3.370 \(\int \frac{x^2 (d-c^2 d x^2)}{(a+b \cosh ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=340 \[ \frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{2 d x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]

[Out]

(2*d*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b
*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3) + (d*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sq
rt[b]])/(16*b^(3/2)*c^3) - (d*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3
/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3*E^(a/b)) + (d*Sqrt[3*Pi]*Erfi[(S
qrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3*E^((3*a)/b)) - (d*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a +
b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3*E^((5*a)/b))

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Rubi [A]  time = 1.77248, antiderivative size = 350, normalized size of antiderivative = 1.03, number of steps used = 32, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {5776, 5781, 5448, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{2 d x^2 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]

[Out]

(-2*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (d*E^(a/b)*Sqrt[Pi]*Erf
[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3) + (d*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCos
h[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) - (d*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]
])/(16*b^(3/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3*E^(a/b)) + (d*Sqrt[3*
Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3*E^((3*a)/b)) - (d*Sqrt[5*Pi]*Erfi[(Sqrt[
5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3*E^((5*a)/b))

Rule 5776

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(
f*m*(-d)^p)/(b*c*(n + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n
+ 1), x], x] - Dist[(c*(-d)^p*(m + 2*p + 1))/(b*f*(n + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(
p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && IGtQ[m, -3] && IGtQ[p, 0]

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 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 3307

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

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]

Rubi steps

\begin{align*} \int \frac{x^2 \left (d-c^2 d x^2\right )}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(4 d) \int \frac{x \sqrt{-1+c x} \sqrt{1+c x}}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}-\frac{(10 c d) \int \frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}-\frac{(10 d) \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{a+b x}}+\frac{\cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}-\frac{(10 d) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 \sqrt{a+b x}}+\frac{\cosh (3 x)}{16 \sqrt{a+b x}}+\frac{\cosh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}-\frac{d \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}-\frac{d \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}-\frac{d \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(5 d) \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^3}-\frac{(5 d) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^3}-\frac{(5 d) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^3}-\frac{(5 d) \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^3}+\frac{d \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^2 c^3}-\frac{d \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^2 c^3}-\frac{d \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^2 c^3}+\frac{d \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^2 c^3}+\frac{(5 d) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^3}+\frac{(5 d) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^3}\\ &=-\frac{2 d x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{d e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{d e^{\frac{5 a}{b}} \sqrt{5 \pi } \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{d e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{d e^{-\frac{5 a}{b}} \sqrt{5 \pi } \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}\\ \end{align*}

Mathematica [A]  time = 1.48846, size = 384, normalized size = 1.13 \[ \frac{d e^{-\frac{5 a}{b}} \left (-2 e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sqrt{5} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt{3} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-\sqrt{3} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt{5} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-4 c x e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-4 e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-2 e^{\frac{5 a}{b}} \sinh \left (3 \cosh ^{-1}(c x)\right )+2 e^{\frac{5 a}{b}} \sinh \left (5 \cosh ^{-1}(c x)\right )\right )}{16 b c^3 \sqrt{a+b \cosh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^2*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]

[Out]

(d*(-4*E^((5*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*c*E^((5*a)/b)*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*E^((6*a)/b)*S
qrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] - Sqrt[5]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-
5*(a + b*ArcCosh[c*x]))/b] + Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh
[c*x]))/b] + 2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - Sqrt[3]*E^(
(8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + Sqrt[5]*E^((10*a)/b)*Sqrt[a/b + Arc
Cosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b] - 2*E^((5*a)/b)*Sinh[3*ArcCosh[c*x]] + 2*E^((5*a)/b)*Sinh[5*
ArcCosh[c*x]]))/(16*b*c^3*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c*x]])

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Maple [F]  time = 0.275, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( -{c}^{2}d{x}^{2}+d \right ) \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x)

[Out]

int(x^2*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (c^{2} d x^{2} - d\right )} x^{2}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxima")

[Out]

-integrate((c^2*d*x^2 - d)*x^2/(b*arccosh(c*x) + a)^(3/2), 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*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - d \left (\int - \frac{x^{2}}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx + \int \frac{c^{2} x^{4}}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*d*x**2+d)/(a+b*acosh(c*x))**(3/2),x)

[Out]

-d*(Integral(-x**2/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**2*x**4/(
a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), 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*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")

[Out]

sage0*x