3.5.92 \(\int \frac {x (a+b \cosh ^{-1}(c x))}{d+e x^2} \, dx\) [492]

Optimal. Leaf size=449 \[ -\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.50, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5959, 5962, 5681, 2221, 2317, 2438} \begin {gather*} \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5959

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

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

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Mathematica [A]
time = 0.10, size = 447, normalized size = 1.00 \begin {gather*} -\frac {b \cosh ^{-1}(c x)^2}{2 e}+\frac {b \cosh ^{-1}(c x) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \cosh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \cosh ^{-1}(c x) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \cosh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {a \log \left (d+e x^2\right )}{2 e}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 6.04, size = 2845, normalized size = 6.34

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2845\)
default	\(\text {Expression too large to display}\)	\(2845\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(1/2*b*c^2/(c^2*d+e)*arccosh(c*x)^2-1/4*b*c^2/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/
(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))+1/2*b*c^2/e*sum((_R1^2*e+4*c^2*d+2*e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x
)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e
*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-b*c^2*arccosh(c*x)^2/e+1/2*a*c^2/e*ln(c^2*e*x^2+c^2*d)-b*c^2*((c^2*d+e)*c^2*d)^(1
/2)/e/(c^2*d+e)*arccosh(c*x)^2-b*c^2/e^2*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2
*d)^(1/2)-e))*arccosh(c*x)*((c^2*d+e)*c^2*d)^(1/2)-1/4*b*((c^2*d+e)*c^2*d)^(1/2)/d/(c^2*d+e)*arccosh(c*x)*ln(1
-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*((c^2*d+e)*c^2*d)^(1/2)-e))-3*b*c^4*((c^2*d+e)*c^2*d)^(1/2)
/e^2*d/(c^2*d+e)*arccosh(c*x)^2-2*b*c^6/e^3*d^2/(c^2*d+e)*arccosh(c*x)^2*((c^2*d+e)*c^2*d)^(1/2)-1/2*b*c^2*((c
^2*d+e)*c^2*d)^(1/2)/e/(c^2*d+e)*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*((c^2*d+e
)*c^2*d)^(1/2)-e))+3/2*b*c^2/e/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2
*d)^(1/2)-e))*arccosh(c*x)*((c^2*d+e)*c^2*d)^(1/2)-2*b*c^8/e^3*d^3/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)
^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)-2*b*c^4/e^3*d*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)
^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)*((c^2*d+e)*c^2*d)^(1/2)-1/2*b*c^2/(c^2*d+e)*ln(
1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)+b*c^2/e^2*arccosh
(c*x)^2*((c^2*d+e)*c^2*d)^(1/2)-1/2*b*c^2/e^2*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^
2*d+e)*c^2*d)^(1/2)-e))*((c^2*d+e)*c^2*d)^(1/2)-1/8*b*((c^2*d+e)*c^2*d)^(1/2)/d/(c^2*d+e)*polylog(2,e*(c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*((c^2*d+e)*c^2*d)^(1/2)-e))+1/8*b/d/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)
^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*((c^2*d+e)*c^2*d)^(1/2)+2*b*c^8/e^3*d^3/(c^2*d
+e)*arccosh(c*x)^2+1/4*b/d/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^
(1/2)-e))*arccosh(c*x)*((c^2*d+e)*c^2*d)^(1/2)-5/2*b*c^4/e/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^
2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)*d-4*b*c^6/e^2/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)*d^2+3/2*b*c^4/e^2/(c^2*d+e)*polylog(2,e*(c*x+
(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*d*((c^2*d+e)*c^2*d)^(1/2)+b*c^6/e^3*d^2
/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*((c^2*d+e)*
c^2*d)^(1/2)+3/4*b*c^2/e/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*
d)^(1/2)-e))*((c^2*d+e)*c^2*d)^(1/2)-2*b*c^6/e^2/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2
*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*d^2-5/4*b*c^4/e/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2
/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*d+2*b*c^6/e^3*d^2*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*
d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)+2*b*c^4/e^2*d*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*
d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)+5/2*b*c^4/e/(c^2*d+e)*arccosh(c*x)^2*d-1/4*b*c^2*((c^2*d+e)*c^2*d
)^(1/2)/e/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*((c^2*d+e)*c^2*d)^(1/2)-e))+4*
b*c^6/e^2*d^2/(c^2*d+e)*arccosh(c*x)^2-b*c^8/e^3*d^3/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2
/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))-b*c^4/e^3*d*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d
-2*((c^2*d+e)*c^2*d)^(1/2)-e))*((c^2*d+e)*c^2*d)^(1/2)+2*b*c^4/e^3*d*arccosh(c*x)^2*((c^2*d+e)*c^2*d)^(1/2)+1/
2*b*c^2/e*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)+b*c^
6/e^3*d^2*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))+b*c^4/e^2*d*
polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))-2*b*c^6/e^3*d^2*arccos
h(c*x)^2-2*b*c^4*d*arccosh(c*x)^2/e^2+1/4*b*c^2/e*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*
((c^2*d+e)*c^2*d)^(1/2)-e))+2*b*c^6/e^3*d^2/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(
(c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)*((c^2*d+e)*c^2*d)^(1/2)+3*b*c^4/e^2/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2
)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)-e))*arccosh(c*x)*d*((c^2*d+e)*c^2*d)^(1/2))

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

[Out]

1/2*a*e^(-1)*log(x^2*e + d) + b*integrate(x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(x^2*e + d), 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(x*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*acosh(c*x))/(d + e*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(x*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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