\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d-c^2 d x^2)^2} \, dx\) [44]

Optimal result

Integrand size = 25, antiderivative size = 152 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 (a+b \text {arccosh}(c x))}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d^2}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5932, 5936, 5916, 5569, 4267, 2317, 2438, 39, 105, 12} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {4 c^2 \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^2}+\frac {c^2 (a+b \text {arccosh}(c x))}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d^2}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{d^2}-\frac {b c}{2 d^2 x \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^2),x]

[Out]

-1/2*(b*c)/(d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c^2*(a + b*ArcCosh[c*x]))/(d^2*(1 - c^2*x^2)) - (a + b*ArcC
osh[c*x])/(2*d^2*x^2*(1 - c^2*x^2)) + (4*c^2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^2 + (b*c^2*Po
lyLog[2, -E^(2*ArcCosh[c*x])])/d^2 - (b*c^2*PolyLog[2, E^(2*ArcCosh[c*x])])/d^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 105

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

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

Mathematica [A] (warning: unable to verify)

Time = 1.16 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.90 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {a}{x^2}+\frac {a c^2}{1-c^2 x^2}+4 a c^2 \log (x)-2 a c^2 \log \left (1-c^2 x^2\right )+\frac {1}{2} b \left (\frac {2 c x \sqrt {-1+c x} \sqrt {1+c x}-2 \text {arccosh}(c x)}{x^2}+c^2 \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-c^2 \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+4 c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+2 c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+2 c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )\right )}{2 d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.84

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )+\frac {1}{4 c x +4}-\ln \left (c x +1\right )-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )\right )}{d^{2}}+\frac {b \left (-\frac {2 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\right )\)	\(280\)
default	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )+\frac {1}{4 c x +4}-\ln \left (c x +1\right )-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )\right )}{d^{2}}+\frac {b \left (-\frac {2 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\right )\)	\(280\)
parts	\(\frac {a \left (\frac {c^{2}}{4 c x +4}-c^{2} \ln \left (c x +1\right )-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )-\frac {c^{2}}{4 \left (c x -1\right )}-c^{2} \ln \left (c x -1\right )\right )}{d^{2}}+\frac {b \,c^{2} \left (-\frac {2 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(289\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arccosh(c*x) + a)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(b*acosh(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3), x))
/d**2

Maxima [F]

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

[In]

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

[Out]

-1/2*a*(2*c^2*log(c*x + 1)/d^2 + 2*c^2*log(c*x - 1)/d^2 - 4*c^2*log(x)/d^2 + (2*c^2*x^2 - 1)/(c^2*d^2*x^4 - d^
2*x^2)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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