3.46 \(\int \frac{\cosh ^{-1}(a x)}{(c+d x^2)^2} \, dx\)

Optimal. Leaf size=774 \[ \frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}+1\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}+1\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a x+1} \sqrt{a \sqrt{-c}-\sqrt{d}}}{\sqrt{a x-1} \sqrt{a \sqrt{-c}+\sqrt{d}}}\right )}{2 c \sqrt{d} \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a x+1} \sqrt{a \sqrt{-c}+\sqrt{d}}}{\sqrt{a x-1} \sqrt{a \sqrt{-c}-\sqrt{d}}}\right )}{2 c \sqrt{d} \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}}} \]

[Out]

-ArcCosh[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCosh[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) + (a*Ar
cTanh[(Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[1 + a*x])/(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqr
t[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d]) - (a*ArcTanh[(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[1 + a*x])/(
Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d
]) - (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d])
+ (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) -
(ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + (
ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + Pol
yLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sq
rt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])]/(4*(-c)^(3/2)*Sqrt[d]) + PolyLog[2, -((Sqrt[d]*E^ArcC
osh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*
Sqrt[-c] + Sqrt[-(a^2*c) - d])]/(4*(-c)^(3/2)*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 1.1014, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5707, 5802, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}+1\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}+1\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a x+1} \sqrt{a \sqrt{-c}-\sqrt{d}}}{\sqrt{a x-1} \sqrt{a \sqrt{-c}+\sqrt{d}}}\right )}{2 c \sqrt{d} \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a x+1} \sqrt{a \sqrt{-c}+\sqrt{d}}}{\sqrt{a x-1} \sqrt{a \sqrt{-c}-\sqrt{d}}}\right )}{2 c \sqrt{d} \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c + d*x^2)^2,x]

[Out]

-ArcCosh[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCosh[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) + (a*Ar
cTanh[(Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[1 + a*x])/(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqr
t[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d]) - (a*ArcTanh[(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[1 + a*x])/(
Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d
]) - (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d])
+ (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) -
(ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + (
ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + Pol
yLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sq
rt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])]/(4*(-c)^(3/2)*Sqrt[d]) + PolyLog[2, -((Sqrt[d]*E^ArcC
osh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*
Sqrt[-c] + Sqrt[-(a^2*c) - d])]/(4*(-c)^(3/2)*Sqrt[d])

