3.157 \(\int \frac{x^2}{(a+b \cosh ^{-1}(c x))^{5/2}} \, dx\)

Optimal. Leaf size=276 \[ -\frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}-\frac{\sqrt{3 \pi } e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{\sqrt{3 \pi } e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \]

[Out]

(-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcCosh
[c*x]]) - (4*x^3)/(b^2*Sqrt[a + b*ArcCosh[c*x]]) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(6
*b^(5/2)*c^3) - (E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3) + (Sq
rt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(6*b^(5/2)*c^3*E^(a/b)) + (Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*
ArcCosh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3*E^((3*a)/b))

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Rubi [A]  time = 1.33041, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205, 5658} \[ -\frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}-\frac{\sqrt{3 \pi } e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{\sqrt{3 \pi } e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcCosh
[c*x]]) - (4*x^3)/(b^2*Sqrt[a + b*ArcCosh[c*x]]) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(6
*b^(5/2)*c^3) - (E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3) + (Sq
rt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(6*b^(5/2)*c^3*E^(a/b)) + (Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*
ArcCosh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3*E^((3*a)/b))

Rule 5668

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

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5658

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x]
, x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{4 \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}+\frac{(2 c) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{b}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{12 \int \frac{x^2}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b^2}-\frac{8 \int \frac{1}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{3 b^2 c^2}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac{12 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{3 b^3 c^3}-\frac{4 \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac{12 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{\sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{8 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{3 b^3 c^3}-\frac{8 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{3 b^3 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}-\frac{4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}-\frac{4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}-\frac{3 \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}-\frac{3 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}+\frac{3 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}+\frac{3 \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}\\ &=-\frac{2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}-\frac{e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}+\frac{e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}\\ \end{align*}

Mathematica [A]  time = 2.04002, size = 340, normalized size = 1.23 \[ \frac{e^{-3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )} \left (-6 \sqrt{3} b e^{3 \cosh ^{-1}(c x)} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-2 b e^{\frac{2 a}{b}+3 \cosh ^{-1}(c x)} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+2 e^{\frac{4 a}{b}+3 \cosh ^{-1}(c x)} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )+e^{\frac{3 a}{b}} \left (6 \sqrt{3} e^{3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-\left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a \left (-4 e^{2 \cosh ^{-1}(c x)}+6 e^{4 \cosh ^{-1}(c x)}+6\right )+b \left (-4 e^{2 \cosh ^{-1}(c x)} \cosh ^{-1}(c x)+6 \cosh ^{-1}(c x)+e^{4 \cosh ^{-1}(c x)} \left (6 \cosh ^{-1}(c x)+1\right )-1\right )\right )\right )\right )}{12 b^2 c^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(2*E^((4*a)/b + 3*ArcCosh[c*x])*Sqrt[a/b + ArcCosh[c*x]]*(a + b*ArcCosh[c*x])*Gamma[1/2, a/b + ArcCosh[c*x]] -
6*Sqrt[3]*b*E^(3*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] - 2*
b*E^((2*a)/b + 3*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + E^((3
*a)/b)*(-((1 + E^(2*ArcCosh[c*x]))*(a*(6 - 4*E^(2*ArcCosh[c*x]) + 6*E^(4*ArcCosh[c*x])) + b*(-1 + 6*ArcCosh[c*
x] - 4*E^(2*ArcCosh[c*x])*ArcCosh[c*x] + E^(4*ArcCosh[c*x])*(1 + 6*ArcCosh[c*x])))) + 6*Sqrt[3]*E^(3*(a/b + Ar
cCosh[c*x]))*Sqrt[a/b + ArcCosh[c*x]]*(a + b*ArcCosh[c*x])*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b]))/(12*b^2*c^
3*E^(3*(a/b + ArcCosh[c*x]))*(a + b*ArcCosh[c*x])^(3/2))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*arccosh(c*x))^(5/2),x)

[Out]

int(x^2/(a+b*arccosh(c*x))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*arccosh(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate(x^2/(b*arccosh(c*x) + a)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*arccosh(c*x))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*acosh(c*x))**(5/2),x)

[Out]

Integral(x**2/(a + b*acosh(c*x))**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*arccosh(c*x))^(5/2),x, algorithm="giac")

[Out]

sage0*x