3.1.45 \(\int \frac {\cosh ^{-1}(a x)}{c+d x^2} \, dx\) [45]

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

[Out]

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

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Rubi [A]
time = 0.55, antiderivative size = 481, normalized size of antiderivative = 1.00, 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 = {5909, 5962, 5681, 2221, 2317, 2438} \begin {gather*} -\frac {\text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-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 (\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}} \end {gather*}

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 5681

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 5909

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 5962

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]

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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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*}

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Mathematica [A]
time = 0.27, size = 375, normalized size = 0.78 \begin {gather*} \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 )+\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{-a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )-\cosh ^{-1}(a x) \log \left (1+\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)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )-\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)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\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}} \end {gather*}

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])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 14.84, size = 222, normalized size = 0.46

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\mathrm {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{d \,\textit {\_R1}^{2}+2 a^{2} c +d}\right )}{2}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\mathrm {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (d \,\textit {\_R1}^{2}+2 a^{2} c +d \right )}\right )}{2}}{a}\)	\(222\)
default	\(\frac {\frac {a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\mathrm {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{d \,\textit {\_R1}^{2}+2 a^{2} c +d}\right )}{2}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\mathrm {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (d \,\textit {\_R1}^{2}+2 a^{2} c +d \right )}\right )}{2}}{a}\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(1/2*a^2*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((_R
1-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^2*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}{\left (a x \right )}}{c + d x^{2}}\, dx \end {gather*}

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {acosh}\left (a\,x\right )}{d\,x^2+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)/(c + d*x^2),x)

[Out]

int(acosh(a*x)/(c + d*x^2), x)

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