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5802

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rule 5800

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

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\int \left (-\frac{d \cosh ^{-1}(a x)}{4 c \left (\sqrt{-c} \sqrt{d}-d x\right )^2}-\frac{d \cosh ^{-1}(a x)}{4 c \left (\sqrt{-c} \sqrt{d}+d x\right )^2}-\frac{d \cosh ^{-1}(a x)}{2 c \left (-c d-d^2 x^2\right )}\right ) \, dx\\ &=-\frac{d \int \frac{\cosh ^{-1}(a x)}{\left (\sqrt{-c} \sqrt{d}-d x\right )^2} \, dx}{4 c}-\frac{d \int \frac{\cosh ^{-1}(a x)}{\left (\sqrt{-c} \sqrt{d}+d x\right )^2} \, dx}{4 c}-\frac{d \int \frac{\cosh ^{-1}(a x)}{-c d-d^2 x^2} \, dx}{2 c}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \left (\sqrt{-c} \sqrt{d}-d x\right )} \, dx}{4 c}-\frac{a \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \left (\sqrt{-c} \sqrt{d}+d x\right )} \, dx}{4 c}-\frac{d \int \left (-\frac{\sqrt{-c} \cosh ^{-1}(a x)}{2 c d \left (\sqrt{-c}-\sqrt{d} x\right )}-\frac{\sqrt{-c} \cosh ^{-1}(a x)}{2 c d \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx}{2 c}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{\int \frac{\cosh ^{-1}(a x)}{\sqrt{-c}-\sqrt{d} x} \, dx}{4 (-c)^{3/2}}+\frac{\int \frac{\cosh ^{-1}(a x)}{\sqrt{-c}+\sqrt{d} x} \, dx}{4 (-c)^{3/2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{a \sqrt{-c} \sqrt{d}+d-\left (a \sqrt{-c} \sqrt{d}-d\right ) x^2} \, dx,x,\frac{\sqrt{1+a x}}{\sqrt{-1+a x}}\right )}{2 c}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a \sqrt{-c} \sqrt{d}-d-\left (a \sqrt{-c} \sqrt{d}+d\right ) x^2} \, dx,x,\frac{\sqrt{1+a x}}{\sqrt{-1+a x}}\right )}{2 c}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \frac{x \sinh (x)}{a \sqrt{-c}-\sqrt{d} \cosh (x)} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{x \sinh (x)}{a \sqrt{-c}+\sqrt{d} \cosh (x)} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \frac{e^x x}{a \sqrt{-c}-\sqrt{-a^2 c-d}-\sqrt{d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x x}{a \sqrt{-c}+\sqrt{-a^2 c-d}-\sqrt{d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x x}{a \sqrt{-c}-\sqrt{-a^2 c-d}+\sqrt{d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x x}{a \sqrt{-c}+\sqrt{-a^2 c-d}+\sqrt{d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{d} e^x}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{d} e^x}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{d} e^x}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{d} e^x}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{d} x}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{d} x}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{d} x}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{d} x}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}\\ &=-\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cosh ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{1+a x}}{\sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{-1+a x}}\right )}{2 c \sqrt{a \sqrt{-c}-\sqrt{d}} \sqrt{a \sqrt{-c}+\sqrt{d}} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\text{Li}_2\left (-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\text{Li}_2\left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\text{Li}_2\left (-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\text{Li}_2\left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt{d}}\\ \end{align*}

Mathematica [C]  time = 1.22343, size = 687, normalized size = 0.89 \[ \frac{i \left (2 \text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}-i a \sqrt{c}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+i a \sqrt{c}}\right )+\cosh ^{-1}(a x) \left (-\cosh ^{-1}(a x)+2 \left (\log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{-\sqrt{a^2 (-c)-d}+i a \sqrt{c}}\right )+\log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+i a \sqrt{c}}\right )\right )\right )\right )-i \left (2 \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}-i a \sqrt{c}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+i a \sqrt{c}}\right )+\cosh ^{-1}(a x) \left (-\cosh ^{-1}(a x)+2 \left (\log \left (1+\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}-i a \sqrt{c}}\right )+\log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+i a \sqrt{c}}\right )\right )\right )\right )+2 \sqrt{c} \left (\frac{a \log \left (\frac{2 d \left (-i \sqrt{a x-1} \sqrt{a x+1} \sqrt{a^2 (-c)-d}+a^2 \sqrt{c} x+i \sqrt{d}\right )}{a \sqrt{a^2 (-c)-d} \left (\sqrt{c}+i \sqrt{d} x\right )}\right )}{\sqrt{a^2 (-c)-d}}+\frac{\cosh ^{-1}(a x)}{\sqrt{d} x-i \sqrt{c}}\right )-2 \sqrt{c} \left (-\frac{a \log \left (\frac{2 d \left (\sqrt{a x-1} \sqrt{a x+1} \sqrt{a^2 (-c)-d}-i a^2 \sqrt{c} x-\sqrt{d}\right )}{a \sqrt{a^2 (-c)-d} \left (\sqrt{d} x+i \sqrt{c}\right )}\right )}{\sqrt{a^2 (-c)-d}}-\frac{\cosh ^{-1}(a x)}{\sqrt{d} x+i \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[c]*(ArcCosh[a*x]/((-I)*Sqrt[c] + Sqrt[d]*x) + (a*Log[(2*d*(I*Sqrt[d] + a^2*Sqrt[c]*x - I*Sqrt[-(a^2*c)
- d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x]))/(a*Sqrt[-(a^2*c) - d]*(Sqrt[c] + I*Sqrt[d]*x))])/Sqrt[-(a^2*c) - d]) - 2*
Sqrt[c]*(-(ArcCosh[a*x]/(I*Sqrt[c] + Sqrt[d]*x)) - (a*Log[(2*d*(-Sqrt[d] - I*a^2*Sqrt[c]*x + Sqrt[-(a^2*c) - d
]*Sqrt[-1 + a*x]*Sqrt[1 + a*x]))/(a*Sqrt[-(a^2*c) - d]*(I*Sqrt[c] + Sqrt[d]*x))])/Sqrt[-(a^2*c) - d]) + I*(Arc
Cosh[a*x]*(-ArcCosh[a*x] + 2*(Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] - Sqrt[-(a^2*c) - d])] + Log[1 + (
Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqrt[-(a^2*c) - d])])) + 2*PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/((-I)*a*
Sqrt[c] + Sqrt[-(a^2*c) - d])] + 2*PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqrt[-(a^2*c) - d]))])
- I*(ArcCosh[a*x]*(-ArcCosh[a*x] + 2*(Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/((-I)*a*Sqrt[c] + Sqrt[-(a^2*c) - d])]
+ Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqrt[-(a^2*c) - d])])) + 2*PolyLog[2, -((Sqrt[d]*E^ArcCosh[
a*x])/((-I)*a*Sqrt[c] + Sqrt[-(a^2*c) - d]))] + 2*PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqrt[-(a^
2*c) - d])]))/(8*c^(3/2)*Sqrt[d])

