\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d-c^2 d x^2)^{3/2}} \, dx\) [122]

Optimal result

Integrand size = 27, antiderivative size = 329 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}} \]

[Out]

3/2*c^2*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)+1/2*(-a-b*arccosh(c*x))/d/x^2/(-c^2*d*x^2+d)^(1/2)+1/2*b*c*(
c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x/(-c^2*d*x^2+d)^(1/2)+3*c^2*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^
(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)+b*c^2*arctanh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c
^2*d*x^2+d)^(1/2)-3/2*I*b*c^2*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-
c^2*d*x^2+d)^(1/2)+3/2*I*b*c^2*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-
c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5932, 5936, 5946, 4265, 2317, 2438, 35, 213, 84} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {3 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x*Sqrt[d - c^2*d*x^2]) + (3*c^2*(a + b*ArcCosh[c*x]))/(2*d*Sqrt[d - c^
2*d*x^2]) - (a + b*ArcCosh[c*x])/(2*d*x^2*Sqrt[d - c^2*d*x^2]) + (3*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*Ar
cCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x]
)/(d*Sqrt[d - c^2*d*x^2]) - (((3*I)/2)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d*
Sqrt[d - c^2*d*x^2]) + (((3*I)/2)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(d*Sqrt[d -
c^2*d*x^2])

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 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])

Rule 5946

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

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

Mathematica [A] (warning: unable to verify)

Time = 3.68 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.33 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (-\frac {a \left (-1+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{d^2 x^2 \left (-1+c^2 x^2\right )}+\frac {3 a c^2 \log (x)}{d^{3/2}}-\frac {3 a c^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{3/2}}-\frac {b c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\left (-1+\frac {1}{c^2 x^2}\right ) \text {arccosh}(c x)-2 \text {arccosh}(c x) \cosh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \sinh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{d \sqrt {d-c^2 d x^2}}\right ) \]

[In]

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

[Out]

(-((a*(-1 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^2*x^2*(-1 + c^2*x^2))) + (3*a*c^2*Log[x])/d^(3/2) - (3*a*c^2*Lo
g[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^(3/2) - (b*c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x)) + (-1 +
1/(c^2*x^2))*ArcCosh[c*x] - 2*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)
*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^
ArcCosh[c*x]] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] + 2*Sqrt[(-1 + c*x)/(1 + c*x)
]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c
*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*Sinh[ArcCosh[c
*x]/2]^2))/(d*Sqrt[d - c^2*d*x^2]))/2

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.79

method	result	size
default	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )\)	\(588\)
parts	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )\)	\(588\)

[In]

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

[Out]

a*(-1/2/d/x^2/(-c^2*d*x^2+d)^(1/2)+3/2*c^2*(1/d/(-c^2*d*x^2+d)^(1/2)-1/d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d
)^(1/2))/x)))+b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(3*c^2*x^2*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-arccosh(c
*x))/d^2/(c^2*x^2-1)/x^2-(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*ln(1+c*x+(c*x-1)^(
1/2)*(c*x+1)^(1/2))*c^2+(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*ln((c*x-1)^(1/2)*(c
*x+1)^(1/2)+c*x-1)*c^2+3/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*dilog(1+I*(c*x
+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+3/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*ar
ccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2-3/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1
/2)/(c^2*x^2-1)/d^2*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2-3/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2
)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2)

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(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]

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

[In]

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

[Out]

Integral((a + b*acosh(c*x))/(x**3*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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