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

Optimal. Leaf size=481 \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}+1\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}+1\right )}{2 \sqrt{-c} \sqrt{d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.757693, antiderivative size = 481, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}+1\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}+1\right )}{2 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c + d*x^2),x]

[Out]

(ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d]) - (Arc
Cosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCosh
[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCosh[a*x
]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sq
rt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*E^ArcCosh
[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])]/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt
[-c] + Sqrt[-(a^2*c) - d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a
^2*c) - d])]/(2*Sqrt[-c]*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 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)}{c+d x^2} \, dx &=\int \left (\frac{\sqrt{-c} \cosh ^{-1}(a x)}{2 c \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\sqrt{-c} \cosh ^{-1}(a x)}{2 c \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\cosh ^{-1}(a x)}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{-c}}-\frac{\int \frac{\cosh ^{-1}(a x)}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{-c}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x \sinh (x)}{a \sqrt{-c}-\sqrt{d} \cosh (x)} \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{-c}}-\frac{\operatorname{Subst}\left (\int \frac{x \sinh (x)}{a \sqrt{-c}+\sqrt{d} \cosh (x)} \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{-c}}\\ &=-\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 )}{2 \sqrt{-c}}-\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 )}{2 \sqrt{-c}}-\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 )}{2 \sqrt{-c}}-\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 )}{2 \sqrt{-c}}\\ &=\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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \sqrt{d}}-\frac{\text{Li}_2\left (-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{-a^2 c-d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\text{Li}_2\left (-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}+\sqrt{-a^2 c-d}}\right )}{2 \sqrt{-c} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.37311, size = 375, normalized size = 0.78 \[ \frac{\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}-a \sqrt{-c}}\right )-\text{PolyLog}\left (2,-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )-\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{a \sqrt{-c}-\sqrt{a^2 (-c)-d}}+1\right )+\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}-a \sqrt{-c}}+1\right )+\cosh ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}\right )-\cosh ^{-1}(a x) \log \left (\frac{\sqrt{d} e^{\cosh ^{-1}(a x)}}{\sqrt{a^2 (-c)-d}+a \sqrt{-c}}+1\right )}{2 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]/(c + d*x^2),x]

[Out]

(-(ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])]) + ArcCosh[a*x]*Log[1 + (S
qrt[d]*E^ArcCosh[a*x])/(-(a*Sqrt[-c]) + Sqrt[-(a^2*c) - d])] + ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(
a*Sqrt[-c] + Sqrt[-(a^2*c) - d])] - ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c)
- d])] + PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])] - PolyLog[2, (Sqrt[d]*E^ArcCos
h[a*x])/(-(a*Sqrt[-c]) + Sqrt[-(a^2*c) - d])] - PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2
*c) - d]))] + PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d])

________________________________________________________________________________________

Maple [C]  time = 0.623, size = 214, normalized size = 0.4 \begin{align*}{\frac{a}{2}\sum _{{\it \_R1}={\it RootOf} \left ( d{{\it \_Z}}^{4}+ \left ( 4\,{a}^{2}c+2\,d \right ){{\it \_Z}}^{2}+d \right ) }{\frac{{\it \_R1}}{{{\it \_R1}}^{2}d+2\,{a}^{2}c+d} \left ({\rm arccosh} \left (ax\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) } \right ) \right ) }}-{\frac{a}{2}\sum _{{\it \_R1}={\it RootOf} \left ( d{{\it \_Z}}^{4}+ \left ( 4\,{a}^{2}c+2\,d \right ){{\it \_Z}}^{2}+d \right ) }{\frac{1}{{\it \_R1}\, \left ({{\it \_R1}}^{2}d+2\,{a}^{2}c+d \right ) } \left ({\rm arccosh} \left (ax\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*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))-1/2*a*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))

________________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccosh(a*x)/(d*x^2 + c), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)/(c + d*x**2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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