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Maple [C]  time = 1.255, size = 1632, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)/(d*x^2+c)^2,x)

[Out]

1/2*a^2*arccosh(a*x)*x/c/(a^2*d*x^2+a^2*c)+1/4*a/c*sum(_R1/(_R1^2*d+2*a^2*c+d)*(arccosh(a*x)*ln((_R1-a*x-(a*x-
1)^(1/2)*(a*x+1)^(1/2))/_R1)+dilog((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)
*_Z^2+d))+a^5*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-
2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))*c/(a^2*c+d)/d^3+a^3*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^
(1/2)*arctanh(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/(a^2*c+d)/
d^3*(a^2*c*(a^2*c+d))^(1/2)+a^3*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(a*x+(a*x-1)^(1/2)*
(a*x+1)^(1/2))/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/(a^2*c+d)/d^2+1/2*a*(-(2*a^2*c-2*(a^2*c*(a^2*
c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)
^(1/2))/c/(a^2*c+d)/d^2*(a^2*c*(a^2*c+d))^(1/2)-a^3*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d
*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/d^3-a*(-(2*a^2*c-2*(a^2*c
*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)
-d)*d)^(1/2))/c/d^3*(a^2*c*(a^2*c+d))^(1/2)-1/2*a*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(
a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/c/d^2+a^5*((2*a^2*c+2*(a^2*
c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+
d)*d)^(1/2))*c/(a^2*c+d)/d^3-a^3*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*x-1)^(1/2)*(
a*x+1)^(1/2))/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/(a^2*c+d)/d^3*(a^2*c*(a^2*c+d))^(1/2)+a^3*((2*a
^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((2*a^2*c+2*(a^2*c*(a^2*
c+d))^(1/2)+d)*d)^(1/2))/(a^2*c+d)/d^2-1/2*a*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*
x-1)^(1/2)*(a*x+1)^(1/2))/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/c/(a^2*c+d)/d^2*(a^2*c*(a^2*c+d))^(
1/2)-a^3*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((2*a^2*c+
2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/d^3+a*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*
x-1)^(1/2)*(a*x+1)^(1/2))/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/c/d^3*(a^2*c*(a^2*c+d))^(1/2)-1/2*a
*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((2*a^2*c+2*(a^2*c
*(a^2*c+d))^(1/2)+d)*d)^(1/2))/c/d^2-1/4*a/c*sum(1/_R1/(_R1^2*d+2*a^2*c+d)*(arccosh(a*x)*ln((_R1-a*x-(a*x-1)^(
1/2)*(a*x+1)^(1/2))/_R1)+dilog((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)*_Z^
2+d))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcosh}\left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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