frac {a \left (\frac {1}{c x}-\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right )}{c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(146\)
default	\(c \left (-\frac {a \left (\frac {1}{c x}-\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right )}{c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(146\)

[In]

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

[Out]

-a/d*(1/x-1/2*c*ln(c*x+1)+1/2*c*ln(c*x-1))-b/d*c*(arccosh(c*x)/c/x-2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-d
ilog(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-dilog(1+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)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/8*(24*c^3*integrate(1/4*x*log(c*x - 1)/(c^2*d*x^2 - d), x) - 4*c^2*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d))
- 8*c^2*integrate(1/4*log(c*x - 1)/(c^2*d*x^2 - d), x) - (c*x*log(c*x + 1)^2 + 2*c*x*log(c*x + 1)*log(c*x - 1
) - 4*(c*x*log(c*x + 1) - c*x*log(c*x - 1) - 2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(d*x) + 8*integrate(1/
2*(c^2*x*log(c*x + 1) - c^2*x*log(c*x - 1) - 2*c)/(c^3*d*x^4 - c*d*x^2 + (c^2*d*x^3 - d*x)*sqrt(c*x + 1)*sqrt(
c*x - 1)), x))*b + 1/2*a*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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